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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##?
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##?