= Gradient in orthogonal coordinates
{title2=$\nabla\Phi=\sum_i h_i^{-1}\mathbf e_i\,\partial_{q_i}\Phi$}
For a regular chart of <orthogonal coordinates> $\mathbf r(q)$, define $h_i=|\partial_{q_i}\mathbf r|>0$ and $\mathbf e_i=h_i^{-1}\partial_{q_i}\mathbf r$. The tangent <unit vectors> form an <orthonormal basis>. Matching $d\Phi=\nabla\Phi\cdot d\mathbf r$ with $d\mathbf r=\sum_i h_i\mathbf e_i dq_i$ proves the formula. Dividing by a scale factor converts a derivative per coordinate increment into a derivative per physical distance in that direction.
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