Integrating $\iint \hat{n} \, dS $ over a closed surface?

Question

One of the exercises in the book Div, Grad, Curl, and All That is to show that $$ \iint_S \hat{n} \hspace{1mm} dS = 0$$ for every closed surface $S$, using the divergence theorem.

I know the theorem, namely that $$\iint_S {F \cdot \hat{n}} \hspace{1mm} dS = \iiint_V \nabla \cdot F \hspace{1mm} dV,$$

but I'm not sure how to proceed in this case. What confuses me is the fact that I'm not integrating a scalar function anymore but a vector function... And I have no idea what should I do with the right side of the equation in this case...

Any hints?

Hint: what is $\nabla \cdot \vec{k}$ when $\vec{k}$ is a constant vector? — achille hui, Aug 24 '13 at 12:24
Doesn't it matter that $\hat{n}$ isn't a constant vector but a vector function (unit normal vector) ? — lel, Aug 24 '13 at 12:27
Maybe I should ask what is $\nabla \cdot F$ when $F$ is a constant vector. — achille hui, Aug 24 '13 at 12:34
It sure is $0$. So, if $F(x,y,z)=constant$, we can write $F \cdot \hat{n}$ under the integral instead of $\hat{n}$ ?
If yes, why are we formally allowed to do this? — lel, Aug 24 '13 at 12:37
@SchlomoSteinbergerstein: the surface integral of the vector field is performed componentwise. What is the expression for the $x$ component of $\hat n$? — Anthony Carapetis, Aug 24 '13 at 12:46
Does this answer your question? Is this problem ill-defined? — amWhy, Apr 29 '21 at 18:45

score 3 · Accepted Answer · answered Aug 24 '13 at 12:58

3

Let $\vec{k}$ be the vector $\displaystyle \iint_S \hat{n}\,dS$, we have:

$$|\vec{k}|^2 = \vec{k} \cdot \left( \iint_S \hat{n}\,dS\right) = \iint_S \vec{k} \cdot \hat{n}\,dS = \iiint_V \nabla \cdot \vec{k}\,dV = \iiint_{V} 0\,dV = 0\\ \implies \iint_S \hat{n} dS = \vec{k} = \vec{0} $$

answered Aug 24 '13 at 12:58

achille hui

122,701

Excellent and clear, thank you!! :D – lel Aug 24 '13 at 13:01

Integrating $\iint \hat{n} \, dS $ over a closed surface?

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