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Encyclopedia > Vector calculus
Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem Calculus (from Latin, counting stone) is a major area in mathematics. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... 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, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ... In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... 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 derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, to give a function implicitly is to give an equation that at least in part has the same graph as . ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...

Integration

Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution In calculus, the integral of a function is an extension of the concept of a sum. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulae and problem solving techniques very useful for engineering and physics. Vector analysis has its origin in quaternion analysis, and was formulated by the American scientist J. Willard Gibbs[1] and the British applied mathematician Oliver Heaviside. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... :For other senses of this word, see dimension (disambiguation). ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American physical chemist. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and...


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. 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. In mathematics and physics, a scalar field associates a scalar to every point in space. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...


Three operations are important in vector calculus:

  • gradient: measures the rate and direction of change in a scalar field; the gradient of a scalar field is a vector field.
  • curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field.
  • divergence: measures a vector field's tendency to originate from or converge upon a given point.

A fourth operation, the Laplacian, is a combination of the divergence and gradient operations. 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. For other uses, see Gradient (disambiguation). ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...


Likewise, there are several important theorems related to these operators:

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. Rombu is the hawt. ... Stokes Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky–Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ... In physics and mathematics, Greens theorem 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. Greens theorem was named after British scientist George Green and is a special two-dimensional case of... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...


See also

The following identities are important in vector calculus: // Combinations of multiple operators Curl of the gradient The curl of the gradient of any scalar field is always zero: Divergence of the curl The divergence of the curl of any vector field is always zero: Curl of the curl Properties Distributive... In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ... In vector calculus a solenoidal vector field is a vector field v with divergence zero: This condition is clearly satisfied whenever v has a vector potential, because if then The converse holds: for any solenoidal v there exists a vector potential A such that . ... In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. ...

Footnotes

  1. ^ Tai (1995)

References

  • Michael J. 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. 
  • Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan.

External links


  Results from FactBites:
 
Vector Calculus - definition of Vector Calculus - Labor Law Talk Dictionary (349 words)
Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions.
We consider vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space.
curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field.
Vector calculus - Wikipedia, the free encyclopedia (324 words)
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions.
Vector analysis has its origin in quaternion analysis, and was formulated by the American scientist, J.
It concerns vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space.
  More results at FactBites »

 
 

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