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Prove the integral $\int\int\vec{F}\cdot \hat{n}\,dS = [\vec{F}(x_0)\cdot \hat{n}(x_0)] \,A(s)$ where A is the surface area and $F$ is continuous vector field.

I have been trying to prove this problem but I really don't know how to prove it. Anyone knows how to prove this mean value theorem proof?

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  • $\begingroup$ You need some assumption on you surface $s$... this is not true in general. $\endgroup$ – Emanuele Paolini Mar 3 '13 at 19:12
  • $\begingroup$ sorry, it's connected. Can you show me how to prove this? $\endgroup$ – user61676 Mar 3 '13 at 19:14
  • $\begingroup$ connected is not enough... $\endgroup$ – Emanuele Paolini Mar 3 '13 at 19:15
  • $\begingroup$ maybe $S$ is a closed surface? $\endgroup$ – Emanuele Paolini Mar 3 '13 at 19:15
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It is enough that $S$ is compact smooth and connected.

Define $g(x) = F(x)\cdot n(x)$ for $x\in S$. Let $x_0$ and $x_1$ be the points on $S$ where $g$ has, respectively, the minimum and maximum value. Hence $$ g(x_0) \le \frac{\iint_S g(s)\, ds }{A(S)} \le g(x_1). $$ Then take a curve $\gamma$ joining $x_0$ and $x_1$ on $S$ and notice that $g(\gamma(t))$ is continuous and hence assumes all intermediate values between $g(x_0)$ and $g(x_1)$.

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  • $\begingroup$ how can you prove g is continuous? $\endgroup$ – user61676 Mar 3 '13 at 19:36
  • $\begingroup$ because it is the composition of continuous functions: $\gamma$, $F$ and $n$ $\endgroup$ – Emanuele Paolini Mar 4 '13 at 6:51
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I believe the assumptions should be that the surface $S$ must be connected, compact and Jordan-measurable. With these assumptions you can do as Emanuele did and define a continuous function $g$ such that $m$ and $M$ are the minimum and maximum values of $g$ on S (this is by the extreme-value theorem). Then use the intermediate-value theorem (since $S$ is connected and $g$ is continuous).

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