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Vector Calculus




Vector calculus is a field of Mathematics concerned with multivariate Real Analysis of Vector s in two or more Dimension s. It consists of a suite of Formula s and problem solving techniques very useful for Engineering and Physics .

It concerns Vector Field s, which associate a vector to every point in space, and Scalar Field s, 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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