A matrix calculus approach to changing variables in Laplace’s equation

laplaceIn three-dimensional rectangular coordinates, the partial differential equation known as Laplace’s equation takes the form

\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2} = 0

This equation is applicable to a wide range of problems in physics but it is often necessary to make a change of variables from rectangular to spherical polar coordinates in order to better match the spherical symmetry of particular contexts. Until now I had only ever worked out this change of variables using a scale factor method based on the formula for the Laplacian of a function F in general orthogonal curvilinear coordinates:

\nabla^2 F = \frac{1}{h_1 h_2 h_3}\bigg[ \frac{\partial }{\partial x_1}\big( \frac{h_2 h_3}{h_1}\frac{\partial F}{\partial x_1} \big) + \frac{\partial }{\partial x_2}\big( \frac{h_1 h_3}{h_2}\frac{\partial F}{\partial x_2} \big) + \frac{\partial }{\partial x_3}\big( \frac{h_1 h_2}{h_3}\frac{\partial F}{\partial x_3} \big) \bigg]

This formula is derived using basic results from vector calculus and allows Laplace’s equation to be easily expressed in any orthogonal coordinate system once the form of the Euclidean metric is known in that coordinate system. (Orthogonal coordinate systems are those for which the coordinate basis vectors are always orthogonal at each point). The Euclidean metric in general orthogonal coordinates x_1, x_2, x_3 will take the form

ds^2 = h_1^2 dx_1^2 + h_2^2 dx_2^2 + h_3^2 dx_3^2

The coefficients h_1, h_2 and h_3 are the scale factors appearing in the Laplacian formula. In the metric, these convert the coordinate differentials dx_i into lengths h_i dx_i.

In the case of rectangular coordinates we have x_1 = x, x_2 = y, x_3 = z and h_1 = h_2 = h_3 = 1 so the Euclidean metric and the Laplacian formula reduce to the familiar forms

ds^2 = dx^2 + dy^2 + dz^2

and

\nabla^2 F = \frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2}

The scale factors are all equal to 1 in this case since the coordinate differentials are already in units of length. In the case of spherical polar coordinates (an orthogonal coordinate system) we can find the scale factors and hence the Laplacian by using the standard conversion equations

x = r \sin \theta \cos \phi

y = r \sin \theta \sin \phi

z = r \cos \theta

Taking the differentials of these, squaring them and adding them we find that the Euclidean metric in spherical polar coordinates takes the form

ds^2 = dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta d \phi^2

The scale factors in this case are therefore h_1 = 1, h_2 = r and h_3 = r \sin \theta. Putting these into the Laplacian formula we immediately get

\nabla^2 F = \frac{1}{r^2 \sin \theta} \bigg[ \frac{\partial }{\partial r}\big( \frac{r^2 \sin \theta}{1}\frac{\partial F}{\partial r} \big) + \frac{\partial }{\partial \theta}\big( \frac{r \sin \theta}{r}\frac{\partial F}{\partial \theta} \big) + \frac{\partial }{\partial \phi}\big( \frac{r}{r \sin \theta}\frac{\partial F}{\partial \phi} \big) \bigg]

= \frac{1}{r^2} \frac{\partial }{\partial r}\big( r^2 \frac{\partial F}{\partial r}\big) + \frac{1}{r^2 \sin \theta}\frac{\partial }{\partial \theta}\big( \sin \theta \frac{\partial F}{\partial \theta} \big) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 F}{\partial \phi^2}

Therefore Laplace’s equation in spherical polar coordinates is

\frac{1}{r^2} \frac{\partial }{\partial r}\big( r^2 \frac{\partial F}{\partial r}\big) + \frac{1}{r^2 \sin \theta}\frac{\partial }{\partial \theta}\big( \sin \theta \frac{\partial F}{\partial \theta} \big) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 F}{\partial \phi^2} = 0

I recently had occasion to use Laplace’s equation in spherical polar coordinates again and this time I decided to have a go at justifying the change of variables to myself using a matrix calculus approach instead of the above scalar factor method. This turned out to be a laborious but interesting process. In the present note I want to record my calculations using the matrix calculus approach in detail as I have never seen something like this elsewhere. The advantage of the matrix calculus approach is that it works for any coordinate system irrespective of whether it is orthogonal or not.

In the matrix calculus approach we use the standard conversion equations for spherical polar coordinates

x = r \sin \theta \cos \phi

y = r \sin \theta \sin \phi

z = r \cos \theta

to work out the coefficient matrix in the system

\begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi}\end{bmatrix} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r}\\ \ \\ \frac{\partial x}{\partial \theta} & \frac{\partial y}{\partial \theta} & \frac{\partial z}{\partial \theta}\\ \ \\ \frac{\partial x}{\partial \phi} & \frac{\partial y}{\partial \phi} & \frac{\partial z}{\partial \phi}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial x}\\ \ \\ \frac{\partial F}{\partial y} \\ \ \\ \frac{\partial F}{\partial z} \end{bmatrix}

\big(which of course is equivalent to the set of equations

\frac{\partial F}{\partial r} = \frac{\partial F}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial r} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial r}

\frac{\partial F}{\partial \theta} = \frac{\partial F}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial \theta} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial \theta}

\frac{\partial F}{\partial \phi} = \frac{\partial F}{\partial x} \frac{\partial x}{\partial \phi} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial \phi} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial \phi} \big)

We get the system

\begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi}\end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta\\ \ \\ r \cos \theta \cos \phi & r \cos \theta \sin \phi & - r \sin \theta\\ \ \\ - r \sin \theta \sin \phi & r \sin \theta \cos \phi & 0\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial x}\\ \ \\ \frac{\partial F}{\partial y} \\ \ \\ \frac{\partial F}{\partial z} \end{bmatrix}

We can solve this system either by inverting the coefficient matrix or by using Cramer’s rule. Using Cramer’s rule we get

\frac{\partial F}{\partial x} = \frac{\begin{vmatrix} \frac{\partial F}{\partial r} & \sin \theta \sin \phi & \cos \theta\\ \ \\ \frac{\partial F}{\partial \theta} & r \cos \theta \sin \phi & - r \sin \theta\\ \ \\ \frac{\partial F}{\partial \phi} & r \sin \theta \cos \phi & 0\end{vmatrix}}{\begin{vmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta\\ \ \\ r \cos \theta \cos \phi & r \cos \theta \sin \phi & - r \sin \theta\\ \ \\ - r \sin \theta \sin \phi & r \sin \theta \cos \phi & 0\end{vmatrix}} = \frac{r^2 \sin^2 \theta \cos \phi \frac{\partial F}{\partial r} + r \cos \theta \sin \theta \cos \phi \frac{\partial F}{\partial \theta} - r \sin \phi \frac{\partial F}{\partial \phi} }{r^2 \sin \theta}
\
= \sin \theta \cos \phi \frac{\partial F}{\partial r} + \frac{\cos \theta \cos \phi}{r} \frac{\partial F}{\partial \theta} - \frac{\sin \phi}{r \sin \theta} \frac{\partial F}{\partial \phi}

and similarly

\frac{\partial F}{\partial y} = \frac{\begin{vmatrix} \sin \theta \cos \phi & \frac{\partial F}{\partial r} & \cos \theta\\ \ \\ r \cos \theta \cos \phi & \frac{\partial F}{\partial \theta} & - r \sin \theta\\ \ \\ - r \sin \theta \sin \phi & \frac{\partial F}{\partial \phi} & 0\end{vmatrix}}{\begin{vmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta\\ \ \\ r \cos \theta \cos \phi & r \cos \theta \sin \phi & - r \sin \theta\\ \ \\ - r \sin \theta \sin \phi & r \sin \theta \cos \phi & 0\end{vmatrix}} = \frac{r^2 \sin^2 \theta \sin \phi \frac{\partial F}{\partial r} + r \cos \theta \sin \theta \sin \phi \frac{\partial F}{\partial \theta} + r \cos \phi \frac{\partial F}{\partial \phi} }{r^2 \sin \theta}
\
= \sin \theta \sin \phi \frac{\partial F}{\partial r} + \frac{\cos \theta \sin \phi}{r} \frac{\partial F}{\partial \theta} + \frac{\cos \phi}{r \sin \theta} \frac{\partial F}{\partial \phi}

and

\frac{\partial F}{\partial z} = \frac{\begin{vmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \frac{\partial F}{\partial r}\\ \ \\ r \cos \theta \cos \phi & r \cos \theta \sin \phi & \frac{\partial F}{\partial \theta}\\ \ \\ - r \sin \theta \sin \phi & r \sin \theta \cos \phi & \frac{\partial F}{\partial \phi}\end{vmatrix}}{\begin{vmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta\\ \ \\ r \cos \theta \cos \phi & r \cos \theta \sin \phi & - r \sin \theta\\ \ \\ - r \sin \theta \sin \phi & r \sin \theta \cos \phi & 0\end{vmatrix}} = \frac{r^2 \cos \theta \sin \theta \frac{\partial F}{\partial r} - r \sin^2 \theta \frac{\partial F}{\partial \theta}}{r^2 \sin \theta}
\
= \cos \theta \frac{\partial F}{\partial r} - \frac{\sin \theta}{r} \frac{\partial F}{\partial \theta}

We can write these solutions more concisely in matrix form as

\begin{bmatrix} \frac{\partial F}{\partial x}\\ \ \\ \frac{\partial F}{\partial y} \\ \ \\ \frac{\partial F}{\partial z}\end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\\ \ \\ \sin \theta \sin \phi & \frac{\cos \theta \sin \theta}{r} & \frac{\cos \phi}{r \sin \theta}\\ \ \\ \cos \theta & -\frac{\sin \theta}{r} & 0\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

The coefficient matrix here is, of course, the inverse of the coefficient matrix in the original system.

To find the second-order partials \frac{\partial^2F}{\partial x^2}, \frac{\partial^2F}{\partial y^2} and \frac{\partial^2F}{\partial z^2}, let G \equiv \frac{\partial F}{\partial x}, H \equiv \frac{\partial F}{\partial y} and I \equiv \frac{\partial F}{\partial z}. Then we need to find \frac{\partial G}{x}, \frac{\partial H}{\partial y} and \frac{\partial I}{\partial z}. We can use the first, second and third equations respectively in the above matrix system to find these, with F replaced by G, H and I respectively. Thus,

\frac{\partial G}{\partial x} =  \begin{bmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial G}{\partial r}\\ \ \\ \frac{\partial G}{\partial \theta} \\ \ \\ \frac{\partial G}{\partial \phi} \end{bmatrix}

\frac{\partial H}{\partial y} =  \begin{bmatrix} \sin \theta \sin \phi & \frac{\cos \theta \sin \phi}{r} & \frac{\cos \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial H}{\partial r}\\ \ \\ \frac{\partial H}{\partial \theta} \\ \ \\ \frac{\partial H}{\partial \phi} \end{bmatrix}

\frac{\partial I}{\partial z} =  \begin{bmatrix} \cos \theta & -\frac{\sin \theta}{r} \end{bmatrix} \begin{bmatrix} \frac{\partial I}{\partial r}\\ \ \\ \frac{\partial I}{\partial \theta} \end{bmatrix}

But differentiating

G \equiv \begin{bmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

with respect to r, \theta and \phi we get

\begin{bmatrix} \frac{\partial G}{\partial r}\\ \ \\ \frac{\partial G}{\partial \theta} \\ \ \\ \frac{\partial G}{\partial \phi}\end{bmatrix} = \begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta} & \frac{\partial^2 F}{\partial r \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} & \frac{\partial^2 F}{\partial \theta \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \phi \partial r} & \frac{\partial^2 F}{\partial \phi \partial \theta} & \frac{\partial^2 F}{\partial \phi^2}\end{bmatrix} \begin{bmatrix} \sin \theta \cos \phi\\ \ \\ \frac{\cos \theta \cos \phi}{r} \\ \ \\ -\frac{\sin \phi}{r \sin \theta} \end{bmatrix}  + \begin{bmatrix} 0 & -\frac{\cos \theta \cos \phi}{r^2} & \frac{\sin \phi}{r^2 \sin \theta}\\ \ \\ \cos \theta \cos \phi & -\frac{\sin \theta \cos \phi}{r} & \frac{\sin \phi}{r \sin^2 \theta \cos \theta}\\ \ \\ -\sin \theta \sin \phi & -\frac{\cos \theta \sin \phi}{r} & -\frac{\cos \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

Similarly, differentiating

H \equiv \begin{bmatrix} \sin \theta \sin \phi & \frac{\cos \theta \sin \phi}{r} & \frac{\cos \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

with respect to r, \theta and \phi we get

\begin{bmatrix} \frac{\partial H}{\partial r}\\ \ \\ \frac{\partial H}{\partial \theta} \\ \ \\ \frac{\partial H}{\partial \phi}\end{bmatrix} = \begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta} & \frac{\partial^2 F}{\partial r \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} & \frac{\partial^2 F}{\partial \theta \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \phi \partial r} & \frac{\partial^2 F}{\partial \phi \partial \theta} & \frac{\partial^2 F}{\partial \phi^2}\end{bmatrix} \begin{bmatrix} \sin \theta \sin \phi\\ \ \\ \frac{\cos \theta \sin \phi}{r} \\ \ \\ \frac{\cos \phi}{r \sin \theta} \end{bmatrix}  + \begin{bmatrix} 0 & -\frac{\cos \theta \sin \phi}{r^2} & -\frac{\cos \phi}{r^2 \sin \theta}\\ \ \\ \cos \theta \sin \phi & -\frac{\sin \theta \sin \phi}{r} & -\frac{\cos \phi}{r \sin^2 \theta \cos \theta}\\ \ \\ \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

and differentiating

I \equiv \begin{bmatrix} \cos \theta & -\frac{\sin \theta}{r}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \end{bmatrix}

with respect to r and \theta we get

\begin{bmatrix} \frac{\partial I}{\partial r}\\ \ \\ \frac{\partial I}{\partial \theta} \end{bmatrix} = \begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} \end{bmatrix} \begin{bmatrix} \cos \theta \\ \ \\ -\frac{\sin \theta}{r} \end{bmatrix}  + \begin{bmatrix} 0 & \frac{\sin \theta}{r^2}\\ \ \\ -\sin \theta & -\frac{\cos \theta}{r}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \end{bmatrix}

Therefore we have

\frac{\partial G}{\partial x} =  \begin{bmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix}\begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta} & \frac{\partial^2 F}{\partial r \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} & \frac{\partial^2 F}{\partial \theta \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \phi \partial r} & \frac{\partial^2 F}{\partial \phi \partial \theta} & \frac{\partial^2 F}{\partial \phi^2}\end{bmatrix} \begin{bmatrix} \sin \theta \cos \phi\\ \ \\ \frac{\cos \theta \cos \phi}{r} \\ \ \\ -\frac{\sin \phi}{r \sin \theta} \end{bmatrix}

+ \begin{bmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix}\begin{bmatrix} 0 & -\frac{\cos \theta \cos \phi}{r^2} & \frac{\sin \phi}{r^2 \sin \theta}\\ \ \\ \cos \theta \cos \phi & -\frac{\sin \theta \cos \phi}{r} & \frac{\sin \phi}{r \sin^2 \theta \cos \theta}\\ \ \\ -\sin \theta \sin \phi & -\frac{\cos \theta \sin \phi}{r} & -\frac{\cos \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

and similarly

\frac{\partial H}{\partial y} =  \begin{bmatrix} \sin \theta \sin \phi & \frac{\cos \theta \sin \phi}{r} & \frac{\cos \phi}{r \sin \theta}\end{bmatrix}\begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta} & \frac{\partial^2 F}{\partial r \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} & \frac{\partial^2 F}{\partial \theta \partial \phi}\\ \ \\ \frac{\partial^2 F}{\partial \phi \partial r} & \frac{\partial^2 F}{\partial \phi \partial \theta} & \frac{\partial^2 F}{\partial \phi^2}\end{bmatrix} \begin{bmatrix} \sin \theta \sin \phi\\ \ \\ \frac{\cos \theta \sin \phi}{r} \\ \ \\ \frac{\cos \phi}{r \sin \theta} \end{bmatrix}

+ \begin{bmatrix} \sin \theta \sin \phi & \frac{\cos \theta \sin \phi}{r} & \frac{\cos \phi}{r \sin \theta}\end{bmatrix}\begin{bmatrix} 0 & -\frac{\cos \theta \sin \phi}{r^2} & -\frac{\cos \phi}{r^2 \sin \theta}\\ \ \\ \cos \theta \sin \phi & -\frac{\sin \theta \sin \phi}{r} & -\frac{\cos \phi}{r \sin^2 \theta \cos \theta}\\ \ \\ \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \\ \ \\ \frac{\partial F}{\partial \phi} \end{bmatrix}

and

\frac{\partial I}{\partial z} =  \begin{bmatrix} \cos \theta & -\frac{\sin \theta}{r} \end{bmatrix}\begin{bmatrix} \frac{\partial^2 F}{\partial r^2} & \frac{\partial^2 F}{\partial r \partial \theta}\\ \ \\ \frac{\partial^2 F}{\partial \theta \partial r} & \frac{\partial^2 F}{\partial \theta^2} \end{bmatrix} \begin{bmatrix} \cos \theta \\ \ \\ -\frac{\sin \theta}{r} \end{bmatrix}  + \begin{bmatrix} \cos \theta & -\frac{\sin \theta}{r} \end{bmatrix}\begin{bmatrix} 0 & \frac{\sin \theta}{r^2}\\ \ \\ -\sin \theta & -\frac{\cos \theta}{r}\end{bmatrix} \begin{bmatrix} \frac{\partial F}{\partial r}\\ \ \\ \frac{\partial F}{\partial \theta} \end{bmatrix}

Laplace’s equation is given by \frac{\partial G}{\partial x} + \frac{\partial H}{\partial y} + \frac{\partial I}{\partial z} = 0 so we need to work out the three matrix equations above and sum them. The result will be a sum involving the nine partials \frac{\partial^F}{\partial r^2}, \frac{\partial^F}{\partial \theta^2}, \frac{\partial^F}{\partial \phi^2}, \frac{\partial^F}{\partial r \partial \theta}, \frac{\partial^F}{\partial r \partial \phi}, \frac{\partial^F}{\partial \theta \phi}, \frac{\partial F}{\partial r}, \frac{\partial F}{\partial \theta} and \frac{\partial F}{\partial \phi}. I found that the most convenient way to work out the sum was by working out the coefficients of each of these nine partials individually in turn. The result is

\frac{\partial G}{\partial x} + \frac{\partial H}{\partial y} + \frac{\partial I}{\partial z} =

\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2} =

\big \{\sin^2 \theta \cos^2 \phi + \sin^2 \theta \sin^2 \phi + \cos^2 \theta \big \} \frac{\partial^2 F}{\partial r^2}

+ \big \{ \frac{\cos^2 \theta \cos^2 \phi}{r^2} + \frac{\cos^2 \theta \sin^2 \phi}{r^2} + \frac{\sin^2 \theta}{r^2} \big \} \frac{\partial^2 F}{\partial \theta^2}

+ \big \{ \frac{\sin^2 \phi}{r^2 \sin^2 \theta} + \frac{\cos^2 \phi}{r^2 \sin^2 \theta}\big \} \frac{\partial^2 F}{\partial \phi^2}

+ \big \{ \frac{\sin \theta \cos \theta \cos^2 \phi}{r} + \frac{\sin \theta \cos \theta \sin^2 \phi}{r} - \frac{\sin \theta \cos \theta}{r}\big \} \frac{\partial^2 F}{\partial r \partial \theta}

+ \big \{ -\frac{\cos \phi \sin \phi}{r} + \frac{\cos \phi \sin \phi}{r} \big \} \frac{\partial^2 F}{\partial r \partial \phi}

+ \big \{ -\frac{\cos \theta \cos \phi \sin \phi}{r^2 \sin \theta} -\frac{\cos \theta \cos \phi \sin \phi}{r^2 \sin \theta} + \frac{\cos \theta \cos \phi \sin \phi}{r^2 \sin \theta} + \frac{\cos \theta \cos \phi \sin \phi}{r^2 \sin \theta} \big \} \frac{\partial^2 F}{\partial \theta \partial \phi}

+ \big \{ \frac{\cos^2 \theta \cos^2 \phi}{r} + \frac{\sin^2 \phi}{r} + \frac{\cos^2 \theta \sin^2 \phi}{r} + \frac{\cos^2 \phi}{r} + \frac{\sin^2 \theta}{r}\big \} \frac{\partial F}{\partial r}

+ \big \{ -\frac{2\sin \theta \cos \theta \cos^2 \phi}{r^2} + \frac{\cos \theta \sin^2 \phi}{r^2 \sin \theta} -\frac{2\sin \theta \cos \theta \sin^2 \phi}{r^2} + \frac{\cos \theta \cos^2 \phi}{r^2 \sin \theta} + \frac{2\cos \theta \sin \theta}{r^2} \big \} \frac{\partial F}{\partial \theta}

+ \big \{ \frac{\cos \phi \sin \phi}{r^2} + \frac{2 \cos \phi \sin \phi}{r^2 \sin^2 \theta} - \frac{\cos \phi \sin \phi}{r^2} - \frac{2 \cos \phi \sin \phi}{r^2 \sin^2 \theta} \big \} \frac{\partial F}{\partial \phi}

which simplifies to

\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2} =

\frac{\partial^2 F}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 F}{\partial \theta^2} + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 F}{\partial \phi^2} + \frac{2}{r} \frac{\partial F}{\partial r} + \frac{\cos \theta}{r^2 \sin \theta} \frac{\partial F}{\partial \theta}

= \frac{1}{r^2} \frac{\partial }{\partial r}\big( r^2 \frac{\partial F}{\partial r}\big) + \frac{1}{r^2 \sin \theta}\frac{\partial }{\partial \theta}\big( \sin \theta \frac{\partial F}{\partial \theta} \big) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 F}{\partial \phi^2}

Therefore Laplace’s equation in spherical polar coordinates is

\frac{1}{r^2} \frac{\partial }{\partial r}\big( r^2 \frac{\partial F}{\partial r}\big) + \frac{1}{r^2 \sin \theta}\frac{\partial }{\partial \theta}\big( \sin \theta \frac{\partial F}{\partial \theta} \big) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 F}{\partial \phi^2} = 0

as before.