curvilinear coordinates

A point can be represented as (x,y,z).Consider 3 independent,smooth and unique invertible(and hopefully orthogonal) functions: u1=f1(x,y,z), u2=f2(x,y,z),u3=f3(x,y,z).The intersection of constant u1,u2,u3 defines a point in a space. The coordinate (u1,u2,u3) is called curvilinear coordinates.Base on inversion, we have x=f1'(u1,u2,u3),y=f2'(u1,u2,u3),z=f3'(u1,u2,u3)

\begin{align*}
\pmb{r} & (u_1,u_2,u_3) =x(u_1,u_2,u_3)\hat{x}+y(u_1,u_2,u_3)\hat{y}+z(u_1,u_2,u_3)\hat{z}\\
d &\pmb{r}=\frac{\partial \pmb{r}}{\partial u_1}du_1+\frac{\partial \pmb{r}}{\partial u_2}du_2+\frac{\partial \pmb{r}}{\partial u_3}du_3\\
=& \pmb{a}_1'du_1+\pmb{a}_2'du_2+\pmb{a}_3'du_3\\
i&f ~ \pmb{a}_1' \perp \pmb{a}_2' \perp \pmb{a}_3', this ~ is ~ orthogonal ~ curvilinear ~ coordinates ~ system\\
d \pmb{r} & = h_1\pmb{a}_1du_1+h_2\pmb{a}_2du_2+h_3\pmb{a}_3du_3\\
w&here ~ \pmb{a}_1,\pmb{a}_2,\pmb{a}_3 ~ are ~ unit ~ vectors
\end{align*}

Similarly,for small surface and volume

\begin{align*}
d\pmb{S}_i &  =  d\pmb{s}_j \times d\pmb{s}_k \\
wh&ere ~ d\pmb{s}_j=h_jdu_j\pmb{a}_j,d\pmb{s}_k=h_kdu_k\pmb{a}_k,\pmb{a}_j \times \pmb{a}_k=\pmb{a}_i\\
dV  = & d\pmb{s}_i \cdot d\pmb{S}_i =(h_idu_i\pmb{a}_i)\cdot (h_jh_kdu_jdu_k\pmb{a}_i)\\
= &(h_1h_2h_3)du_1du_2du_3
\end{align*}

For Cartesian coordinate,

\begin{align*}
u_1 & =x,u_2=y,u_3=z \\
\pmb{a}_1 & =\hat{x},\pmb{a}_2=\hat{y},\pmb{a}_z=\hat{z}\\
d\pmb{r} & =d\pmb{s}=\hat{x}dx+\hat{y}dy+\hat{z}dz \\
h_1 & =h_2=h_3=1\\
d\pmb{S} & =\hat{x}dS_x+\hat{y}dS_y+\hat{z}dS_z=dydz\hat{x}+dxdz\hat{y}+dxdy\hat{z} \\
dV & =dxdydz
\end{align*}

For cylindrical coordinates,

\begin{align*}
u_1 & =\rho, u_2=\psi, u_3=z \\
\pmb{a}_1 & =\hat{u}_{\rho},\pmb{a}_2=\hat{u}_{\psi},\pmb{a}_3=\hat{u}_{z}\\
d\pmb{r} & =\hat{u}_{\rho}d\rho+\hat{u}_{\psi}\rho d\psi+\hat{u}_{\z}dz\\
h_1 &=1, h_2=\rho, h_3=1 \\
d\pmb{S} & =\rho d\psi dz\hat{u}_{\rho} +d\rho dz \hat{u}_{\psi}+\rho d\rho d\psi \hat{u}_{z}\\
dV & =\rho d\rho d\psi dz
\end{align*}

For spherical coordinates,

\begin{align*}
u_1 & =\rho, u_2=\psi, u_3=\theta \\
\pmb{a}_1 & =\hat{u}_{\rho},\pmb{a}_2=\hat{u}_{\psi},\pmb{a}_3=\hat{u}_{\theta}\\
d\pmb{r} & =\hat{u}_{\rho}d\rho+\hat{u}_{\psi}\rho sin\theta d\psi+\hat{u}_{\theta}\rho d\theta \\ 
h_1 & =1, h_2=\rho, h_3=\rho sin \theta \\
d\pmb{S} &=dS_r\hat{u}_r+dS_{psi}\hat{u}_\psi+dS_{\theta}\hat{u}_{\theta} \\
d\pmb{S} &=\rho^2sin\theta d\psi d \theta\hat{u}_r+\rho sin\theta d\rho d\theta \hat{u}_\psi+\rho d\rho d\psi \hat{u}_{\theta}\\
dV &=\rho^2sin\theta d\rho d\psi d\theta
\end{align*}

For gradient,

\begin{align*}
d\phi & (u_1,u_2,u_3)=\nabla \phi (u_1,u_2,u_3) \cdot d\pmb{s} \\
d\phi & =\frac{\partial \phi}{\partial u_1}du_1+\frac{\partial \phi}{\partial u_2}du_2+\frac{\partial \phi}{\partial u_3}du_3\\
d\pmb{s} & =h_1du_1\pmb{a}_1+h_2du_2\pmb{a}_2+h_3du_3\pmb{a}_3 ~~\Rightarrow \\
\nabla & \phi (u_1,u_2,u_3)=\frac{\partial \phi}{h_1\partial u_1}\pmb{a}_1+\frac{\partial \phi}{h_2\partial u_2}\pmb{a}_2+\frac{\partial \phi}{h_3\partial u_3}\pmb{a}_3
\end{align*}

For divergence,

\begin{align*}
\nabla \cdot \pmb{A}=\frac{1}{h_1h_2h_3}[\frac{\partial(A_1h_2h_3)}{\partial u_1}+\frac{\partial(A_2h_1h_3)}{\partial u_2}+\frac{\partial(A_3h_1h_2)}{\partial u_3}]
\end{align*}

For curl,

\begin{align*}
\nabla \times \pmb{A}=\frac{1}{h_1h_2h_3}
\begin{vmatrix}
h_1\pmb{a}_1 & h_2\pmb{a}_2 & h_3\pmb{a}_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{vmatrix}
\end{align*}

For laplacian,

\begin{align*}
\nabla ^2 \phi  & =\nabla \cdot (\nabla \phi)
=\frac{1}{h_1h_2h_3}[\frac{\partial(A_1h_2h_3)}{\partial u_1}+\frac{\partial(A_2h_1h_3)}{\partial u_2}+\frac{\partial(A_3h_1h_2)}{\partial u_3}] \\
wh & ere ~ A_1=\frac{\partial \phi}{h_1\partial u_1},A_2=\frac{\partial \phi}{h_2\partial u_2},A_3=\frac{\partial \phi}{h_3\partial u_3} \Rightarrow \\
\nabla ^2 \phi & =\frac{1}{h_1h_2h_3}[\frac{\partial}{\partial u_1}(\frac{h_2h_3}{h_1}\frac{\partial \phi}{\partial u_1})+\frac{\partial}{\partial u_2}(\frac{h_1h_3}{h_2}\frac{\partial \phi}{\partial u_2})+\frac{\partial}{\partial u_3}(\frac{h_1h_2}{h_2}\frac{\partial \phi}{\partial u_3})

\end{align*}