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## Note on ﬁnding the Laplacian in Polar, Cylinderical and Spherical coordinates using Tensor calculus

April 10, 2019   Compiled on January 30, 2024 at 5:58pm

### 1 Introduction

I wrote this note to help me learn tensors. The goal is to derive the Laplacian $$\nabla ^{2}$$ using tensor calculus for 2D Polar, 3D Cylindrical and in 3D Spherical coordinates.

The Laplacian in Cartesian coordinates is given by $$\nabla ^{2}=\frac {\partial ^{2}}{\partial x^{2}}+\frac {\partial ^{2}}{\partial y^{2}}+\frac {\partial ^{2}}{\partial z^{2}}$$. The following diagram shows $$\nabla ^{2}u$$ in Polar, Cylindrical and Spherical coordinates

The Laplacian operator in any orthogonal coordinate system is given by \begin {equation} \nabla ^{2}=\frac {1}{\sqrt {\det \left ( g\right ) }}\frac {\partial }{\partial x_{i}}\left ( \frac {\sqrt {\det \left ( g\right ) }}{g_{ii}}\frac {\partial }{\partial x^{i}}\right ) \tag {1} \end {equation}

Where $$g$$ is the metric tensor and $$\left \vert g\right \vert$$ is the determinant of $$g$$.  The derivation of the above is not shown here. References contains the derivation of the above. Only the use of (1) is shown here.

### 2 2D Polar

The coordinates in the Cartesian system are $$\zeta ^{1}=x,\zeta ^{2}=y$$ and the coordinates in the other system (Polar) are $$x^{1}=r,x^{2}=\theta$$. The relation between these must be known and invertible also, meaning $$\zeta \equiv \zeta \left ( x\right )$$ and $$x\equiv x\left ( \zeta \right )$$. This relation can be found from geometry as\begin {align*} \zeta ^{1} & =r\cos \theta \\ \zeta ^{2} & =r\sin \theta \end {align*}

The ﬁrst step is to determine the metric tensor $$g$$ for the Polar coordinates. This is given by$g_{kl}=\delta _{ij}\frac {\partial \zeta ^{i}}{\partial x^{k}}\frac {\partial \zeta ^{j}}{\partial x^{l}}$ The  above using Einstein summation notation. Since the coordinate system is orthogonal, $$g_{kl}$$ will be diagonal, hence only $$g_{11},g_{22}$$ are non zero. This is not the case for all coordinates systems. For general curvilinear coordinates system, $$g$$ can contain all components. But for the coordinates systems used here, $$g$$ will always be diagonal.\begin {align*} g_{11} & =\frac {\partial \zeta ^{1}}{\partial x^{1}}\frac {\partial \zeta ^{1}}{\partial x^{1}}+\frac {\partial \zeta ^{2}}{\partial x^{1}}\frac {\partial \zeta ^{2}}{\partial x^{1}}\\ & =\frac {\partial \zeta ^{1}}{\partial r}\frac {\partial \zeta ^{1}}{\partial r}+\frac {\partial \zeta ^{2}}{\partial r}\frac {\partial \zeta ^{2}}{\partial r}\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial r}\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial r}\right ) ^{2}\\ & =\cos ^{2}\theta +\sin ^{2}\theta \\ & =1 \end {align*}

And\begin {align*} g_{22} & =\frac {\partial \zeta ^{1}}{\partial x^{2}}\frac {\partial \zeta ^{1}}{\partial x^{2}}+\frac {\partial \zeta ^{2}}{\partial x^{2}}\frac {\partial \zeta ^{2}}{\partial x^{2}}\\ & =\frac {\partial \zeta ^{1}}{\partial \theta }\frac {\partial \zeta ^{1}}{\partial \theta }+\frac {\partial \zeta ^{2}}{\partial \theta }\frac {\partial \zeta ^{2}}{\partial \theta }\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial \theta }\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial \theta }\right ) ^{2}\\ & =\left ( -r\sin \theta \right ) ^{2}+\left ( r\cos \theta \right ) ^{2}\\ & =r^{2} \end {align*}

Hence $$ds^{2}$$ in polar coordinates is\begin {align*} ds^{2} & =g_{kl}dx^{k}dx^{l}\\ & =g_{11}\left ( dx^{1}\right ) ^{2}+g_{22}\left ( dx^{2}\right ) ^{2}\\ & =g_{11}\left ( dr\right ) ^{2}+g_{22}\left ( d\theta \right ) ^{2}\\ & =\left ( dr\right ) ^{2}+r^{2}\left ( d\theta \right ) ^{2} \end {align*}

From the above we see that$g=\begin {pmatrix} g_{11} & g_{12}\\ g_{21} & g_{22}\end {pmatrix} =\begin {pmatrix} 1 & 0\\ 0 & r^{2}\end {pmatrix}$ Hence $$\det \left ( g\right ) =r^{2}$$. We are now ready to apply (1)\begin {align*} \nabla ^{2} & =\frac {1}{\sqrt {\det \left ( g\right ) }}\frac {\partial }{\partial x_{i}}\left ( \frac {\sqrt {\det \left ( g\right ) }}{g_{ii}}\frac {\partial }{\partial x^{i}}\right ) \\ & =\frac {1}{\sqrt {r^{2}}}\frac {\partial }{\partial x_{1}}\left ( \frac {\sqrt {r^{2}}}{g_{11}}\frac {\partial }{\partial x^{1}}\right ) +\frac {1}{\sqrt {r^{2}}}\frac {\partial }{\partial x_{2}}\left ( \frac {\sqrt {r^{2}}}{g_{22}}\frac {\partial }{\partial x^{2}}\right ) \\ & =\frac {1}{r}\frac {\partial }{\partial r}\left ( r\frac {\partial }{\partial r}\right ) +\frac {1}{r}\frac {\partial }{\partial \theta }\left ( \frac {r}{r^{2}}\frac {\partial }{\partial \theta }\right ) \\ & =\frac {1}{r}\frac {\partial }{\partial r}\left ( r\frac {\partial }{\partial r}\right ) +\frac {1}{r}\frac {\partial }{\partial \theta }\left ( \frac {1}{r}\frac {\partial }{\partial \theta }\right ) \\ & =\frac {1}{r}\left ( \frac {\partial }{\partial r}+r\frac {\partial ^{2}}{\partial r^{2}}\right ) +\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \theta ^{2}}\\ & =\frac {\partial ^{2}}{\partial r^{2}}+\frac {1}{r}\frac {\partial }{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \theta ^{2}} \end {align*}

Therefore\begin {align*} \nabla ^{2}u & =\frac {\partial ^{2}u}{\partial r^{2}}+\frac {1}{r}\frac {\partial u}{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}u}{\partial \theta ^{2}}\\ & =u_{rr}+\frac {1}{r}u_{r}+\frac {1}{r^{2}}u_{\theta \theta } \end {align*}

### 3 3D Spherical

The coordinates in the Cartesian system are $$\zeta ^{1}=x,\zeta ^{2}=y,\zeta ^{3}=z$$. And the coordinates in the Spherical system are $$x^{1}=\phi ,x^{2}=r,x^{3}=\theta$$. The relation between these is known as (Note that the following depends on convention used for which is $$\theta$$ and which is $$\phi$$. Physics convention as shown in the diagram above is used here).\begin {align*} \zeta ^{1} & =r\sin \theta \cos \phi \\ \zeta ^{2} & =r\sin \theta \sin \phi \\ \zeta ^{3} & =r\cos \theta \end {align*}

The ﬁrst step is to determine the metric tensor $$g$$ for the Spherical coordinates. This is given by$g_{kl}=\delta _{ij}\frac {\partial \zeta ^{i}}{\partial x^{k}}\frac {\partial \zeta ^{j}}{\partial x^{l}}$ Since the coordinate system are orthogonal, $$g_{kl}$$ will be diagonal. Hence only $$g_{11},g_{22},g_{33}$$ are non zero.\begin {align*} g_{11} & =\frac {\partial \zeta ^{1}}{\partial x^{1}}\frac {\partial \zeta ^{1}}{\partial x^{1}}+\frac {\partial \zeta ^{2}}{\partial x^{1}}\frac {\partial \zeta ^{2}}{\partial x^{1}}+\frac {\partial \zeta ^{3}}{\partial x^{1}}\frac {\partial \zeta ^{3}}{\partial x^{1}}\\ & =\frac {\partial \zeta ^{1}}{\partial \phi }\frac {\partial \zeta ^{1}}{\partial \phi }+\frac {\partial \zeta ^{2}}{\partial \phi }\frac {\partial \zeta ^{2}}{\partial \phi }+\frac {\partial \zeta ^{3}}{\partial \phi }\frac {\partial \zeta ^{3}}{\partial \phi }\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial \phi }\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial \phi }\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial \phi }\right ) ^{2}\\ & =\left ( -r\sin \theta \sin \phi \right ) ^{2}+\left ( r\sin \theta \cos \phi \right ) ^{2}+\left ( 0\right ) ^{2}\\ & =r^{2}\sin ^{2}\theta \sin ^{2}\phi +r^{2}\sin ^{2}\theta \cos ^{2}\phi \\ & =r^{2}\sin ^{2}\theta \left ( \sin ^{2}\phi +\cos ^{2}\phi \right ) \\ & =r^{2}\sin ^{2}\theta \end {align*}

And\begin {align*} g_{22} & =\frac {\partial \zeta ^{1}}{\partial x^{2}}\frac {\partial \zeta ^{1}}{\partial x^{2}}+\frac {\partial \zeta ^{2}}{\partial x^{2}}\frac {\partial \zeta ^{2}}{\partial x^{2}}+\frac {\partial \zeta ^{3}}{\partial x^{2}}\frac {\partial \zeta ^{3}}{\partial x^{2}}\\ & =\frac {\partial \zeta ^{1}}{\partial r}\frac {\partial \zeta ^{1}}{\partial r}+\frac {\partial \zeta ^{2}}{\partial r}\frac {\partial \zeta ^{2}}{\partial r}+\frac {\partial \zeta ^{3}}{\partial r}\frac {\partial \zeta ^{3}}{\partial r}\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial r}\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial r}\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial r}\right ) ^{2}\\ & =\left ( \sin \theta \cos \phi \right ) ^{2}+\left ( \sin \theta \sin \phi \right ) ^{2}+\left ( \cos \theta \right ) ^{2}\\ & =\sin ^{2}\theta \cos ^{2}\phi +\sin ^{2}\theta \sin ^{2}\phi +\cos ^{2}\theta \\ & =\sin ^{2}\theta \left ( \cos ^{2}\phi +\sin ^{2}\phi \right ) +\cos ^{2}\theta \\ & =\sin ^{2}\theta +\cos ^{2}\theta \\ & =1 \end {align*}

And\begin {align*} g_{33} & =\frac {\partial \zeta ^{1}}{\partial x^{3}}\frac {\partial \zeta ^{1}}{\partial x^{3}}+\frac {\partial \zeta ^{2}}{\partial x^{3}}\frac {\partial \zeta ^{2}}{\partial x^{3}}+\frac {\partial \zeta ^{3}}{\partial x^{3}}\frac {\partial \zeta ^{3}}{\partial x^{3}}\\ & =\frac {\partial \zeta ^{1}}{\partial \theta }\frac {\partial \zeta ^{1}}{\partial \theta }+\frac {\partial \zeta ^{2}}{\partial \theta }\frac {\partial \zeta ^{2}}{\partial \theta }+\frac {\partial \zeta ^{3}}{\partial \theta }\frac {\partial \zeta ^{3}}{\partial \theta }\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial \theta }\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial \theta }\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial \theta }\right ) ^{2}\\ & =\left ( r\cos \theta \cos \phi \right ) ^{2}+\left ( r\cos \theta \sin \phi \right ) ^{2}+\left ( -r\sin \theta \right ) ^{2}\\ & =r^{2}\cos ^{2}\theta \left ( \cos ^{2}\phi +\sin ^{2}\phi \right ) +r^{2}\sin ^{2}\theta \\ & =r^{2}\cos ^{2}\theta +r^{2}\sin ^{2}\theta \\ & =r^{2} \end {align*}

Hence $$ds^{2}$$ in Spherical coordinates is\begin {align*} ds^{2} & =g_{kl}dx^{k}dx^{l}\\ & =g_{11}\left ( dx^{1}\right ) ^{2}+g_{22}\left ( dx^{2}\right ) ^{2}+g_{33}\left ( dx^{3}\right ) ^{2}\\ & =g_{11}\left ( d\phi \right ) ^{2}+g_{22}\left ( dr\right ) ^{2}+g_{33}\left ( d\theta \right ) ^{2}\\ & =r^{2}\sin ^{2}\theta \left ( d\phi \right ) ^{2}+\left ( dr\right ) ^{2}+r^{2}\left ( d\theta \right ) ^{2} \end {align*}

From the above we see that$g=\begin {pmatrix} g_{11} & g_{12} & g_{13}\\ g_{21} & g_{22} & g_{23}\\ g_{31} & g_{32} & g_{33}\end {pmatrix} =\begin {pmatrix} r^{2}\sin ^{2}\theta & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & r^{2}\end {pmatrix}$ Hence $$\det \left ( g\right ) =r^{4}\sin ^{2}\theta$$. We are now ready to apply (1)\begin {align*} \nabla ^{2} & =\frac {1}{\sqrt {\det \left ( g\right ) }}\frac {\partial }{\partial x_{i}}\left ( \frac {\sqrt {\det \left ( g\right ) }}{g_{ii}}\frac {\partial }{\partial x^{i}}\right ) \\ & =\frac {1}{\sqrt {r^{4}\sin ^{2}\theta }}\frac {\partial }{\partial x_{1}}\left ( \frac {\sqrt {r^{4}\sin ^{2}\theta }}{g_{11}}\frac {\partial }{\partial x^{1}}\right ) +\frac {1}{\sqrt {r^{4}\sin ^{2}\theta }}\frac {\partial }{\partial x_{2}}\left ( \frac {\sqrt {r^{4}\sin ^{2}\theta }}{g_{22}}\frac {\partial }{\partial x^{2}}\right ) +\frac {1}{\sqrt {r^{4}\sin ^{2}\theta }}\frac {\partial }{\partial x_{3}}\left ( \frac {\sqrt {r^{4}\sin ^{2}\theta }}{g_{33}}\frac {\partial }{\partial x^{3}}\right ) \\ & =\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial \phi }\left ( \frac {r^{2}\sin \theta }{r^{2}\sin ^{2}\theta }\frac {\partial }{\partial \phi }\right ) +\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial r}\left ( \frac {r^{2}\sin \theta }{1}\frac {\partial }{\partial r}\right ) +\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial \theta }\left ( \frac {r^{2}\sin \theta }{r^{2}}\frac {\partial }{\partial \theta }\right ) \\ & =\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial \phi }\left ( \frac {1}{\sin \theta }\frac {\partial }{\partial \phi }\right ) +\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial r}\left ( r^{2}\sin \theta \frac {\partial }{\partial r}\right ) +\frac {1}{r^{2}\sin \theta }\frac {\partial }{\partial \theta }\left ( \sin \theta \frac {\partial }{\partial \theta }\right ) \\ & =\frac {1}{r^{2}\sin ^{2}\theta }\frac {\partial ^{2}}{\partial \phi ^{2}}+\frac {1}{r^{2}}\left ( 2r\frac {\partial }{\partial r}+r^{2}\frac {\partial ^{2}}{\partial r^{2}}\right ) +\frac {1}{r^{2}\sin \theta }\left ( \cos \theta \frac {\partial }{\partial \theta }+\sin \theta \frac {\partial ^{2}}{\partial \theta ^{2}}\right ) \\ & =\frac {1}{r^{2}\sin ^{2}\theta }\frac {\partial ^{2}}{\partial \phi ^{2}}+\frac {2}{r}\frac {\partial }{\partial r}+\frac {\partial ^{2}}{\partial r^{2}}+\frac {\cos \theta }{r^{2}\sin \theta }\frac {\partial }{\partial \theta }+\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \theta ^{2}}\\ & =\frac {\partial ^{2}}{\partial r^{2}}+\frac {2}{r}\frac {\partial }{\partial r}+\frac {1}{r^{2}}\left ( \frac {\cos \theta }{\sin \theta }\frac {\partial }{\partial \theta }+\frac {\partial ^{2}}{\partial \theta ^{2}}\right ) +\frac {1}{r^{2}\sin ^{2}\theta }\frac {\partial ^{2}}{\partial \phi ^{2}} \end {align*}

Therefore\begin {align*} \nabla ^{2}u & =\frac {\partial ^{2}u}{\partial r^{2}}+\frac {2}{r}\frac {\partial u}{\partial r}+\frac {1}{r^{2}}\left ( \frac {\cos \theta }{\sin \theta }\frac {\partial u}{\partial \theta }+\frac {\partial ^{2}u}{\partial \theta ^{2}}\right ) +\frac {1}{r^{2}\sin ^{2}\theta }\frac {\partial ^{2}u}{\partial \phi ^{2}}\\ & =u_{rr}+\frac {2}{r}u_{r}+\frac {1}{r^{2}}\left ( \frac {\cos \theta }{\sin \theta }u_{\theta }+u_{\theta \theta }\right ) +\frac {1}{r^{2}\sin ^{2}\theta }u_{\phi \phi } \end {align*}

### 4 3D Cylindrical

The coordinates in the Cartesian system are $$\zeta ^{1}=x,\zeta ^{2}=y,\zeta ^{3}=z$$. And the coordinates in the Cylindrical system are $$x^{1}=\phi ,x^{2}=r,x^{3}=z$$. The relation between these is known as \begin {align*} \zeta ^{1} & =r\cos \phi \\ \zeta ^{2} & =r\sin \phi \\ \zeta ^{3} & =z \end {align*}

The ﬁrst step is to determine the metric tensor $$g$$ for the Spherical coordinates. This is given by$g_{kl}=\delta _{ij}\frac {\partial \zeta ^{i}}{\partial x^{k}}\frac {\partial \zeta ^{j}}{\partial x^{l}}$ Since the coordinate system are orthogonal, $$g_{kl}$$ will be diagonal. Hence only $$g_{11},g_{22},g_{33}$$ are non zero.\begin {align*} g_{11} & =\frac {\partial \zeta ^{1}}{\partial x^{1}}\frac {\partial \zeta ^{1}}{\partial x^{1}}+\frac {\partial \zeta ^{2}}{\partial x^{1}}\frac {\partial \zeta ^{2}}{\partial x^{1}}+\frac {\partial \zeta ^{3}}{\partial x^{1}}\frac {\partial \zeta ^{3}}{\partial x^{1}}\\ & =\frac {\partial \zeta ^{1}}{\partial \phi }\frac {\partial \zeta ^{1}}{\partial \phi }+\frac {\partial \zeta ^{2}}{\partial \phi }\frac {\partial \zeta ^{2}}{\partial \phi }+\frac {\partial \zeta ^{3}}{\partial \phi }\frac {\partial \zeta ^{3}}{\partial \phi }\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial \phi }\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial \phi }\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial \phi }\right ) ^{2}\\ & =\left ( -r\sin \phi \right ) ^{2}+\left ( r\cos \phi \right ) ^{2}+\left ( 0\right ) ^{2}\\ & =r^{2} \end {align*}

And\begin {align*} g_{22} & =\frac {\partial \zeta ^{1}}{\partial x^{2}}\frac {\partial \zeta ^{1}}{\partial x^{2}}+\frac {\partial \zeta ^{2}}{\partial x^{2}}\frac {\partial \zeta ^{2}}{\partial x^{2}}+\frac {\partial \zeta ^{3}}{\partial x^{2}}\frac {\partial \zeta ^{3}}{\partial x^{2}}\\ & =\frac {\partial \zeta ^{1}}{\partial r}\frac {\partial \zeta ^{1}}{\partial r}+\frac {\partial \zeta ^{2}}{\partial r}\frac {\partial \zeta ^{2}}{\partial r}+\frac {\partial \zeta ^{3}}{\partial r}\frac {\partial \zeta ^{3}}{\partial r}\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial r}\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial r}\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial r}\right ) ^{2}\\ & =\left ( \cos \phi \right ) ^{2}+\left ( \sin \phi \right ) ^{2}+\left ( 1\right ) ^{2}\\ & =1 \end {align*}

And\begin {align*} g_{33} & =\frac {\partial \zeta ^{1}}{\partial x^{3}}\frac {\partial \zeta ^{1}}{\partial x^{3}}+\frac {\partial \zeta ^{2}}{\partial x^{3}}\frac {\partial \zeta ^{2}}{\partial x^{3}}+\frac {\partial \zeta ^{3}}{\partial x^{3}}\frac {\partial \zeta ^{3}}{\partial x^{3}}\\ & =\frac {\partial \zeta ^{1}}{\partial z}\frac {\partial \zeta ^{1}}{\partial z}+\frac {\partial \zeta ^{2}}{\partial z}\frac {\partial \zeta ^{2}}{\partial z}+\frac {\partial \zeta ^{3}}{\partial z}\frac {\partial \zeta ^{3}}{\partial z}\\ & =\left ( \frac {\partial \zeta ^{1}}{\partial z}\right ) ^{2}+\left ( \frac {\partial \zeta ^{2}}{\partial z}\right ) ^{2}+\left ( \frac {\partial \zeta ^{3}}{\partial z}\right ) ^{2}\\ & =\left ( 0\right ) ^{2}+\left ( 0\right ) ^{2}+\left ( 1\right ) ^{2}\\ & =1 \end {align*}

Hence $$ds^{2}$$ in Cylindrical coordinates is\begin {align*} ds^{2} & =g_{kl}dx^{k}dx^{l}\\ & =g_{11}\left ( dx^{1}\right ) ^{2}+g_{22}\left ( dx^{2}\right ) ^{2}+g_{33}\left ( dx^{3}\right ) ^{2}\\ & =g_{11}\left ( d\phi \right ) ^{2}+g_{22}\left ( dr\right ) ^{2}+g_{33}\left ( dz\right ) ^{2}\\ & =r^{2}\left ( d\phi \right ) ^{2}+\left ( dr\right ) ^{2}+\left ( dz\right ) ^{2} \end {align*}

From the above we see that$g=\begin {pmatrix} g_{11} & g_{12} & g_{13}\\ g_{21} & g_{22} & g_{23}\\ g_{31} & g_{32} & g_{33}\end {pmatrix} =\begin {pmatrix} r^{2} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end {pmatrix}$ Hence $$\det \left ( g\right ) =r^{2}$$. We are now ready to apply (1)\begin {align*} \nabla ^{2} & =\frac {1}{\sqrt {\det \left ( g\right ) }}\frac {\partial }{\partial x_{i}}\left ( \frac {\sqrt {\det \left ( g\right ) }}{g_{ii}}\frac {\partial }{\partial x^{i}}\right ) \\ & =\frac {1}{\sqrt {r^{2}}}\frac {\partial }{\partial x_{1}}\left ( \frac {\sqrt {r^{2}}}{g_{11}}\frac {\partial }{\partial x^{1}}\right ) +\frac {1}{\sqrt {r^{2}}}\frac {\partial }{\partial x_{2}}\left ( \frac {\sqrt {r^{2}}}{g_{22}}\frac {\partial }{\partial x^{2}}\right ) +\frac {1}{\sqrt {r^{2}}}\frac {\partial }{\partial x_{3}}\left ( \frac {\sqrt {r^{2}}}{g_{33}}\frac {\partial }{\partial x^{3}}\right ) \\ & =\frac {1}{r}\frac {\partial }{\partial \phi }\left ( \frac {1}{r}\frac {\partial }{\partial \phi }\right ) +\frac {1}{r}\frac {\partial }{\partial r}\left ( r\frac {\partial }{\partial r}\right ) +\frac {1}{r}\frac {\partial }{\partial z}\left ( r\frac {\partial }{\partial z}\right ) \\ & =\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \phi ^{2}}+\frac {1}{r}\frac {\partial }{\partial r}\left ( r\frac {\partial }{\partial r}\right ) +\frac {\partial ^{2}}{\partial z^{2}}\\ & =\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \phi ^{2}}+\frac {1}{r}\frac {\partial }{\partial r}+\frac {\partial ^{2}}{\partial r^{2}}+\frac {\partial ^{2}}{\partial z^{2}} \end {align*}

Therefore\begin {align*} \nabla ^{2}u & =\frac {\partial ^{2}u}{\partial r^{2}}+\frac {1}{r}\frac {\partial u}{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}u}{\partial \phi ^{2}}+\frac {\partial ^{2}u}{\partial z^{2}}\\ & =u_{rr}+\frac {1}{r}u_{r}+\frac {1}{r^{2}}u_{\phi \phi }+u_{zz} \end {align*}

#### 4.1 References

1. Lecture notes, Physics 5041. UMN Spring 2019 by Professor Kapusta
2. Appendix A, Einstein’s Theory, A rigorous introduction. By Gron and Naess. Springer publisher.