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

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 quantifies the correlation between the vectors a and b .
2. The cross product is the area of the parallelogram spanned by the vectors a and b.
Notice that Unlike the dot product, which works in all dimensions, the cross product is special to three dimensions.
3. The triple product has the value of the determinant of the matrix consisting of a, b, and c as row vectors. It is unchanged by cyclic permutation: Although the cross product is strictly three-dimensional, the generalization of the triple product as a determinant is useful in all dimensions.

4. Other multiple products.  5. The gradient is defined on a scalar field f and produces a vector field, denoted It quantifies the rate of change and points in the direction of greatest change.
6. The divergence is defined on a vector field v and produces a scalar field, denoted It quantifies the tendency of neighboring vectors to point away from one another (or towards one another, if negative)
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: or, equivalently, grad (f g) = f grad g + g grad f  or, equivalently, div(f v) = f div v + grad f . v or, equivalently, curl(f v) = f curl v + grad f X v  10. Chain rules or, equivalently, grad f(g(x)) = f'(g(x)) grad g(x) or, equivalently, df(w(t))/dt = grad f(w(t)) w'(t)   11. Integral identities ( Green's, Gauss's, and Stokes's identities):
Green's identities:
1. 2. Gauss's divergence theorem:
• Stokes's theorem:

• 12. Relationships among the common three-dimensional coordinate systems.
• Cartesian in spherical • Cartesian in cylindrical • spherical in Cartesian • spherical in cylindrical • cylindrical in Cartesian • cylindrical in spherical 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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