If $Ω$ is any open bounded domain in $R^n$ , we then have the identity $\int_Ω f (x)dx_1 ∧ . . . ∧ dx_n = \int_Ω f (x) dx$
where on the left we have an integral of a differential form (with $Ω$ viewed as a positively oriented n-dimensional manifold), and on the right we have the Riemann or Lebesgue integral of $f$ on $Ω$.
From Wikipedia (basically same as in baby Rudin):
Let
$\omega=\sum a_{i_1,\dots,i_k}({\mathbf x})\,dx^{i_1} \wedge \cdots \wedge dx^{i_k} $
be a differential form and $S$ a differentiable $k$-manifold over which we wish to integrate, where $S$ has the parameterization
$S({\mathbf u})=(x^1({\mathbf u}),\dots,x^n({\mathbf u}))$
for $u$ in the parameter domain $D$. Then (Rudin 1976) defines the integral of the differential form over $S$ as
$\int_S \omega =\int_D \sum a_{i_1,\dots,i_k}(S({\mathbf u})) \frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}\,du^1\ldots du^k$
where
$\frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}$
is the determinant of the Jacobian.
I wonder if in the case of Wikipedia, the change of variable can be eliminated just as in Terence Tao's, for example,
\begin{align} \int_S \omega &=\int_D \sum a_{i_1,\dots,i_k}(S({\mathbf u})) \frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}\,du^1\ldots du^k \\ &=\int_S \sum a_{i_1,\dots,i_k}(x) \,dx^{i_1}\ldots dx^{i_k} ? \end{align}
If not, when can it be?
- If the manifold $S$ is not a subset of $R^n$, the Jacobian will not make sense. Can $\int_S \omega $ still be represented by Riemann/Lebesgue integral? How is that like if yes?
Thanks and regards!