Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariable real analysis of vectors in an inner product space of two or more dimensions (some results — those that involve the cross product — can only be applied to three dimensions). Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which It consists of a suite of formulae and problem solving techniques very useful for engineering and physics. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Vector analysis has its origin in quaternion analysis, and was formulated by the American engineer and scientist J. Willard Gibbs and the British engineer Oliver Heaviside. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist
Vector calculus is concerned with scalar fields, which associate a scalar to every point in space, and vector fields, which associate a vector to every point in space. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.
Vector operations
Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (
). In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator &nablaDel The four most important operations in vector calculus are:
| Operation | Notation | Description | Domain/Range |
|---|---|---|---|
| Gradient |  |
Measures the rate and direction of change in a scalar field. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar | Maps scalar fields to vector fields. |
| Curl |  |
Measures the tendency to rotate about a point in a vector field. cURL is a Command line tool for transferring files with URL syntax. | Maps vector fields to vector fields. |
| Divergence |  |
Measures the magnitude of a source or sink at a given point in a vector field. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the | Maps vector fields to scalar fields. |
| Laplacian |  |
A composition of the divergence and gradient operations. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after | Maps scalar fields to scalar fields. |
A quantity called the Jacobian is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.
Theorems
Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus to higher dimensions:
| Theorem | Statement | Description |
|---|---|---|
| Gradient theorem |  |
The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. The gradient theorem, sometimes also known as the fundamental theorem of calculus for line integrals, says that a Line integral through a Gradient field In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object |
| Green's theorem |  |
The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the curve bounding the region. In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral |
| Stokes' theorem |  |
The integral of the curl of a vector field over a surface equals the line integral of the vector field over the curve bounding the surface. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. |
| Divergence theorem |  |
The integral of the divergence of a vector field over some solid equals the integral of the flux through the surface bounding the solid. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks |
The use of vector calculus may require the handedness of the coordinate system to be taken into account (see cross product and handedness for more detail). In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry
See also
- Vector calculus identities
- Irrotational vector field
- Solenoidal vector field
- Laplacian vector field
- Vector Analysis (Gibbs/Wilson)
References
- Michael J. The following identities are important in Vector calculus: Single operators (summary This section explicitly lists what some symbols mean for clarity In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero In Vector calculus, a Laplacian vector field is a Vector field which is both irrotational and incompressible. Vector Analysis is a book on Vector calculus first published in 1901 by Edwin Bidwell Wilson. Crowe (1994). A History of Vector Analysis : The Evolution of the Idea of a Vectorial System. Dover Publications; Reprint edition. ISBN 0-486-67910-1. (Summary)
- H. M. Schey (2005). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-92516-1.
- J. E. Marsden (1976). Vecor Calculus. W. H. Freeman & Company. ISBN 0-7167-o462-5.
- Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan.
External links
- Expanding vector analysis to a non-orthogonal space
- Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson, published 1902. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist Edwin Bidwell Wilson ( April 25 1879 – December 28 1964) was a Mathematician and Polymath.
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