I'm reading a paper on tension in a membrane and am currently stuck at this part.
The paper so far reads: We consider a portion $S^M$ of a membrane $\Omega^M$, where $\hat k$ denotes the unit vector pointing out of $S^M$ which is normal to $\partial S^M$ and $\hat n$ denoting the unit normal to $\Omega^M$.
Now the paper states that, using the divergence theorem,
$$\int_{\partial S^M} \hat k\,ds=\int_{S^M} (-\nabla\cdot \hat n)\,\hat n\,dS$$
The proof in the paper uses the idea of local parameters, which I am unable to understand (despite reading the outstanding answer here.
Is there:
- Any proof that does not require applying the idea of local parameters, or
- Any text that I should read first to better understand the abstract algebra behind the idea of local parameters?
As a side note, the paper also states that $-\nabla\cdot \hat n$ is equal to the mean curvature of the membrane. The wikipedia link suggests that it in fact is equal to twice the mean curvature. Which is correct?
I don't know the answer to your main question. I do know the answer to your side note. Depending on the community/journal where the paper was published, the definition of mean curvature may be $H = k_1 + k_2$ or $2H = k_1 + k_2$, where $k_1$ and $k_2$ are the two principal curvatures in 3D. The latter is technically the correct one, the former is used for convenience (often in the CFD world).
Source: my own experience with these differing definitions.