Gradient of a Function

Introduction

In this post, I find an expression for the gradient of a function, in terms of spherical coordinates. This is a continuation of previous posts, such as this one. This post has a lot of symbols, but there is a lot of repetition.

Formalism

Recall that

$ \vec{e}_r = \frac{\partial x(r)}{\partial r}\big|_{r^+} \vec{e}_x + \frac{\partial y(r)}{\partial r}\big|_{r^+} \vec{e}_y + \frac{\partial z(r)}{\partial r}\big|_{r^+} \vec{e}_z $

$ \vec{e}_{\theta} = \frac{1}{r} \frac{\partial x(\theta)}{\partial\theta} \big|_{\theta^+} \vec{e}_x + \frac{1}{r} \frac{\partial y(\theta)}{\partial \theta} \big|_{\theta^+}  \vec{e}_y  + \frac{1}{r} \frac{\partial z(\theta)}{\partial\theta} \big|_{\theta^+}  \vec{e}_z $

$\vec{e}_{\phi} = \frac{1}{r \sin\theta} \frac{\partial x(\phi)}{\partial\phi}\big|_{\phi^+}  \vec{e}_x + \frac{1}{r \sin\theta} \frac{\partial y(\phi)}{\partial\phi}\big|_{\phi^+}  \vec{e}_{y}$.

Multiply the first equation by $\partial r|_{r^+}$, multiply the second equation by $r \partial\theta |_{\theta^+}$, and multiply the third equation by $ r\sin\theta \partial\phi |_{\phi^+}$.

$ \partial r|_{r^+} \vec{e}_r = {\partial x(r)} |_{r^+} \vec{e}_x + {\partial y(r)} |_{r^+} \vec{e}_y + {\partial z(r)} |_{r^+} \vec{e}_z $

$r \partial\theta |_{\theta^+} \vec{e}_{\theta} =  {\partial x(\theta)} |_{\theta^+} \vec{e}_x +  {\partial y(\theta)} |_{\theta^+}  \vec{e}_y  + {\partial z(\theta)} |_{\theta^+}  \vec{e}_z $

$r\sin\theta \partial\phi |_{\phi^+} \vec{e}_{\phi} = {\partial x(\phi)} |_{\phi^+}  \vec{e}_x +  {\partial y(\phi)} |_{\phi^+}  \vec{e}_{y}$.

Next, multiply the first equation by $\frac{1}{\partial f|_{r^+}}$, multiply the second equation by $\frac{1}{\partial f |_{\theta^+}}$, and multiply the third equation by $\frac{1}{\partial f|_{\phi^+}}$. Here, $f$ is a function. I use $df$ instead of just $d$ because $df$ is a quantity that can be simply treated as a factor whereas $d$ is an operator.

$\frac{\partial f}{ \partial r }\big|_{r^+} \vec{e}_r  = \frac{\partial f}{\partial x(r)} \big|_{r^+} \vec{e}_x + \frac{\partial f}{\partial y(r)} \big|_{r^+} \vec{e}_y + \frac{\partial f}{\partial z(r)} \big|_{r^+} \vec{e}_z $

$\frac{\partial f}{r \partial \theta }\big|_{\theta^+} \vec{e}_{\theta} = \frac{\partial f}{\partial x(\theta)} \big|_{\theta^+} \vec{e}_x + \frac{\partial f}{\partial y(\theta)} \big|_{\theta^+}  \vec{e}_y  + \frac{\partial f}{\partial z(\theta)} \big|_{\theta^+}  \vec{e}_z $

$\frac{\partial f}{r\sin\theta \partial \phi } \big|_{\phi^+} \vec{e}_{\phi} = \frac{\partial f}{\partial x(\phi)} \big|_{\phi^+}  \vec{e}_x + \frac{\partial f}{\partial y(\phi)} \big |_{\phi^+}  \vec{e}_{y}$.

Simply the left hand sides.

$\frac{\partial f}{ \partial r }\big|_{r^+} \vec{e}_r  = \frac{\partial f}{\partial x(r)} \big|_{r^+} \vec{e}_x + \frac{\partial f}{\partial y(r)} \big|_{r^+} \vec{e}_y + \frac{\partial f}{\partial z(r)} \big|_{r^+} \vec{e}_z $

$\frac{1}{r}\frac{\partial f}{ \partial \theta }\big|_{\theta^+} \vec{e}_{\theta} = \frac{\partial f}{\partial x(\theta)} \big|_{\theta^+} \vec{e}_x + \frac{\partial f}{\partial y(\theta)} \big|_{\theta^+}  \vec{e}_y  + \frac{\partial f}{\partial z(\theta)} \big|_{\theta^+}  \vec{e}_z $

$\frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi } \big|_{\phi^+} \vec{e}_{\phi} = \frac{\partial f}{\partial x(\phi)} \big|_{\phi^+}  \vec{e}_x + \frac{\partial f}{\partial y(\phi)} \big |_{\phi^+}  \vec{e}_{y}$.

On the right sides, the 1st column comprises the variation of $f$ with respect to $x$ in the $\vec{e}_x$-direction, the 2nd column comprises the variation of $f$ with respect to $y$ in the $\vec{e}_y$-direction, and the third column comprises the variation of $f$ with respect to $z$ in the $\vec{e}_z$-direction. Adding all the terms yields

$\frac{\partial f}{ \partial r }\big|_{r^+} \vec{e}_r + \frac{1}{r}\frac{\partial f}{ \partial \theta }\big|_{\theta^+} \vec{e}_{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi } \big|_{\phi^+} \vec{e}_{\phi} = \frac{\partial f}{\partial x}  \vec{e}_x + \frac{\partial f}{\partial y}  \vec{e}_y + \frac{\partial f}{\partial z} \vec{e}_z$.

I am not sure whether $\frac{\partial f}{\partial x}$ should be written as $\frac{\partial f}{\partial x} \big|_{x}$ , $\frac{\partial f}{\partial x} \big|_{x^+}$, or $\frac{\partial f}{\partial x} \big|_{x^-}$, so I leave this point ambiguous. Since

$\vec{\nabla} f \equiv \frac{\partial f}{\partial x}  \vec{e}_x + \frac{\partial f}{\partial y} \vec{e}_y + \frac{\partial f}{\partial z} \vec{e}_z$,

one has

$ \boxed{ \vec{\nabla} f \equiv \vec{e}_r \frac{\partial f}{ \partial r }\big|_{r^+}  + \vec{e}_{\theta} \frac{1}{r}\frac{\partial f}{ \partial \theta }\big|_{\theta^+}  + \vec{e}_{\phi} \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi } \big|_{\phi^+}  } $.

This is the gradient of a function, in terms of spherical coordinates.

References

[1] https://en.wikipedia.org/wiki/Gradient

[2] http://www.thphys.nuim.ie/Notes/MP469/Laplace.pdf

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Christina Daniel

I am a research assistant in theoretical physics. This website, derive-it.com, serves to organize my ongoing learning and research as well as to provide a resource to other learners around the world.

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