Fluid mechanics is the study of fluids and the forces on them. (Fluids include liquids, gases, and plasmas.) Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms. Fluid mechanics, especially fluid dynamics, is an active field of research with many unsolved or partly solved problems. Fluid mechanics can be mathematically complex. Sometimes it can best be solved by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Also taking advantage of the highly visual nature of fluid flow is particle image velocimetry, an experimental method for visualizing and analyzing fluid flow.
Contents
1 Brief history
2 Relationship to continuum mechanics
3 Assumptions
3.1 The continuum hypothesis
4 Navier–Stokes equations
4.1 General form of the equation
5 Newtonian versus non-Newtonian fluids
5.1 Equations for a Newtonian fluid
6 See also
7 Notes
8 References
9 External links
//
[edit] Brief history
Main article: History of fluid mechanics
The study of fluid mechanics goes back at least to the days of ancient Greece, when Archimedes investigated fluid statics and buoyancy. Medieval Persian & Arab natural philosophers, including Abū Rayhān al-Bīrūnī and Al-Khazini, combined that earlier work with dynamics[1] to presage the later development of fluid dynamics. Rapid advancement in fluid mechanics began with Leonardo da Vinci (observation and experiment), Evangelista Torricelli (barometer), Isaac Newton (viscosity) and Blaise Pascal (hydrostatics), and was continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics in Hydrodynamica (1738). Inviscid flow was further analyzed by various mathematicians (Leonhard Euler, d'Alembert, Lagrange, Laplace, Poisson) and viscous flow was explored by a multitude of engineers including Poiseuille and Gotthilf Heinrich Ludwig Hagen. Further mathematical justification was provided by Claude-Louis Navier and George Gabriel Stokes in the Navier–Stokes equations, and boundary layers were investigated (Ludwig Prandtl), while various scientists (Osborne Reynolds, Andrey Kolmogorov, Geoffrey Ingram Taylor) advanced the understanding of fluid viscosity and turbulence.
[edit] Relationship to continuum mechanics
Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table.
Continuum mechanicsThe study of the physics of continuous materials
Solid mechanicsThe study of the physics of continuous materials with a defined rest shape.
ElasticityDescribes materials that return to their rest shape after an applied stress.
PlasticityDescribes materials that permanently deform after a sufficient applied stress.
RheologyThe study of materials with both solid and fluid characteristics.
Fluid mechanicsThe study of the physics of continuous materials which take the shape of their container.
Non-Newtonian fluids
Newtonian fluids
In a mechanical view, a fluid is a substance that does not support shear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.
[edit] Assumptions
Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface.
Fluid mechanics assumes that every fluid obeys the following:
Conservation of mass
Conservation of energy
Conservation of momentum
The continuum hypothesis, detailed below.
Further, it is often useful (at subsonic conditions) to assume a fluid is incompressible – that is, the density of the fluid does not change. Liquids can often be modelled as incompressible fluids, whereas gases cannot.
Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between the fluid and the boundary results also in a zero velocity for the fluid at the boundary. This is called the no-slip condition. For a porous media otherwise, in the frontier of the containing vessel, the slip condition is not zero velocity, and the fluid has a discontinuous velocity field between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition).
[edit] The continuum hypothesis
Main article: Continuum mechanics
Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.
The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.
Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above unity are best evaluated using statistical mechanics for reliable solutions.
[edit] Navier–Stokes equations
Main article: Navier–Stokes equations
The Navier–Stokes equations (named after Claude-Louis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier–Stokes equations describe the balance of forces acting at any given region of the fluid.
The Navier–Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change of the variables of interest. For example, the Navier–Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.
This means that solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.
For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.
[edit] General form of the equation
The general form of the Navier–Stokes equations for the conservation of momentum is:
where
is the fluid density,
is the substantive derivative (also called the material derivative),
is the velocity vector,
is the body force vector, and
is a tensor that represents the surface forces applied on a fluid particle (the comoving stress tensor).
Unless the fluid is made up of spinning degrees of freedom like vortices, is a symmetric tensor. In general, (in three dimensions) has the form:
where
are normal stresses,
are tangential stresses (shear stresses).
The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.
[edit] Newtonian versus non-Newtonian fluids
A Newtonian fluid (named after Isaac Newton) is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction). Important fluids, like water as well as most gases, behave — to good approximation — as a Newtonian fluid under normal conditions on Earth.[2]
By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time – this behaviour is seen in materials such as pudding, oobleck, or sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property — for example, most fluids with long molecular chains can react in a non-Newtonian manner.[2]
[edit] Equations for a Newtonian fluid
Main article: Newtonian fluid
The constant of proportionality between the shear stress and the velocity gradient is known as the viscosity. A simple equation to describe Newtonian fluid behaviour is
where
τ is the shear stress exerted by the fluid ("drag")
μ is the fluid viscosity – a constant of proportionality
is the velocity gradient perpendicular to the direction of shear.
For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure, not on the forces acting upon it. If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is
where
τij is the shear stress on the ith face of a fluid element in the jth direction
vi is the velocity in the ith direction
xj is the jth direction coordinate.
If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.
Among fluids, two rough broad divisions can be made: ideal and non-ideal fluids. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. An Ideal fluid is non viscous- offers no resistance whatsoever to a shearing force.
One can group real fluids into Newtonian and non-Newtonian. Newtonian fluids agree with Newton's law of viscosity. Non-Newtonian fluids can be either plastic, bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelatic.
[edit] See also
Physics portal
Wikibooks has more on the topic of
Fluid mechanics
Applied mechanics
Secondary flow
Bernoulli's principle
Communicating vessels
Tuesday, September 7, 2010
Is It More Expensive To Be Single?
Have you ever considered how much it costs to be single versus the cost of being in a relationship? It may seem strange, but being single can actually cost you more money than being with someone.
You always hear stories about the guy spending a ton of money on the girl. So it is natural to think that there is no way being in a relationship can be cheaper. At a quick glance, this seems to be very true. Looking around you, there are couples going on dates all of the time, buying gifts, and every little thing adds up.
These are important but you need to consider other aspects of the relationship. If you share an apartment or a loft, the cost of the apartment is much cheaper. It is cut in half if you share the same sized place. You can easily share a bedroom, so a 1 bedroom apartment would cost half what you would originally pay. This amount adds up to a lot over time.
More savings comes from travel too. You can do and share more things that you could not do if you were single. Sharing rooms, again is very easy. It is not entirely essential to have a whole room to yourself for the whole day even if you are traveling alone. Significant savings are made if you share with a partner.
Sharing food also saves a lot of money. Aside from the fact that you can share food, you can also cook at home. Couples are more likely to cook at home than single people are. Eating out costs a lot more money and hurts savings possibilities couples would have at home.
You always hear stories about the guy spending a ton of money on the girl. So it is natural to think that there is no way being in a relationship can be cheaper. At a quick glance, this seems to be very true. Looking around you, there are couples going on dates all of the time, buying gifts, and every little thing adds up.
These are important but you need to consider other aspects of the relationship. If you share an apartment or a loft, the cost of the apartment is much cheaper. It is cut in half if you share the same sized place. You can easily share a bedroom, so a 1 bedroom apartment would cost half what you would originally pay. This amount adds up to a lot over time.
More savings comes from travel too. You can do and share more things that you could not do if you were single. Sharing rooms, again is very easy. It is not entirely essential to have a whole room to yourself for the whole day even if you are traveling alone. Significant savings are made if you share with a partner.
Sharing food also saves a lot of money. Aside from the fact that you can share food, you can also cook at home. Couples are more likely to cook at home than single people are. Eating out costs a lot more money and hurts savings possibilities couples would have at home.
DESIGN
No generally-accepted definition of “design” exists[1], and the term has different connotations in different fields (see design disciplines below). Informally, “a design” (noun) refers to a plan for the construction of an object (as in architectural blueprints, circuit diagrams and sewing patterns) and “to design” (verb) refers to making this plan[2]. However, one can also design by directly constructing an object (as in pottery, cowboy coding and graphic design).
More formally, design has been defined as follows.
(noun) a specification of an object, manifested by an agent, intended to accomplish goals, in a particular environment, using a set of primitive components, satisfying a set of requirements, subject to constraints;
(verb, transitive) to create a design, in an environment (where the designer operates)[3]
Here, a "specification" can be manifested as either a plan or a finished product and "primitives" are the elements from which the design object is composed.
With such a broad denotation, there is no universal language or unifying institution for designers of all disciplines. This allows for many differing philosophies and approaches toward the subject (see Philosophies and studies of design, below).
The person designing is called a designer, which is also a term used for people who work professionally in one of the various design areas, usually also specifying which area is being dealt with (such as a fashion designer, concept designer or web designer). A designer’s sequence of activities is called a design process[4]. The scientific study of design is called design science [5].
Designing often necessitates considering the aesthetic, functional, economic and sociopolitical dimensions of both the design object and design process. It may involve considerable research, thought, modeling, interactive adjustment, and re-design. Meanwhile, diverse kinds of objects may be designed, including clothing, graphical user interfaces, skyscrapers, corporate identities, business processes and even methods of designing[6].
Contents
1 Design as a process
1.1 Typical steps
2 Philosophies and studies of design
2.1 Philosophies for guiding design
2.2 Approaches to design
2.3 Methods of designing
2.4 Philosophies for the purpose of designs
3 Terminology
3.1 Design and art
3.2 Design and engineering
3.3 Design and production
3.4 Process design
4 See also
4.1 Design disciplines
4.2 Design approaches and methods
4.3 Other design related topics
5 External links
6 Footnotes
//
[edit] Design as a process
Design, as a process, can take many forms depending on the object being designed and the individual(s) participating. A simple definition is that design is the process of giving form to an idea. "Form" could be a plan of action or a description of a physical thing.
[edit] Typical steps
A design process may include a series of steps followed by designers. Depending on the product or service, some of these stages may be irrelevant, ignored in real-world situations in order to save time, reduce cost, or because they may be redundant in the situation.
Typical stages of the design process include:
Pre-production design
Design brief or Parti – an early (often the beginning) statement of design goals
Analysis – analysis of current design goals
Research – investigating similar design solutions in the field or related topics
Specification – specifying requirements of a design solution for a product (product design specification[7]) or service.
Problem solving – conceptualizing and documenting design solutions
Presentation – presenting design solutions
Design during production
Development – continuation and improvement of a designed solution
Testing – in situ testing a designed solution
Post-production design feedback for future designs
Implementation – introducing the designed solution into the environment
Evaluation and conclusion – summary of process and results, including constructive criticism and suggestions for future improvements
Redesign – any or all stages in the design process repeated (with corrections made) at any time before, during, or after production.
These stages are not universally accepted but do relate typical design process activities. For each activity there are many best practices for completing them.[8]
[edit] Philosophies and studies of design
There are countless philosophies for guiding design as the design values and its accompanying aspects within modern design vary, both between different schools of thought and among practicing designers.[9] Design philosophies are usually for determining design goals. A design goal may range from solving the least significant individual problem of the smallest element, to the most holistic influential utopian goals. Design goals are usually for guiding design. However, conflicts over immediate and minor goals may lead to questioning the purpose of design, perhaps to set better long term or ultimate goals.
A 1938 Bugatti Type 57SC Atlantic from the Ralph Lauren collection. "Form follows function" can be an aesthetic point of view that a design can heighten, as often seen in the work of the Bugattis, Ettore, Rembrandt, and Jean.
[edit] Philosophies for guiding design
A design philosophy is a guide to help make choices when designing such as ergonomics, costs, economics, functionality and methods of re-design. An example of a design philosophy is “dynamic change” to achieve the elegant or stylish look you need.
[edit] Approaches to design
A design approach is a general philosophy that may or may not include a guide for specific methods. Some are to guide the overall goal of the design. Other approaches are to guide the tendencies of the designer. A combination of approaches may be used if they don't conflict.
Some popular approaches include:
KISS principle, (Keep it Simple Stupid, etc.), which strives to eliminate unnecessary complications.
There is more than one way to do it (TIMTOWTDI), a philosophy to allow multiple methods of doing the same thing.
Use-centered design, which focuses on the goals and tasks associated with the use of the artifact, rather than focusing on the end user.
User-centered design, which focuses on the needs, wants, and limitations of the end user of the designed artifact.
[edit] Methods of designing
Main article: Design methods
Design Methods is a broad area that focuses on:
Exploring possibilities and constraints by focusing critical thinking skills to research and define problem spaces for existing products or services—or the creation of new categories; (see also Brainstorming)
Redefining the specifications of design solutions which can lead to better guidelines for traditional design activities (graphic, industrial, architectural, etc.);
Managing the process of exploring, defining, creating artifacts continually over time
Prototyping possible scenarios, or solutions that incrementally or significantly improve the inherited situation
Trendspotting; understanding the trend process.
[edit] Philosophies for the purpose of designs
In philosophy, the abstract noun "design" refers to a pattern with a purpose. Design is thus contrasted with purposelessness, randomness, or lack of complexity.
To study the purpose of designs, beyond individual goals (e.g. marketing, technology, education, entertainment, hobbies), is to question the controversial politics, morals, ethics and needs such as Maslow's hierarchy of needs. "Purpose" may also lead to existential questions such as religious morals and teleology. These philosophies for the "purpose of" designs are in contrast to philosophies for guiding design or methodology.
Often a designer (especially in commercial situations) is not in a position to define purpose. Whether a designer is, is not, or should be concerned with purpose or intended use beyond what they are expressly hired to influence, is debatable, depending on the situation. In society, not understanding or disinterest in the wider role of design might also be attributed to the commissioning agent or client, rather than the designer.
In structuration theory, achieving consensus and fulfillment of purpose is as continuous as society. Raised levels of achievement often lead to raised expectations. Design is both medium and outcome, generating a Janus-like face, with every ending marking a new beginning.
[edit] Terminology
Look up design in Wiktionary, the free dictionary.
The word "design" is often considered ambiguous depending on the application.
The new terminal at Barajas airport in Madrid, Spain
[edit] Design and art
Design is often viewed as a more rigorous form of art, or art with a clearly defined purpose. The distinction is usually made when someone other than the artist is defining the purpose. In graphic arts the distinction is often made between fine art and commercial art. Applied art and decorative arts are other terms, the latter mostly used for objects from the past.
In the realm of the arts, design is more relevant to the "applied" arts, such as architecture and industrial design. In fact today the term design is widely associated to modern industrial product design as initiated by Raymond Loewy and teachings at the Bauhaus and Ulm School of Design (HfG Ulm) in Germany during the 20th Century.
Design implies a conscious effort to create something that is both functional and aesthetically pleasing. For example, a graphic artist may design an advertisement poster. This person's job is to communicate the advertisement message (functional aspect) and to make it look good (aesthetically pleasing).
The distinction between pure and applied arts is not completely clear, but one may consider Jackson Pollock's (often criticized as "splatter") paintings as an example of pure art. One may assume his art does not convey a message based on the obvious differences between an advertisement poster and the mere possibility of an abstract message of a Jackson Pollock painting. One may speculate that Pollock, when painting, worked more intuitively than would a graphic artist, when consciously designing a poster. However, Mark Getlein suggests the principles of design are "almost instinctive", "built-in", "natural", and part of "our sense of 'rightness'."[10] Pollock, as a trained artist, may have utilized design whether conscious or not.
A drawing for a booster engine for steam locomotives. Engineering is applied to design, with emphasis on function and the utilization of mathematics and science.
[edit] Design and engineering
Engineering is often viewed as a more rigorous form of design. Contrary views suggest that design is a component of engineering aside from production and other operations which utilize engineering. A neutral view may suggest that design and engineering simply overlap, depending on the discipline of design. The American Heritage Dictionary defines design as: "To conceive or fashion in the mind; invent," and "To formulate a plan", and defines engineering as: "The application of scientific and mathematical principles to practical ends such as the design, manufacture, and operation of efficient and economical structures, machines, processes, and systems.".[11][12] Both are forms of problem-solving with a defined distinction being the application of "scientific and mathematical principles". How much science is applied in a design is a question of what is considered "science". Along with the question of what is considered science, there is social science versus natural science. Scientists at Xerox PARC made the distinction of design versus engineering at "moving minds" versus "moving atoms".
Jonathan Ive has received several awards for his design of Apple Inc. products like this laptop. In some design fields, personal computers are also used for both design and production
[edit] Design and production
The relationship between design and production is one of planning and executing. In theory, the plan should anticipate and compensate for potential problems in the execution process. Design involves problem-solving and creativity. In contrast, production involves a routine or pre-planned process. A design may also be a mere plan that does not include a production or engineering process, although a working knowledge of such processes is usually expected of designers. In some cases, it may be unnecessary and/or impractical to expect a designer with a broad multidisciplinary knowledge required for such designs to also have a detailed specialized knowledge of how to produce the product.
Design and production are intertwined in many creative professional careers, meaning problem-solving is part of execution and the reverse. As the cost of rearrangement increases, the need for separating design from production increases as well. For example, a high-budget project, such as a skyscraper, requires separating (design) architecture from (production) construction. A Low-budget project, such as a locally printed office party invitation flyer, can be rearranged and printed dozens of times at the low cost of a few sheets of paper, a few drops of ink, and less than one hour's pay of a desktop publisher.
This is not to say that production never involves problem-solving or creativity, nor that design always involves creativity. Designs are rarely perfect and are sometimes repetitive. The imperfection of a design may task a production position (e.g. production artist, construction worker) with utilizing creativity or problem-solving skills to compensate for what was overlooked in the design process. Likewise, a design may be a simple repetition (copy) of a known preexisting solution, requiring minimal, if any, creativity or problem-solving skills from the designer.
An example of a business workflow process using Business Process Modeling Notation.
[edit] Process design
"Process design" (in contrast to "design process" mentioned above) refers to the planning of routine steps of a process aside from the expected result. Processes (in general) are treated as a product of design, not the method of design. The term originated with the industrial designing of chemical processes. With the increasing complexities of the information age, consultants and executives have found the term useful to describe the design of business processes as well as manufacturing processes.
[edit] See also
Design portal
Philosophy of design
[edit] Design disciplines
Commerce
Business design
New product development
Packaging design
Product design
Service design
Process Design
Applications
Experience design
Game design
Interaction design[13]
Sonic interaction design
Software design
Software development
Software engineering
System design
User experience design
User interface design
Web accessibility
Web design
Communications
Book design
Communication design
Content design
Exhibition design
Graphic design
Information design
Instructional design
Motion graphic design
News design
Production design
Sound design
Theatrical design
Typeface design
Typography
Visual communication
Scientific and mathematical
Combinatorial design[14]
Design of experiments
Physical
Architectural design
Architectural engineering
Automotive design
Cellular manufacturing
Ceramic and glass design
Design engineer
Environmental design
Fashion design
Floral design
Furniture design
Garden design
Geometric design
Industrial design
Interior design/redesign
Landscape architecture
Mechanical engineering
Sustainable design
Urban design
[edit] Design approaches and methods
Co-Design
Creative problem solving
Creativity techniques
C-K theory
Design for X
Design leadership
Design management
Design-build
Design patterns
Design strategy
Design thinking
Engineering design process
Error-tolerant design
Fault tolerant design
Functional design
Metadesign
Mind mapping
Open design
Participatory design[15]
Reliable system design
Theory of Constraints
Transformation design
TRIZ
Universal design
User innovation
[edit] Other design related topics
Design organizations
Chartered Society of Designers
The Design Association
AIGA
The Design Society
Interaction Design Association
Design Awards
European Design Awards
IF product design award
James Dyson Award
German Design Award
Design tools
Computer-aided design[16]
Graphic organizers
Design as intellectual property
Design patent (US patent law)
Industrial design rights
Industrial design rights in the European Union
Impact of design
Creative industries
Design classic
Design Museums and Education Centres
Design Exchange
Studying design
Critical design
Design education
Design research
Wicked problems[17]
Designs for the future
Concept design
Futurology
More formally, design has been defined as follows.
(noun) a specification of an object, manifested by an agent, intended to accomplish goals, in a particular environment, using a set of primitive components, satisfying a set of requirements, subject to constraints;
(verb, transitive) to create a design, in an environment (where the designer operates)[3]
Here, a "specification" can be manifested as either a plan or a finished product and "primitives" are the elements from which the design object is composed.
With such a broad denotation, there is no universal language or unifying institution for designers of all disciplines. This allows for many differing philosophies and approaches toward the subject (see Philosophies and studies of design, below).
The person designing is called a designer, which is also a term used for people who work professionally in one of the various design areas, usually also specifying which area is being dealt with (such as a fashion designer, concept designer or web designer). A designer’s sequence of activities is called a design process[4]. The scientific study of design is called design science [5].
Designing often necessitates considering the aesthetic, functional, economic and sociopolitical dimensions of both the design object and design process. It may involve considerable research, thought, modeling, interactive adjustment, and re-design. Meanwhile, diverse kinds of objects may be designed, including clothing, graphical user interfaces, skyscrapers, corporate identities, business processes and even methods of designing[6].
Contents
1 Design as a process
1.1 Typical steps
2 Philosophies and studies of design
2.1 Philosophies for guiding design
2.2 Approaches to design
2.3 Methods of designing
2.4 Philosophies for the purpose of designs
3 Terminology
3.1 Design and art
3.2 Design and engineering
3.3 Design and production
3.4 Process design
4 See also
4.1 Design disciplines
4.2 Design approaches and methods
4.3 Other design related topics
5 External links
6 Footnotes
//
[edit] Design as a process
Design, as a process, can take many forms depending on the object being designed and the individual(s) participating. A simple definition is that design is the process of giving form to an idea. "Form" could be a plan of action or a description of a physical thing.
[edit] Typical steps
A design process may include a series of steps followed by designers. Depending on the product or service, some of these stages may be irrelevant, ignored in real-world situations in order to save time, reduce cost, or because they may be redundant in the situation.
Typical stages of the design process include:
Pre-production design
Design brief or Parti – an early (often the beginning) statement of design goals
Analysis – analysis of current design goals
Research – investigating similar design solutions in the field or related topics
Specification – specifying requirements of a design solution for a product (product design specification[7]) or service.
Problem solving – conceptualizing and documenting design solutions
Presentation – presenting design solutions
Design during production
Development – continuation and improvement of a designed solution
Testing – in situ testing a designed solution
Post-production design feedback for future designs
Implementation – introducing the designed solution into the environment
Evaluation and conclusion – summary of process and results, including constructive criticism and suggestions for future improvements
Redesign – any or all stages in the design process repeated (with corrections made) at any time before, during, or after production.
These stages are not universally accepted but do relate typical design process activities. For each activity there are many best practices for completing them.[8]
[edit] Philosophies and studies of design
There are countless philosophies for guiding design as the design values and its accompanying aspects within modern design vary, both between different schools of thought and among practicing designers.[9] Design philosophies are usually for determining design goals. A design goal may range from solving the least significant individual problem of the smallest element, to the most holistic influential utopian goals. Design goals are usually for guiding design. However, conflicts over immediate and minor goals may lead to questioning the purpose of design, perhaps to set better long term or ultimate goals.
A 1938 Bugatti Type 57SC Atlantic from the Ralph Lauren collection. "Form follows function" can be an aesthetic point of view that a design can heighten, as often seen in the work of the Bugattis, Ettore, Rembrandt, and Jean.
[edit] Philosophies for guiding design
A design philosophy is a guide to help make choices when designing such as ergonomics, costs, economics, functionality and methods of re-design. An example of a design philosophy is “dynamic change” to achieve the elegant or stylish look you need.
[edit] Approaches to design
A design approach is a general philosophy that may or may not include a guide for specific methods. Some are to guide the overall goal of the design. Other approaches are to guide the tendencies of the designer. A combination of approaches may be used if they don't conflict.
Some popular approaches include:
KISS principle, (Keep it Simple Stupid, etc.), which strives to eliminate unnecessary complications.
There is more than one way to do it (TIMTOWTDI), a philosophy to allow multiple methods of doing the same thing.
Use-centered design, which focuses on the goals and tasks associated with the use of the artifact, rather than focusing on the end user.
User-centered design, which focuses on the needs, wants, and limitations of the end user of the designed artifact.
[edit] Methods of designing
Main article: Design methods
Design Methods is a broad area that focuses on:
Exploring possibilities and constraints by focusing critical thinking skills to research and define problem spaces for existing products or services—or the creation of new categories; (see also Brainstorming)
Redefining the specifications of design solutions which can lead to better guidelines for traditional design activities (graphic, industrial, architectural, etc.);
Managing the process of exploring, defining, creating artifacts continually over time
Prototyping possible scenarios, or solutions that incrementally or significantly improve the inherited situation
Trendspotting; understanding the trend process.
[edit] Philosophies for the purpose of designs
In philosophy, the abstract noun "design" refers to a pattern with a purpose. Design is thus contrasted with purposelessness, randomness, or lack of complexity.
To study the purpose of designs, beyond individual goals (e.g. marketing, technology, education, entertainment, hobbies), is to question the controversial politics, morals, ethics and needs such as Maslow's hierarchy of needs. "Purpose" may also lead to existential questions such as religious morals and teleology. These philosophies for the "purpose of" designs are in contrast to philosophies for guiding design or methodology.
Often a designer (especially in commercial situations) is not in a position to define purpose. Whether a designer is, is not, or should be concerned with purpose or intended use beyond what they are expressly hired to influence, is debatable, depending on the situation. In society, not understanding or disinterest in the wider role of design might also be attributed to the commissioning agent or client, rather than the designer.
In structuration theory, achieving consensus and fulfillment of purpose is as continuous as society. Raised levels of achievement often lead to raised expectations. Design is both medium and outcome, generating a Janus-like face, with every ending marking a new beginning.
[edit] Terminology
Look up design in Wiktionary, the free dictionary.
The word "design" is often considered ambiguous depending on the application.
The new terminal at Barajas airport in Madrid, Spain
[edit] Design and art
Design is often viewed as a more rigorous form of art, or art with a clearly defined purpose. The distinction is usually made when someone other than the artist is defining the purpose. In graphic arts the distinction is often made between fine art and commercial art. Applied art and decorative arts are other terms, the latter mostly used for objects from the past.
In the realm of the arts, design is more relevant to the "applied" arts, such as architecture and industrial design. In fact today the term design is widely associated to modern industrial product design as initiated by Raymond Loewy and teachings at the Bauhaus and Ulm School of Design (HfG Ulm) in Germany during the 20th Century.
Design implies a conscious effort to create something that is both functional and aesthetically pleasing. For example, a graphic artist may design an advertisement poster. This person's job is to communicate the advertisement message (functional aspect) and to make it look good (aesthetically pleasing).
The distinction between pure and applied arts is not completely clear, but one may consider Jackson Pollock's (often criticized as "splatter") paintings as an example of pure art. One may assume his art does not convey a message based on the obvious differences between an advertisement poster and the mere possibility of an abstract message of a Jackson Pollock painting. One may speculate that Pollock, when painting, worked more intuitively than would a graphic artist, when consciously designing a poster. However, Mark Getlein suggests the principles of design are "almost instinctive", "built-in", "natural", and part of "our sense of 'rightness'."[10] Pollock, as a trained artist, may have utilized design whether conscious or not.
A drawing for a booster engine for steam locomotives. Engineering is applied to design, with emphasis on function and the utilization of mathematics and science.
[edit] Design and engineering
Engineering is often viewed as a more rigorous form of design. Contrary views suggest that design is a component of engineering aside from production and other operations which utilize engineering. A neutral view may suggest that design and engineering simply overlap, depending on the discipline of design. The American Heritage Dictionary defines design as: "To conceive or fashion in the mind; invent," and "To formulate a plan", and defines engineering as: "The application of scientific and mathematical principles to practical ends such as the design, manufacture, and operation of efficient and economical structures, machines, processes, and systems.".[11][12] Both are forms of problem-solving with a defined distinction being the application of "scientific and mathematical principles". How much science is applied in a design is a question of what is considered "science". Along with the question of what is considered science, there is social science versus natural science. Scientists at Xerox PARC made the distinction of design versus engineering at "moving minds" versus "moving atoms".
Jonathan Ive has received several awards for his design of Apple Inc. products like this laptop. In some design fields, personal computers are also used for both design and production
[edit] Design and production
The relationship between design and production is one of planning and executing. In theory, the plan should anticipate and compensate for potential problems in the execution process. Design involves problem-solving and creativity. In contrast, production involves a routine or pre-planned process. A design may also be a mere plan that does not include a production or engineering process, although a working knowledge of such processes is usually expected of designers. In some cases, it may be unnecessary and/or impractical to expect a designer with a broad multidisciplinary knowledge required for such designs to also have a detailed specialized knowledge of how to produce the product.
Design and production are intertwined in many creative professional careers, meaning problem-solving is part of execution and the reverse. As the cost of rearrangement increases, the need for separating design from production increases as well. For example, a high-budget project, such as a skyscraper, requires separating (design) architecture from (production) construction. A Low-budget project, such as a locally printed office party invitation flyer, can be rearranged and printed dozens of times at the low cost of a few sheets of paper, a few drops of ink, and less than one hour's pay of a desktop publisher.
This is not to say that production never involves problem-solving or creativity, nor that design always involves creativity. Designs are rarely perfect and are sometimes repetitive. The imperfection of a design may task a production position (e.g. production artist, construction worker) with utilizing creativity or problem-solving skills to compensate for what was overlooked in the design process. Likewise, a design may be a simple repetition (copy) of a known preexisting solution, requiring minimal, if any, creativity or problem-solving skills from the designer.
An example of a business workflow process using Business Process Modeling Notation.
[edit] Process design
"Process design" (in contrast to "design process" mentioned above) refers to the planning of routine steps of a process aside from the expected result. Processes (in general) are treated as a product of design, not the method of design. The term originated with the industrial designing of chemical processes. With the increasing complexities of the information age, consultants and executives have found the term useful to describe the design of business processes as well as manufacturing processes.
[edit] See also
Design portal
Philosophy of design
[edit] Design disciplines
Commerce
Business design
New product development
Packaging design
Product design
Service design
Process Design
Applications
Experience design
Game design
Interaction design[13]
Sonic interaction design
Software design
Software development
Software engineering
System design
User experience design
User interface design
Web accessibility
Web design
Communications
Book design
Communication design
Content design
Exhibition design
Graphic design
Information design
Instructional design
Motion graphic design
News design
Production design
Sound design
Theatrical design
Typeface design
Typography
Visual communication
Scientific and mathematical
Combinatorial design[14]
Design of experiments
Physical
Architectural design
Architectural engineering
Automotive design
Cellular manufacturing
Ceramic and glass design
Design engineer
Environmental design
Fashion design
Floral design
Furniture design
Garden design
Geometric design
Industrial design
Interior design/redesign
Landscape architecture
Mechanical engineering
Sustainable design
Urban design
[edit] Design approaches and methods
Co-Design
Creative problem solving
Creativity techniques
C-K theory
Design for X
Design leadership
Design management
Design-build
Design patterns
Design strategy
Design thinking
Engineering design process
Error-tolerant design
Fault tolerant design
Functional design
Metadesign
Mind mapping
Open design
Participatory design[15]
Reliable system design
Theory of Constraints
Transformation design
TRIZ
Universal design
User innovation
[edit] Other design related topics
Design organizations
Chartered Society of Designers
The Design Association
AIGA
The Design Society
Interaction Design Association
Design Awards
European Design Awards
IF product design award
James Dyson Award
German Design Award
Design tools
Computer-aided design[16]
Graphic organizers
Design as intellectual property
Design patent (US patent law)
Industrial design rights
Industrial design rights in the European Union
Impact of design
Creative industries
Design classic
Design Museums and Education Centres
Design Exchange
Studying design
Critical design
Design education
Design research
Wicked problems[17]
Designs for the future
Concept design
Futurology
A Surefire Way to Make Any Guy Fall in Love
Have you been trying to win the love of a really hot guy, but nothing is working and you don't know what to do anymore? Are you a regular on the singles scene and you now want to become one half of a great couple? Is it hard switching over from single thought process to relationship building thought process?
Some women get stuck in that mold of wild and fancy free dating and think that it will carry over well as they seek out a serious contender for a stable relationship. For the most part, the guys who are hanging out in single's hang outs aren't really looking to hook up for real. And if you're maintaining your wild and sexy attitude, the one guy who may have strayed in hoping to find love will most likely look right past you.
Beyond Sexuality
We've been trained, if not completely brainwashed to believe that sex is all there is. So naturally when we try to hook up with a guy it's our first line of defense. Sex draws them in, keeps them looking and keeps them interested.
But for how long?
For a real relationship to grow you need to move beyond this and fast. If a guy is quick to assess that all you're good for is a fun lay, whatever more you may have to offer is not going to interest him. Try switching things around and work to create a great bond with him first.
Don't jump into bed, but have this great conversation with him. Get to know him and let him see that you truly are a fabulous woman with lots to offer.
Keep Pressure Out
If things are going along nicely, but you're feeling that strong desire to move things more quickly, resist. This isn't something you really want to push a guy into. Let his emotions build on their own. Let him realize the gem he's found in you.
The more you try to push him into a relationship the harder time you'll have. Hold back, be patient and let the love grow. Without pressure, he'll fall in love with you a lot sooner.
Some women get stuck in that mold of wild and fancy free dating and think that it will carry over well as they seek out a serious contender for a stable relationship. For the most part, the guys who are hanging out in single's hang outs aren't really looking to hook up for real. And if you're maintaining your wild and sexy attitude, the one guy who may have strayed in hoping to find love will most likely look right past you.
Beyond Sexuality
We've been trained, if not completely brainwashed to believe that sex is all there is. So naturally when we try to hook up with a guy it's our first line of defense. Sex draws them in, keeps them looking and keeps them interested.
But for how long?
For a real relationship to grow you need to move beyond this and fast. If a guy is quick to assess that all you're good for is a fun lay, whatever more you may have to offer is not going to interest him. Try switching things around and work to create a great bond with him first.
Don't jump into bed, but have this great conversation with him. Get to know him and let him see that you truly are a fabulous woman with lots to offer.
Keep Pressure Out
If things are going along nicely, but you're feeling that strong desire to move things more quickly, resist. This isn't something you really want to push a guy into. Let his emotions build on their own. Let him realize the gem he's found in you.
The more you try to push him into a relationship the harder time you'll have. Hold back, be patient and let the love grow. Without pressure, he'll fall in love with you a lot sooner.
CALCULUS
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the branch of mathematics. For other uses, see Calculus (disambiguation).
It has been suggested that infinitesimal calculus be merged into this article or section. (Discuss)
Topics in Calculus
Fundamental theoremLimits of functionsContinuityMean value theorem
Differential calculus
DerivativeChange of variablesImplicit differentiationTaylor's theoremRelated ratesIdentitiesRules:
Power rule, Product rule, Quotient rule, Chain rule
Integral calculus
Integral
Lists of integralsImproper integralsIntegration by:parts, disks, cylindricalshells, substitution,trigonometric substitution,partial fractions, changing order
Vector calculus
GradientDivergenceCurlLaplacianGradient theoremGreen's theoremStokes' theoremDivergence theorem
Multivariable calculus
Matrix calculusPartial derivativeMultiple integralLine integralSurface integralVolume integralJacobian
Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change[1], in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
Contents
1 History
1.1 Ancient
1.2 Medieval
1.3 Modern
1.4 Significance
1.5 Foundations
2 Principles
2.1 Limits and infinitesimals
2.2 Differential calculus
2.3 Leibniz notation
2.4 Integral calculus
2.5 Fundamental theorem
3 Applications
4 See also
4.1 Lists
4.2 Related topics
5 References
5.1 Notes
5.2 Books
6 Other resources
6.1 Further reading
6.2 Online books
6.3 External links
//
[edit] History
Main article: History of calculus
[edit] Ancient
Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation.
The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[2][3] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[4] The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method which would later be called Cavalieri's principle to find the volume of a sphere.[3]
[edit] Medieval
This section may contain inappropriate or misinterpreted citations that do not verify the text. Please help improve this article by checking for inaccuracies. (help, talk, get involved!) (August 2010)
Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that is readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration.[5] In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form of Rolle's theorem.[6] Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Indian mathematician Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[8] which are treated in the text Yuktibhasa.[9][10][11]
[edit] Modern
This section does not cite any references or sources.Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2009)
In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1675.
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work (only in Britain, not in continental Europe), but is now regarded as an independent inventor of and contributor to calculus.
These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism—he often spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass (see (ε, δ)-definition of limit). It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.
Calculus is a ubiquitous topic in most modern high schools and universities around the world.[12]
[edit] Significance
While some of the ideas of calculus were developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.
Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.
[edit] Foundations
In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.
There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.
[edit] Principles
[edit] Limits and infinitesimals
Main articles: Limit of a function and Infinitesimal
Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.
In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
[edit] Differential calculus
Main article: Differential calculus
Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative, the doubling function.
If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = mx + b, where:
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:
Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.
The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.
The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.
[edit] Leibniz notation
Main article: Leibniz's notation
A common notation, introduced by Leibniz, for the derivative in the example above is
In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
In this usage, the dx in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.
[edit] Integral calculus
Main article: Integral
Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators.
The indefinite integral is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.)
The definite integral inputs a function and outputs a number, which gives the area between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.
A motivating example is the distances traveled in a given time.
If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
If f(x) in the diagram on the left represents speed as it varies over time, the distance traveled (between the times represented by a and b) is the area of the shaded region s.
To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero.
The symbol of integration is , an elongated S (the S stands for "sum"). The definite integral is written as:
and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation
is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; dx is not a number, and is not being multiplied by f(x).
The indefinite integral, or antiderivative, is written:
Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:
An undetermined constant like C in the antiderivative is known as a constant of integration.
[edit] Fundamental theorem
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
The Fundamental Theorem of Calculus states: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval (a, b), then
Furthermore, for every x in the interval (a, b),
This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
[edit] Applications
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.
Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are interrelated through calculus. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term "rate of change" which refers to the derivative saying The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Commonly expressed today as Force = Mass × acceleration, it involves differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.
Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or it can be used in probability theory to determine the probability of a continuous random variable from an assumed density function. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points.
Green's Theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.
In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it's used to derive dosing laws. In nuclear medicine, it's used to build models of radiation transport in targeted tumor therapies.
In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
Calculus is also used to find approximate solutions to equations; in practice it's the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.
[edit] See also
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This article is about the branch of mathematics. For other uses, see Calculus (disambiguation).
It has been suggested that infinitesimal calculus be merged into this article or section. (Discuss)
Topics in Calculus
Fundamental theoremLimits of functionsContinuityMean value theorem
Differential calculus
DerivativeChange of variablesImplicit differentiationTaylor's theoremRelated ratesIdentitiesRules:
Power rule, Product rule, Quotient rule, Chain rule
Integral calculus
Integral
Lists of integralsImproper integralsIntegration by:parts, disks, cylindricalshells, substitution,trigonometric substitution,partial fractions, changing order
Vector calculus
GradientDivergenceCurlLaplacianGradient theoremGreen's theoremStokes' theoremDivergence theorem
Multivariable calculus
Matrix calculusPartial derivativeMultiple integralLine integralSurface integralVolume integralJacobian
Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change[1], in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
Contents
1 History
1.1 Ancient
1.2 Medieval
1.3 Modern
1.4 Significance
1.5 Foundations
2 Principles
2.1 Limits and infinitesimals
2.2 Differential calculus
2.3 Leibniz notation
2.4 Integral calculus
2.5 Fundamental theorem
3 Applications
4 See also
4.1 Lists
4.2 Related topics
5 References
5.1 Notes
5.2 Books
6 Other resources
6.1 Further reading
6.2 Online books
6.3 External links
//
[edit] History
Main article: History of calculus
[edit] Ancient
Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation.
The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[2][3] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[4] The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method which would later be called Cavalieri's principle to find the volume of a sphere.[3]
[edit] Medieval
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Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that is readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration.[5] In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form of Rolle's theorem.[6] Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Indian mathematician Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[8] which are treated in the text Yuktibhasa.[9][10][11]
[edit] Modern
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In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1675.
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work (only in Britain, not in continental Europe), but is now regarded as an independent inventor of and contributor to calculus.
These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism—he often spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass (see (ε, δ)-definition of limit). It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.
Calculus is a ubiquitous topic in most modern high schools and universities around the world.[12]
[edit] Significance
While some of the ideas of calculus were developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.
Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.
[edit] Foundations
In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.
There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.
[edit] Principles
[edit] Limits and infinitesimals
Main articles: Limit of a function and Infinitesimal
Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.
In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
[edit] Differential calculus
Main article: Differential calculus
Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative, the doubling function.
If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = mx + b, where:
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:
Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.
The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.
The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.
[edit] Leibniz notation
Main article: Leibniz's notation
A common notation, introduced by Leibniz, for the derivative in the example above is
In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
In this usage, the dx in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.
[edit] Integral calculus
Main article: Integral
Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators.
The indefinite integral is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.)
The definite integral inputs a function and outputs a number, which gives the area between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.
A motivating example is the distances traveled in a given time.
If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
If f(x) in the diagram on the left represents speed as it varies over time, the distance traveled (between the times represented by a and b) is the area of the shaded region s.
To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero.
The symbol of integration is , an elongated S (the S stands for "sum"). The definite integral is written as:
and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation
is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; dx is not a number, and is not being multiplied by f(x).
The indefinite integral, or antiderivative, is written:
Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:
An undetermined constant like C in the antiderivative is known as a constant of integration.
[edit] Fundamental theorem
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
The Fundamental Theorem of Calculus states: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval (a, b), then
Furthermore, for every x in the interval (a, b),
This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
[edit] Applications
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.
Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are interrelated through calculus. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term "rate of change" which refers to the derivative saying The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Commonly expressed today as Force = Mass × acceleration, it involves differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.
Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or it can be used in probability theory to determine the probability of a continuous random variable from an assumed density function. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points.
Green's Theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.
In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it's used to derive dosing laws. In nuclear medicine, it's used to build models of radiation transport in targeted tumor therapies.
In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
Calculus is also used to find approximate solutions to equations; in practice it's the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.
[edit] See also
Find more about Calculus on Wikipedia's sister projects:
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Main article: Outline of calculus
[edit] Lists
List of differentiation identities
List of calculus topics
Publications in calculus
Table of integrals
[edit] Related topics
Calculus of finite differences
Calculus with polynomials
Complex analysis
Differential equation
Differential geometry
Elementary calculus
Fourier series
Integral equation
Mathematical analysis
Mathematics
Multivariable calculus
Non-classical analysis
Non-standard analysis
Non-standard calculus
Precalculus (mathematical education)
Product Integrals
Stochastic calculus
Taylor series
Time-scale calculus
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