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Change of Variable in $\mathbb{R}^k$. Assume that $f: V \to U$ is a diffeomorphism of open sets in $\mathbb{R^k}$ and $a$ is an integrable function on $U$. Then $$\int_U a dx_1 \cdots dx_k = \int_V (a \circ f) \left|\det(df)\right|dy_1 \cdots dy_k.$$

Is $dx_1 \cdots dx_k$ is a short hand for $dx_1 \wedge \cdots \wedge dx_k$? I just don't see a reason why the author suddenly decide to abbreviate it.

Thank you.

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    $\begingroup$ In tha case of $\mathbb{R}$ or $\mathbb{R}^n$ we usually do not use wedge between $dx_i$'s. $\endgroup$ – tessellation Aug 3 '13 at 5:17
  • $\begingroup$ I think you can say yes, in that you can consider the wedge $dx\wedge dy$ as a small rectangle, as it is done in the definition of en.wikipedia.org/wiki/Wedge_product , and this is what you're doing when integrating in $R^n$ , of course with the n-th-dimensional version of a rectangle. $\endgroup$ – DBFdalwayse Aug 3 '13 at 5:27
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No, $dx_1\dots dx_k$ here is not a short hand for $dx_1\wedge \dots \wedge dx_k$. It is a longhand for the $k$-dimensional Lebesgue measure.

There are two similar kinds of integrals in $\mathbb R^k$:

  1. Integrals of functions with respect to the Lebesgue measure.
  2. Integrals of $k$-forms over $k$-manifold $\mathbb R^k$.

The first one uses the measure space structure. The second uses the manifold structure.

Since a $k$-form $\omega$ is visibly associated with a function via $\omega=f\,dx_1\wedge\dots\wedge dx_k$, integrals 1 and 2 can look alike. The difference between them transpires in the change of variables formula.

For integrals of type 1 it is $$\int_U a\, dx_1 \cdots dx_k = \int_V (a \circ f)\, \left|\det(df)\right|\,dy_1 \cdots dy_k.\tag1$$ Here $\left|\det(df)\right|$ arises as the Radon-Nikodym derivative of the pushforward of the measure $ dx_1 \cdots dx_k$ under $f^{-1}$.

For integrals of type 2 it is $$\int_U a\, dx_1 \wedge\cdots\wedge dx_k = \int_V (a \circ f)\, \det(df) \,dy_1\wedge \cdots\wedge dy_k.\tag2$$ Here $\det(df)$ arises from the computation of $f^*(a\, dx_1 \wedge\cdots\wedge dx_k)$, the pullback of $k$-form $dx_1 \wedge\cdots\wedge dx_k$ under $f$.

It is rare to see an honest explanation of the above difference in a calculus textbook. Usually, the authors find it convenient to use (2) in $\mathbb R^1$ and (1) in higher dimensions, with little to no explanation of the reason. Wikipedia follows suit.

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