 Gradient in orthogonal coordinates

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 $d\mathbf{r}={\sum }_i{h}_i{\mathbf{e}}_{i}d{q}_i$ proves the formula. Dividing by a scale factor converts a derivative per coordinate increment into a derivative per physical distance in that direction.