Vector calculus

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Vector calculus 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.

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.

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