I have no education about manifolds, but have had real analysis and a lot of calculus. Basically, I am struggling with an n-dimensional form of Stokes Theorem (the Divergence Theorem?) for a scalar field.
Here's the problem:
We have a function, $F(x): \mathbb{R}^n \rightarrow \Bbb R$, over a compact domain in $\Bbb R^n$. The domain is the $n$-dimensional hypercube, $x_i \in [a,b]$ for $i = 1,\ldots,n$, intersected with the hyperplane defined by $\sum x_i = k$, where $k \in [na, nb]$ so this intersection is non-empty.
Edit: Hypercube (I earlier wrote simplex). I meant to say that I think the intersection is a simplex.
$F$ is differentiable as many times as we need, but is difficult to integrate analytically over all the dimensions ($n$ times), so I am attempting to calculate the surface integral over the boundary of this domain, to get $n-1$ dimensions instead of $n$. Ideally, I can come up with some further simplification/iterative process from there.
This is the best I have come up with for the $n$-dimensional integral (call it $H$):
$$H(x) = \int_{x_i \in [a,b], \, \sum x_i = k} F(x)\,\mathrm{d}A $$
In my particular application, the function $F$ is multiplicatively separable over the different dimensions, $F(x) = \prod_{i=1}^{n} G (x_i)$, with $G$ being the same function for each $x_i$. This allows us to simplify somewhat:
$$H(x) = \int_a^b G(x_n) \int_a^b G(x_n-1) \dots \int_a^b G(x_2) \cdot G(k – \sum_{i \ge 2} x_i)\, \mathrm{d}x_2 \dots\mathrm{d}x_n$$,
so long as we define $G(x) = 0$ for $x \notin [a,b]$. Otherwise, I really don't know how to deal with the hyperplane intersection part (i.e., the $k$ in the innermost integrand).
My main challenge is I am unsure how to parameterize the $n-1$ dimensional curve (surface) for the integral over the boundary, and how to choose the normal vector. Stokes theorem should apply because the boundary is piecewise-smooth, but I am having trouble describing that boundary in a meaningful way. Also, can I pick any normal vector I want? Tips/tricks here would be much appreciated.
Thanks!
Best Answer
I am not allowed to comment as a new user, so here's my response to Christian.
First, thank you for looking. Second, I have submitted an edit. I guess I meant that the domain was the hypercube intersected by a hyperplane, and I was guessing that the resulting figure is a simplex. I have taken some steps to visualize it and I see that polytope may be more appropriate (since it's more general, and I haven't seen proof that we have a simplex).
My goal here is to apply Stokes Theorem to go from volume to surface, and then apply again to go to line integrals tracing the 2-faces. Is there any way to say something about the shape of the 2-faces?
I have found a couple papers exploring this phenomenon, e.g., this ArXiv paper.
...but I am having trouble identifying the shapes of the 2-faces of this domain's boundary.