Verifying the flux transport theorem

[Zack K]

Homework Statement

Let $S_t$ be a uniformly expanding hemisphere described by $x^2+y^2+z^2=(vt)^2, (z\ge0)$
Verify the flux transport theorem in this case

Relevant Equations

flux transport theorem: $$\frac{d\phi}{dt} =\int\int_{S_t}\left(\frac{\partial \textbf{F}}{\partial t} + (\nabla \cdot\textbf{F})\textbf{v} + \nabla \times(\textbf{F}\times\textbf{v})\right)\cdot d\textbf{S}$$ where $\textbf{F}(\textbf{R},t)=\textbf{R}t=(xt,yt,zt)$ (I think?)

Let $S_t$ be a uniformly expanding hemisphere described by $x^2+y^2+z^2=(vt)^2, (z\ge0)$

I assume by verify they just want me to calculate this for the surface. I guess that $\textbf{v}=(x/t,y/t,z/t)$ because $v=\frac{\sqrt{x^2+y^2+z^2}}{t}$. The three terms in the parentheses evaluate quite nicely. We end up getting $$2\int\int_{S_t}(x,y,z)\cdot d\textbf{S}$$ Here is where I don't understand. So I am thinking to change to spherical coordinates, but I do not know what the $d\textbf{S}$ vector looks like in spherical coordinates. I also don't know what to integrate over. Don't we have 3 bounds that are changing? $\theta, \phi$ and $r$? And what would the upper bound of r even look like? Would it be $rt$?

 

[archaic]

https://en.wikipedia.org/wiki/Spher..._and_differentiation_in_spherical_coordinates

 

[archaic]

https://en.wikipedia.org/wiki/Spher..._and_differentiation_in_spherical_coordinates

You can actually figure the surface element intuitively. Imagine $\theta$ changing by $d\theta$, that gives you $dl_1=rd\theta$, then imagine $\varphi$ changing by $d\varphi$, that gives you $dl_2=(r\sin\theta)d\varphi$. We then have $|d\vec S|=dl_1\times dl_2$.
Thinking infinitesimally, the lengths are too small to be considered arcs, so we think of them as straight lines.

 

[Zack K]

I should have clarified. I know $dS= r^2sin\theta d\theta d\phi$. I want to be able to "interpret" it as $d\textbf S=(dr e_r,rd\theta e_\theta,r\sin\theta d\phi e_\phi)$, or some equivalent form (I know that is not how you interpret it as it is not a Cartesian vector), so that I can evaluate $(x,y,z)\cdot d\textbf{S}$

 

[archaic]

I should have clarified. I know $dS= r^2sin\theta d\theta d\phi$. I want to be able to "interpret" it as $d\textbf S=(dr e_r,rd\theta e_\theta,r\sin\theta d\phi e_\phi)$, or some equivalent form (I know that is not how you interpret it as it is not a Cartesian vector), so that I can evaluate $(x,y,z)\cdot d\textbf{S}$

If you find a way to parametrize your surface with $\vec r(u,\,v)$, then you'll have:
$$d\vec S=\left(\frac{\partial\vec r}{\partial u}\times\frac{\partial\vec r}{\partial v}\right)du\,dv$$
https://en.wikipedia.org/wiki/Surface_integral#Surface_integrals_of_vector_fields