Without using differential forms, can we unify or organize all the various multivariable integrals: multiple, line, flux, etc.?

Question

In single dimension calculus, there is one and only one obvious way of integrating: over an interval. Of course, there are multiple ways of defining the integral (Riemann sum, Darboux integral, Lebesgue integral), but, where defined, these definitions agree.

Once we move from $\mathbb R$ to $\mathbb R^n$, there seems to be a plethora of types of integrals, with most sources making no attempt to unify or organize them:

In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at... In this section we will define the third type of line integrals we’ll be looking at: line integrals of vector fields.

The Area under a Curve and its Many Generalizations... The method of doing this used is generalized to define a wide variety of integrals...

Is there a way to unify or at least organize all these types of integrals?

My attempt to do so is below. Is it correct? What revisions does it need? Update: My goal is to do this in a simple way, without invoking the complexity and abstraction of differential forms.

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For $f: \mathbb R \to \mathbb R^m$, the one obvious way to integrate is over an interval of $\mathbb R$. When instead $f$ takes an input in $\mathbb R^n$, we gain one, and in some cases two, new ways to integrate:

Region ("Multiple") Integrals

For multiple dimensions, with $f: \mathbb R^n \to \mathbb R^m$, we can take the integral over any "nice" $k \leq n$ dimensional region of $\mathbb R^n$. We can do this for regions of any dimension $k \leq n$ by partitioning it into very small subregions. However, for $k > 1$, these subregions are not defined by two endpoints such that we can define $\Delta x_i$. Rather, we need to take the size or region measure (hypervolume, volume, area, etc.) appropriate for $k$ (irrespective of $n$) of each subregion. This partition is, unlike intervals, unordered and unoriented, and the size of each subregion is always positive: $$\sum_{r_{i, j,...} \in R}f(\xi_{i,j,...})\cdot \|r_{i,j,...}\|.$$

The integral is the limit of such as $\max_{i,j,...} \|r_{i,j,...}\| \to 0$ (provided it exists). (And the fact that the partition unordered and unoriented, with each subregion having positive size, is the reason why the Change of Variables Theorem uses the absolute value of the determinant and not the determinant itself.)

By Fubini's theorem, this limit is (under reasonable conditions) equal to an iterated integral, and so these are often called multiple integrals. However, it is important to emphasize that there is nothing "multiple" in the definition of multiple integrals: They are limits of single Riemann sums over partitions of regions with dimension $> 1$.

This definition therefore includes volume integrals, area integrals, line integrals over scalar fields; all are simply region integrals, defined the same way, with the region having different dimensions.

Vector Field Integrals: Line (dim = 1) and Flux (codim = 1)

Furthermore, if $n = m$, then $f$ defines a vector field, and we gain the ability to take the dot product $f(x) \cdot x$. This gives us two new ways to take integrals: line integrals and flux integrals.

If the region is a curve (i.e. $k = 1$), we can approximate each subregion $r_i$ by a vector $s_i \in \mathbb R^n$ and take the Riemann sum of $$\sum f(\xi_i) \cdot s_i.$$

And if the region is a surface (i.e. $k = n - 1$), we can construct a flux integral by approximating each $r_i$ by an $k$ dimensional hyperplane $p_i$ and letting $n_i$ be the unique vector that is normal to $p_i$, has magnitude equal to the "size" (volume, area, etc.) of $p_i$, and has a sign determined by convention, giving the Riemann sum of $$\sum f(\xi_i) \cdot n_i.$$

Answer 1

I quite like Terry Tao's comment/explanation that the distinct notions of 'integral'/'integration' are really and essentially about distinct things:

The concept of integration is of course fundamental in single-variable calculus. Actually, there are three concepts of integration which appear in the subject: the indefinite integral $\int f$ (also known as the anti-derivative), the unsigned definite integral $\int_{[a,b]} f(x) dx$ (which one would use to find area under a curve, or the mass of a one-dimensional object of varying density), and the signed definite integral $\int^b_a f(x) dx$ (which one would use for instance to compute the work required to move a particle from $a$ to $b$). [...] These three integration concepts are of course closely related to each other in singlevariable calculus; [...]

When one moves from single-variable calculus to several-variable calculus, though, these three concepts begin to diverge significantly from each other. The indefinite integral generalises to the notion of a solution to a differential equation, or of an integral of a connection, vector field, or bundle. The unsigned definite integral generalises to the Lebesgue integral, or more generally to integration on a measure space. Finally, the signed definite integral generalises to the integration of forms, which will be our focus here. While these three concepts still have some relation to each other, they are not as interchangeable as they are in the single-variable setting.

That they coalesce so nicely in the one-dimensional case may very well be a small miracle

Answer 2

In $\mathbb R$ you integrate a function over an interval, $\mathbb R ^2$ you integrate over and area, in $\mathbb R^3$ a volume. In higher dimensions, the region of integration is often considered a generalized volume, and sometimes this generalized volume even refers to an interval or an area.

So in general, there's integration of a single valued function over some "volume" or sub-region of our space.

If we have a vector valued function integrated over a curve, we have a line integral by definition. Ultimately though this can be transformed into the usual single valued function over an interval via the dot product. There are theorems allowing a transformation of some line integrals into an area integral. For example, the magnetic flux through a surface can be found by explicitly calculating the flux, $\Phi_B = \int\int \vec{B} \cdot \hat{n} dS$ or using the Magnetic Vector Potential $\vec{A}$ where $\vec{B} = \nabla \times \vec{A}$. To wit, $\Phi_B = \int \vec{A} \cdot \vec{dl}$ where the flux is a line integral of the vector potential along the boundary of the surface in question.

Flux is an integral over a surface by definition, and a surface implies area. So the Electric Flux is $\Phi_E=\int\int \vec{E} \cdot \hat{n} dS$. This can be converted to a volume integral so that $\Phi_E = \int\int\int \nabla \cdot \vec{E} d\tau$ .

Both the conversion from a surface integral to a line integral and a volume integral to a surface integral are manifestations of some fundamental principles from the study of differential forms. In both cases the ultimate results can be found by integrating over a boundary.

The above mentioned integrals give you a scalar result, but similar principles apply over more general Tensors. A vector is a Rang 1 tensor. A Tensor can loosely be thought of as a nested vector. For example, consider $T^{\alpha\beta}$ as a momentum flux. In the $\alpha$ direction, the $\beta$ component of momentum $\vec{p}$ is changing at rate $T^{\alpha \beta}$.

So there's a general principle in play regarding Exterior Derivatives that establishes relationships between the integral of a tensor over a generalized volume that gives you the same result as the integral of a tensor derivative over a boundary "surface".

Ultimately the general form results from multiple integrals over some region of the parameter space.