Small changes in position transform nearly linearly from rectangular to spherical coordinates and we can represent this transformation by the matrix L. The inverse of this matrix can be used to transform the gradient matrices in a similar manner. Starting with δr̄ in the spherical coordinate system and using the chain rule again we can evaluate δr, δθ, and δφ in terms of δx, δy, and δz and then substitute the coefficients of δθ and δφ for the partial derivatives to get the components of the transformation matrix L.
Finding the coefficients of δθ and δφ is a little tricky since we have ranges 0≤θ≤π and 0≤φ≤2π in spherical coordinates while the inverse tangent is defined only for -π/2≤θ≤π/2. It helps to rewrite the inverse tangent using expressions that are continuous for the segments of a circle with positive and negative values of y and using δ[tan-1u]=δu/(1+u2) to evaluate δθ and δφ to get the results above.
The determinant of L is equal to 1 which confirms the magnitude of the δr̄ is the same in both coordinate systems.
Multiplying L on the right by the transpose of L verifies that it is the right inverse of L.
One can show that the transpose of L is also the left inverse of L.
So we arrive at the following summary of results.
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