Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod*

Copyright 1994-2000 by Evans M. Harrell II and James V. Herod. All rights reserved.


This document collects some standard vector identities and relationships among coordinate systems in three dimensions. It is assumed that all vector fields are differentiable arbitrarily often; if the vector field is not sufficiently smooth, some of these formulae are in doubt.

Basic formulae

  1. The dot product
  2. The cross product
  3. The triple product
  4. Other multiple products.
  5. The gradient is defined on a scalar field f and produces a vector field, denoted
  6. The divergence is defined on a vector field v and produces a scalar field, denoted
  7. The curl is defined on a vector field and produces another vector field, except that the curl of a vector field is not affected by reflection in the same way as the vector field is. It is denoted Unlike the gradient and the divergence, which work in all dimensions, the curl is special to three dimensions.
  8. The Laplacian is defined as
  9. Product rules:
  10. Chain rules
  11. Integral identities ( Green's, Gauss's, and Stokes's identities):
      Green's identities:
    1. int_U  (f grad^2 g) = int_{border U} (f n dot grad g) - int_U  (grad f dot grad g)
    2. int_U  (f grad^2 g - g grad^2 f) = int_{border U} (f n dot grad g - g n dot grad f)
  12. Relationships among the common three-dimensional coordinate systems.
  13. Cylindrical vector calculus. Let ek denote the unit vector in the direction of increase of coordinate k. Then
  14. Spherical vector calculus. Let ek denote the unit vector in the direction of increase of coordinate k. Then

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