Vector calculus

Vector calculus is a field of mathematics concerned with multivariate real analysis of vectorss in 2 or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics.

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. 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 certain points; the divergence of a vector field is a scalar field.

Most of the analytic results are more easily understood using the machinery of differential geometry, for which vector calculus forms a subset.



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