Definition 5.1.3 (Volume of a $k$-parallelogram in $\R^n$). Let $T = [v_1, \dots, v_k]$ be a $n\times k$ real matrix. Then the k-dimensional volume of $P(v_1, \dots, v_k)$ is $\sqrt{\det (T^TT)}$.

Definition 5.2.1 ($k$-dimensional volume 0 of a subset of $\R^n$).

1. A bounded subset $X \subset \R^n$ has a $k$-dimensional volume 0 if

$$ \lim_{N\to \infty}\sum_{C \in \mathcal{D}_N(\R^n), C\cap X\neq \emptyset} \Big(\frac{1}{2^N}\Big)^k = 0. $$

2. An arbitrary subset $X \subset \R^n$ has $k$-dimensional volume 0 if for all $R$, the bounded set $X\cap B_R(0)$ has $k$-dimensional volume 0.

Proposition 5.2.2 ($k$-dimensional volume 0 of a manifold). If integers $m,k,n$ satisfy $0\leq m < k \leq n$ and $M\subset \R^n$ is a manifold of dimension $m$, any closed subset $X\subset M$ has $k$-dimensional volume 0.

Definition 5.2.3 (Relaxed parametrization of a manifold). Let $M\subset \R^n$ be a $k$-dimensional manifold and let $U\subset \R^k$ be a subet with boudnary of $k$-dimensional volume 0. Let $X \subset U$ be such that $U-X$ is open. Then a continuous mapping $\gamma: U\to \R^n$ parametrizes $M$ if

1. $\gamma(U) \supset M$;
2. $\gamma(U - X) \subset M$;
3. $\gamma : (U- X) \to M$ is one to one, of class $C^1$;
4. the derivative $[D\gamma(u)]$ is one to one for all $u$ in $U-X$
5. $X$ has k-dimensional volume 0, as does $\gamma (X) \cap C$ for any compact subset $C\subset M$.

Theorem 5.2.6 (Existence of parametrizations). All manifolds can be parameterized.

The caveat here is that a manifold does not need to be globally parametrized. It only needs to be locally parametrized.

Change of parametrization

Using the change of variables formula for Lebesgue integrals, we can set up a change of variables for mappings. Suppose we have a $k$-dimensional manifold $M$ and two parametrizations $\gamma_1: \bar U_1 \to M$ and $\gamma_2 : \bar U_2 \to M$ where $U_1$ and $U_2$ are subsets of $\R^k$. Our candidate for the change of variables mapping is $\Phi = \gamma_2^{-1} \circ \gamma_1$, $\bar U_1 \to M \to \bar U_2$.