Vector calculus

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 Tai, Chen-to "A historical study of vector analysis" (1995).

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. 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.

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.

Footnotes

References

(Summary)

External links

Expanding vector analysis to a non-orthogonal space
A historical study of vector analysis (1995) Tai, Chen-to
A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen-to

Vector calculus

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