Curvilinear coordinates

4.6 Curvilinear coordinates

Spherical and cylindrical coordinates are examples of curvilinear coordinates \(u_1\), \(u_2\), and \(u_3\) for which at each point holds

\[ d \vec{r} = \sum_{i=1}^{3} a_i(u_1,u_2,u_3)\; \vec{e}_i(u_1,u_2,u_3)\; du_i \quad .\]

with \(\vec{e}_i \vec{e}_k= \delta_{i\,k}\), i.e. curvilinear coordinates form locally an orthogonal base. The base vectors have a length \(a_i(u_1, u_2, u_3)\) which depends on \(u_k\).
Consequently the Jacobi matrix and determinant and their inverses are

\[ J = \left( \begin{array}{c c c} \uparrow & \uparrow & \uparrow \\ a_1 \vec{e_{1}} & a_2 \vec{e_{2}} & a_3 \vec{e_{3}} \\ \downarrow & \downarrow & \downarrow \end{array}\right) \quad \mbox{det}(J)= a_1\,a_2\,a_3 \quad ; \quad J^{-1} = \left( \begin{array}{c c c} \leftarrow & \frac{\vec{e}_1}{a_1} & \rightarrow \\ \leftarrow & \frac{\vec{e}_2}{a_2} & \rightarrow \\ \leftarrow & \frac{\vec{e}_3}{a_3} & \rightarrow \end{array}\right) \quad \mbox{det}(J^{-1})=\frac{1}{a_1\,a_2\,a_3} \]

According to the chain rule the gradient in curvilinear coordinates can be written as

\[\mbox{grad} f = \vec{\nabla} f = \sum_{i=1}^{3} \frac{\partial f}{a_i\, \partial u_i} \vec{e}_i \quad .\]

It is hard work to find a general expressing for the Laplace operator in curvilinear coordinates. Still we will outline the prove since it summarizes nearly everything we learned about linear algebra and analysis.

\begin{equation*} \begin{split} \mbox{div grad} f = \Delta f & = \sum_{i,k=1}^{3} \frac{\left\langle e_k\right|}{a_k} \frac{\partial}{\partial u_k} \left|e_i \right\rangle \left(\frac{\partial f}{a_i \partial u_i}\right)\\ & = \sum_{i,k} \frac{\left\langle e_k\right|\left|e_i \right\rangle }{a_k} \frac{\partial}{\partial u_k} \left(\frac{\partial f}{a_i \partial u_i}\right) + \sum_{i,k} \frac{\left\langle e_k\right|}{a_k} \left(\frac{\partial f}{a_i \partial u_i}\right) \frac{\partial \left|e_i \right\rangle }{\partial u_k}\\ & = \sum_{i} \frac{1}{a_i} \frac{\partial}{\partial u_i} \left(\frac{\partial f}{a_i \partial u_i}\right) + \sum_{i,k} \left(\frac{\partial f}{a_i \partial u_i}\right) \frac{\left\langle a_k \,e_k\right|}{a_k^2 \,a_i} \frac{a_i \partial \left|e_i \right\rangle }{\partial u_k}\\ \end{split} \end{equation*}

The first sum is already finished. To simplify the second sum we calculate first

\begin{equation*} \begin{split} \left\langle a_k \,e_k\right| \frac{\partial}{\partial u_k} \left|a_i \, e_i \right\rangle & = \left\langle a_k \,e_k\right| \frac{a_i \partial \left|e_i \right\rangle }{\partial u_k} + \left\langle a_k \,e_k\right| \left|e_i \right\rangle \frac{\partial a_i}{\partial u_k}\\ \Rightarrow \quad \left\langle a_k \,e_k\right| \frac{a_i \partial \left|e_i \right\rangle }{\partial u_k} & = \left\langle a_k \,e_k\right| \frac{\partial}{\partial u_k} \left|a_i \, e_i \right\rangle - \delta_{i\,k} \,a_i \frac{\partial a_i}{\partial u_i}\\ \end{split} \end{equation*}

and secondly (since second order derivatives can be interchanged)

\begin{equation*} \begin{split} \left\langle a_k \,e_k\right| \frac{\partial}{\partial u_k} \left|a_i \, e_i \right\rangle & = \frac{\partial \left\langle r\right|}{\partial u_k} \frac{\partial}{\partial u_k} \frac{\partial \left|r\right\rangle }{\partial u_i} = \frac{\partial \left\langle r\right|}{\partial u_k} \frac{\partial}{\partial u_i} \frac{\partial \left|r\right\rangle }{\partial u_k} \\ & = \frac{1}{2} \frac{\partial}{\partial u_i} \frac{\partial \left\langle r\right|}{\partial u_k} \frac{\partial \left|r\right\rangle }{\partial u_k} = \frac{1}{2} \frac{\partial}{\partial u_i} a_k^2 = a_k\, \frac{\partial a_k}{\partial u_i}\\ \end{split} \end{equation*}

Combining all equations we finally get

\begin{equation*} \begin{split} \Delta f = \mbox{div grad} f & = \sum_{i} \frac{1}{a_i} \frac{\partial}{\partial u_i} \left(\frac{\partial f}{a_i \partial u_i}\right) + \sum_{i,k} \left(\frac{\partial f}{a_i \partial u_i}\right) \frac{1}{a_k \,a_i} \frac{\partial a_k}{\partial u_i} - \sum_{i} \left(\frac{\partial f}{a_i \partial u_i}\right) \frac{1}{a_i^2} \frac{\partial a_i}{\partial u_i}\\ & = \frac{1}{a_1\,a_2\,a_3} \left[ \frac{\partial}{\partial u_1} \left(\frac{a_2\,a_3}{a_1} \frac{\partial f}{\partial u_1}\right) + \frac{\partial}{\partial u_2} \left(\frac{a_1\,a_3}{a_2} \frac{\partial f}{\partial u_2}\right) + \frac{\partial}{\partial u_3} \left(\frac{a_1\,a_2}{a_3} \frac{\partial f}{\partial u_3}\right) \right] \end{split} \end{equation*}

thus e.g. for spherical coordinates we get

\[\Delta = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2\,\sin \vartheta} \frac{\partial}{\partial \vartheta} \left(\sin \vartheta \frac{\partial}{\partial \vartheta}\right) + \frac{1}{r^2\,\sin^2 \vartheta} \frac{\partial^2}{\partial \varphi^2} \]