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- TL;DR Summary
- Let ##A:=[a_1,b_1]\times\cdots\times[a_n,b_n]##. Let ##f:A\rightarrow\mathbb{R}##. Let ##g_i:[a_i,b_i]\rightarrow \mathbb{R}##. Assume that ##g_i## defined as ##g_i(x)=f(c_1,\ldots,c_{i-1},x,\ldots,c_n)## is integrable on its domain. Then we can define integration of ##f## as such:

Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##.

Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##.

The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each partition).

Let ##a_0## be a point in the partition.

Let ##a'## be the point in the partition s.t.:

1. Each component of ##a'## is greater than that of ##a_0##.

2. The distance from ##a'## and ##a_0## is the infimum of the set of distances from ##a_0## to another point in the partition whose components exceed those of ##a_0##

The vectors ##a_0## and ##a'## define hyper-rectangle that constitutes the volume element of the volume to be estimated.

Evaluate ##f## at some point ##x_0## in this hyper-rectangle in order to get a proper weight for this region of the domain.

In order to get the volume of that hyper-rectangle, we evaluate the volume element ##dx^1\wedge\ldots\wedge dx^n## at ##(a'-a_0,a'-a_0,\ldots,a'-a_0)##.

This process yields a number ##f(x_0) dx^1\wedge\ldots\wedge dx^n(a'-a_0,\ldots,a'-a_0)## where the latter term is the "volume" of the element of the partition on which ##f## is to be evaluated.

Anyway, I was trying to relate integration of forms to Riemann integration earlier. I was confused by the relationship between the notation ##\int_A f\,dx^1\wedge\,dx^2## and ##\int_A f(x)\,dx^1\,dx^2##, and then I tried to come up with my own explanation for why these two are equivalent, using my rather rusty knowledge of Riemann integration.

Forgive me if this is poorly-explained.

Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##.

The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each partition).

Let ##a_0## be a point in the partition.

Let ##a'## be the point in the partition s.t.:

1. Each component of ##a'## is greater than that of ##a_0##.

2. The distance from ##a'## and ##a_0## is the infimum of the set of distances from ##a_0## to another point in the partition whose components exceed those of ##a_0##

The vectors ##a_0## and ##a'## define hyper-rectangle that constitutes the volume element of the volume to be estimated.

Evaluate ##f## at some point ##x_0## in this hyper-rectangle in order to get a proper weight for this region of the domain.

In order to get the volume of that hyper-rectangle, we evaluate the volume element ##dx^1\wedge\ldots\wedge dx^n## at ##(a'-a_0,a'-a_0,\ldots,a'-a_0)##.

This process yields a number ##f(x_0) dx^1\wedge\ldots\wedge dx^n(a'-a_0,\ldots,a'-a_0)## where the latter term is the "volume" of the element of the partition on which ##f## is to be evaluated.

Anyway, I was trying to relate integration of forms to Riemann integration earlier. I was confused by the relationship between the notation ##\int_A f\,dx^1\wedge\,dx^2## and ##\int_A f(x)\,dx^1\,dx^2##, and then I tried to come up with my own explanation for why these two are equivalent, using my rather rusty knowledge of Riemann integration.

Forgive me if this is poorly-explained.