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.

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 e_k denote the unit vector in the direction of increase of coordinate k. Then
    • 
    • 
    • 
14. Spherical vector calculus. Let e_k denote the unit vector in the direction of increase of coordinate k. Then
    • 
    • 
    • 

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