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Let $U$ be an bounded, open set in $\mathbb{R}^n$ and $\partial U$ be smooth, $u\in C^1 (\bar U)$ and $\nu=(\nu^1,\ldots,\nu^n)$ the outward pointing unit normal vector field. I'm trying to make sense of the integral $$\int_{\partial U}u\nu^idS$$for some fixed $i\in \{1,\ldots,n\}$.

Can some please tell me show me a concrete example (doesn't have to be complicated), so that I can be learning by doing ?

It would be nice if along the way it would also be explained

  • How is the product "$u\nu^i$" defined ?

  • What does integrating on $\partial U$ relative to $dS$ mean ? The explanation should be without the use of differential forms, just purely analytically, as I

As soon as I can I'm willing to offer lots of bounty for this, as soon as I can.

Siminore
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temo
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1 Answers1

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Let $U$ be the open unit disk in $\mathbf{R}^{2}$, so that the outward unit normal is $\nu(x, y) = (x, y)$, and $\partial U = S^{1}$ is the unit circle. Parametrize the boundary by $c(t) = (\cos t, \sin t)$ for $0 \leq t \leq 2\pi$. If $u$ is continuously-differentiable on the closed unit disk, your integrals are $$ \int_{S^{1}} ux\, dS = \int_{0}^{2\pi} u(\cos t, \sin t) \cos t\, dt,\qquad \int_{S^{1}} uy\, dS = \int_{0}^{2\pi} u(\cos t, \sin t) \sin t\, dt. $$

One way to interpret $\int_{\partial U} u\nu^{i}\, dS$ is to introduce the vector fields $F = ue_{1} = (u, 0)$ and $G = ue_{2} = (0, u)$. Since $F \cdot \nu = u\nu^{1}$ and $G \cdot \nu = u\nu^{2}$, the integrals are, respectively, the net fluxes of $F$ and $G$ across the unit circle.

Generally, in your set-up you might consider the $n$ vector fields $F_{i} = ue_{i}$, each everywhere parallel to the $i$th coordinate axis. The function product $u\nu^{i} = F_{i} \cdot \nu$ may be interpreted as the normal component of the fluid velocity at a point of $\partial U$, so the integrand $u\nu^{i}\, dS$ represents the flux through a small piece of the boundary, and the integral represents the net flux of $F_{i}$ across the boundary.

To compute such an integral, you'd typically parametrize the boundary of $U$, i.e., find a region $D$ in $\mathbf{R}^{n-1}$ and a continuously-differentiable function $\phi:D \to \partial U$ that is one-to-one (except possibly at boundary points of $D$) and onto.

To form the integrand, compute the $(n - 1)$ partial derivatives $\partial\phi/\partial x_{i}$ of $\phi$ (each partial derivative is a vector-valued function with values in $\mathbf{R}^{n}$), and form their "(generalized) cross product" $$ N = \frac{\partial\phi}{\partial x_{1}} \times \dots \times \frac{\partial\phi}{\partial x_{n-1}}. $$ (Assemble the partials into rows of a matrix, put the standard basis vectors in the last row, and take the determinant, see the Wikipedia formula.) The cross product is a vector-valued function on $D$ (with values in $\mathbf{R}^{n}$). The desired integral is $$ \int_{D} u\bigl(\phi(x)\bigr) N_{i}\, dx_{1} \dots dx_{n-1}, $$ an ordinary integral over a region in $\mathbf{R}^{n-1}$.

Andrew D. Hwang
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  • Such a great answer! Tomorrow, when I can award the bounty (I have to wait 48 hours after the question has been asked to award it), you will get it. – temo Apr 11 '15 at 10:27
  • @temo: Glad to be of help. But please don't feel there's a need for additional bounty; I answered with the understanding this was "an ordinary question". :) – Andrew D. Hwang Apr 11 '15 at 12:08
  • No, I keep my word, your answer really was so enlightening, that I can at least spare 50 rep points, I just have to wait 22 hours until the reward can be given. I can only hope that for some of my future questions someone like you (or you) is going to be among the answerers. Do you perhaps know of a text in a general integration theory (over manifolds in $\mathbb{R}^n$, though I'd be also happy with some special case of that - like treating only surfaces, i.e. only $(n-1)$-dimensional manifolds in $\mathbb{R}^n$) is presented, like you did, without appealing to differential forms ? – temo Apr 14 '15 at 14:53
  • @temo: Offhand I can't think of a reference for integrating over hypersurfaces. The material in my answer is "folklore", though if memory serves, Spivak's Calculus on Manifolds has an exercise on the generalized cross product, and it's entirely possible he treats integration over hypersurfaces (perhaps in one or more exercises). Incidentally, I found Spivak to be an especially clear introduction to forms. (Do work all the exercises!) Really, differential forms largely boil down to determinants and index combinatorics; one just has to get past the (substantial!) initial intimidation barrier. – Andrew D. Hwang Apr 14 '15 at 15:26
  • Here's your bounty, your answer was well worth it! If some differential-form free reference comes to mind, please let me know. Thank you. – temo Apr 15 '15 at 17:55