Variations of the Radial Unit Vector

  We can use the chain rule to find an expression for the variation of a function, f, in terms of its variables and then factor this into two parts, one involving only positions and the second containing changes in position. This works for both spherical coordinates and rectangular coordinates. The first factor is known as the gradient and is commonly written as ∇f.

For future reference we collect some formulas for spherical coordinates here.

We can use the same trick to factor the changes in the direction cosines of the radial unit vector but in this case the coefficients of the changes in position are vectors. As expected both sets of gradients can be represented by the same formula in both coordinate systems. The matrices G' and G are the coefficients of the changes in position in spherical and rectangular coordinates respectively.