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## Homework Statement

The formula for divergence in the spherical coordinate system can be defined as follows:

[itex]\nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial \phi} [/itex]

where [itex]\vec{f} = f_r \widehat{r} + f_θ \widehat{θ} + f_\phi \widehat{\phi}[/itex]

Derive the divergence formula for the spherical coordinate system.

## Homework Equations

One way to define divergence is as follows:

[itex]\nabla\bullet\vec{f} = \lim_{volume\rightarrow 0} \frac{flux \space through \space the \space volume}{volume}[/itex]

Volume in spherical coordinates can be defined as follows:

[itex] V = volume = r^2 sin(θ) Δθ Δ\phi Δr[/itex]

## The Attempt at a Solution

Just before you read into my solution, I do successfully derive the divergence formula. I am questioning if my methodology is correct though. Without further ado here is my attempted solution.

I first write the basic formula from part 2 down:

[itex]\nabla\bullet\vec{f} = \lim_{V\rightarrow 0}\frac{\hat{dS_r}\bullet\vec{f} + \hat{dS_θ} \bullet\vec{f} + \hat{dS_\phi} \bullet\vec{f}}{V}[/itex]

I fill in all the dS vectors:

[itex]= \lim_{V\rightarrow 0}\frac{\hat{r} r^2 sinθ Δθ Δ\phi \bullet\vec{f} + \hat{θ} r sinθ Δr Δ\phi \bullet\vec{f} + \hat{\phi} r Δr Δθ \bullet\vec{f}}{V}[/itex]

I apply all dot products:

[itex] = \lim_{V\rightarrow 0}\frac{r^2 sinθ Δθ Δ\phi f_r + r sinθ Δr Δ\phi f_θ + r Δr Δθ f_\phi}{V}[/itex]

I take the volume formula from part 2 and put it in the denominator:

[itex] = \lim_{V\rightarrow 0}\frac{r^2 sinθ Δθ Δ\phi f_r + r sinθ Δr Δ\phi f_θ + r Δr Δθ f_\phi}{r^2 sin(θ) Δθ Δ\phi Δr}[/itex]

I cancel like terms with each other:

[itex] = \lim_{V\rightarrow 0}\frac{1}{r^2 Δr} (r^2 f_r) + \frac{1}{r sinθ Δθ} (f_θ sinθ) + \frac{1}{r sinθ Δ\phi} (f_\phi)[/itex]

I apply the limit which makes any part of the volume formula become extremely small:

[itex] = \frac{1}{r^2}\frac{∂}{∂r} (r^2 f_r) + \frac{1}{r sinθ} \frac{∂}{∂θ} (f_θ sinθ) + \frac{1}{r sinθ} \frac{∂}{∂ \phi}(f_\phi)[/itex]

This solution I found definitely matches the formula for divergence is spherical coordinates. I guess my question is is my methodology for solving this problem ok??

Thanks,

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