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Problem

Let us assume surface $ \Sigma$ as a domain, with $\partial \Sigma$ as a moving boundary. $\Sigma$ is a portion of a sphere and $\partial\Sigma$ is a circle and its direction of movement is perpendicular to the plane that the boundary is lying on (See figure 1).

Now calculate $$\frac{\partial}{\partial t} \int_{\Sigma} f(\theta,t) \ d\Sigma$$ where $f(\theta,t)$ is a function defined on the surface $\Sigma$ . Please provide the general form and specific solution.

Work

I've tried to use Leibniz rule. Here is the general form of Leibniz's rule:

$${\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma (t)}\mathbf {F} (\mathbf {r} ,t)\cdot \mathrm {d} \mathbf {A} =\iint _{\Sigma (t)}\left(\mathbf {F} _{t}(\mathbf {r} ,t)+\left[\mathrm {\nabla } \cdot \mathbf {F} (\mathbf {r} ,t)\right]\mathbf {v} \right)\cdot \mathrm {d} \mathbf {A} \,-\,\oint _{\partial \Sigma (t)}\left[\mathbf {v} \times \mathbf {F} (\mathbf {r} ,t)\right]\cdot \mathrm {d} \mathbf {s} $$

I'm not sure whether it can be used in this situation or not. Since in the above equation v is the velocity of the domain (I think) but in this question our domain is stationary and only the boundary is moving.

Figure 1

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    You have that $$\frac{\partial}{\partial t} \int_{\Sigma} f(\theta,t) \ d\Sigma= \int_{\Sigma}\frac{\partial f}{\partial t}\ d\Sigma+\int_{\Sigma}f\frac{\partial\Sigma}{\partial t}.$$ You have to determine $\dfrac{\partial\Sigma}{\partial t}$. – 2017-01-09
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    could you, please provide more details of your solution based on explained problem or even simple 1D Leibniz's rule – 2017-01-09

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