Vtyjkyuk

I'm at a PDE class and my teacher gave a very generic definition of the divergence theorem. I can't find it anywhere. It's something like this:

Definition: let $kin 1,2,cdots,infty$, $Nge 2$

An open $OmegainmathbbR^N$ has border of class $C^k$ if:

$Omega$ is bounded and given any $x_0inpartialOmega$, there is an
open $U_0$ containing $x_0$ and a diffeomorphism $psi:U_0to Q$,
where $Q = ]-1,1[^N$ such that

1. $psi(Omegacap U_0) = tin Q: t_N>0$




2. $psi(x_0) = 0$




3. $psi(partial Omegacap U_0) = tin Q: t_N = 0$

Now he wrote these things:

$X(t) = psi^-1(t_1,cdots,t_N-1, 0)$

$X:t_jto U_0cappartialOmega$

$$vectheta(t') = left(fracpartial(x_2,cdots,x_N)partial(t_1,cdots,t_N-1)(t'),cdots,fracpartial(x_1,cdots,x_N)partial(t_1,cdots,t_N-1)(t')right)$$

then $vectheta(t')$ is normal to $partialOmegacap U_0$ at $X(t')$

What are those derivatives? I don't recognize this notation and I don't know how it can be normal to anything. And why he picked the inverse image of $psi$?

The surface element $dsigma$ of $partialOmega$ is represented in $U_0cap partial Omega$ by

$$|vectheta(t')|dt_1cdots dt_N-1$$

finally we can define the exterior unitary normal in $partial Omega$:

$$vecn:partialOmegatomathbbR^N$$

$$|vecn(p)| = 1, forall pin partial Omega$$

Divergence Theorem

Let $Omega$ be an open with boundary class $C^k$ and $vecX(x) =
(X_1(x), cdots, X_N(x))$ with $X_jin C^1(overlineOmega),
j=1,cdots, N$ then

$$int_Omega(mboxdiv vecX)(x)dx =
int_partialOmegavecX(y)cdotvecn(y)dsigma(y)$$

This seems to be a very generic view of the divergence theorem. And it seems that I need to understand what is an open set with a border and what does it mean for it to have class $C^k$. I also need to understand the surface element $dsigma$.

Is there a place where I can learn about those things but not too deep? They're just tools to solve PDEs

Also, what does $C^n(Omega)$ means?

Also, the integral on $Omega$ is supposed to be on an open of $mathbbR^n$, but $div$ applied to $x$ is a real number, and $dx$ looks like a one dimensional thing.

UPDATE:

I'm downloading some books on vector calculus but I only find ones with a lot of wedge products and differential forms. I needed only a vector explanation of all this. Can someone point me a gook book?

This is just the usual divergence theorem written in an explicit coordinate notation. Any introductory book on multivariate calculus treats it.

First, $C^k(Omega)$ is the space of $C^k$ functions on the set $Omega$. The integral is an $N$-dimensional integral, where $dx$ is an $N$-dimensional volume element.

Here $psi$ is a map that transforms a piece of the domain near its boundary into a domain with flat boundary (in this case the latter is just a half cube). Think of a bite out of an apple, and turning that small portion of the apple into one with flat boundary. This is called flattening out of the boundary. Boundary smoothness is defined by how smooth you can choose $psi$ to be.

Then $X(t)$ is simply a parametrization of a boundary piece, where $tinmathbbR^N-1$. Since $psi$ maps the boundary piece into a domain in $mathbbR^N-1$, its inverse gives you a parametrization of that piece.

The derivative notation means the determinant of the matrix formed by the indicated partial derivatives. Let $phi=psi^-1$. Then $X$ is simply the restriction of $phi$ on $t_N=0$. The Jacobian determinant of $phi$ is
$$
J = fracpartial(x_1,ldots,x_N)partial(t_1,ldots,t_N)=beginvmatrixfracpartial x_1partial t_1&fracpartial x_1partial t_2&cdots&fracpartial x_1partial t_N\fracpartial x_2partial t_1&fracpartial x_2partial t_2&cdots&fracpartial x_2partial t_N\vdots&vdots&ddots&vdots\fracpartial x_Npartial t_1&fracpartial x_Npartial t_2&cdots&fracpartial x_Npartial t_Nendvmatrix.
$$
The vector $theta$ is defined by the expansion coefficients of this determinant with respect to the last column (This is why I think there should be alternating signs in the formula of $theta$. Just think of a surface in 3D space, where we use cross product). Note that the first $N-1$ columns of this matrix constitutes a basis of tangent vectors to the boundary. So taking an inner product of $theta$ with any tangent vector $V$ would be equivalent to computing the above determinant with the last column replaced by $V$. Since $V$ is spanned by the first $N-1$ columns, the determinant will be $0$, meaning that $theta$ is orthogonal to the tangent plane.

Let me elaborate on the tangent basis bit. Fix all $t_2,ldots,t_N$ and look at $phi(t_1,t_2,ldots,t_N)$ as a function of $t_1$. It would draw a curve along the boundary of $Omega$. If you take the derivative of this function, you would get a vector tangent to the boundary:
$$
V_1 = big(fracpartial x_1partial t_1,fracpartial x_2partial t_1,ldots,fracpartial x_Npartial t_1big).
$$
This is simply the first column of the Jacobian matrix of $phi$. By the same reasoning, we see that the first $N-1$ columns of the Jacobian matrix of $phi$ are vectors tangent to the boundary surface. Also, as $phi$ is a diffeomorphism, its Jacobian matrix is invertible, and hence those $N-1$ vectors are linearly independent. This shows that they form a basis of the tangent space.