Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates

If $\Phi$ is a scalar function and $\bold{A} = A_1\bold{e_1} + A_2\bold{e_2} + A_3\bold{e_3}$ a vector function of orthogonal curvilinear coordinates $u_1$ , $u_2$ , $u_3$ , we have the following results.

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1 1.
2 2.
3 3.
4 4.

1. $\displaystyle \nabla\Phi = grad\Phi = \frac{1}{h_1}\frac{\partial\Phi}{\partial u_1}\bold{e_1} + \frac{1}{h_2}\frac{\partial\Phi}{\partial u_2}\bold{e_2} + \frac{1}{h_3}\frac{\partial\Phi}{\partial u_3}\bold{e_3}$

2. $\displaystyle \nabla\cdot\bold{A} = div\bold{A} = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}(h_2h_3A_1) + \frac{\partial}{\partial u_2}(h_3h_1A_2) + \frac{\partial}{\partial u_3}(h_1h_2A_3) \right]$

3. $\displaystyle \nabla\times\bold{A} = curl\bold{A} = \frac{1}{h_1h_2h_3}\left| \begin{array}{ccc} h_1\bold{e_1} & h_2\bold{e_2} & h_3\bold{e_3} \\ \frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{array} \right|$

4. $\displaystyle \nabla^2\Phi = Laplacian\ of\ \Phi\\ = \frac{1}{h_1h_2h_3}\left[ \frac{\partial}{\partial u_1}\left( \frac{h_2h_3}{h_1}\frac{\partial\Phi}{\partial u_1} \right) + \frac{\partial}{\partial u_2}\left( \frac{h_3h_1}{h_2}\frac{\partial\Phi}{\partial u_2} \right) + \frac{\partial}{\partial u_3}\left( \frac{h_1h_2}{h_3}\frac{\partial\Phi}{\partial u_3} \right) \right]$

These reduce to the usual expressions in rectangular coordinates if we replace $(u_1, u_2, u_3)$ by $(x, y, z)$ , in which case $\bold{e_1}$ , $\bold{e_2}$ and $\bold{e_3}$ are replaced by $\bold{i}$ , $\bold{j}$ and $\bold{k}$ and $h_1 = h_2 = h_3 = 1$ .