Vector calculus

From Wikipedia, the free encyclopedia

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus</br>Tensor calculus Mean value theorem
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution

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 formulas 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 <ref>Tai, Chen-to "A historical study of vector analysis" (1995)</ref>.

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.

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 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.

Likewise, there are three 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.

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[edit] References

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.
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)