Generalized Stokes' theorem

Section 6.1 Generalized Stokes' theorem

So far we have seen a number of theorems that take a similar form: the fundamental theorem of calculus, the fundametal theorem of line integrals, Green's theorem, and Stokes' theorem. In this section we show that they are all special cases of the mother of all integral theorems, the “generalized Stokes' theorem”.

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Objectives

You should be able to:

State the generalized Stokes' Theorem.
Show that for integration of an exact one-form over a closed interval in $R,$ it reduces to the Fundamental Theorem of Calculus.
Show that for integration of an exact one-form over a parametric curve in $R^{n},$ it reduces to the Fundamental Theorem of line integrals.
Show that for integration of an exact two-form over a closed bounded region in $R^{2},$ it reduces to Green's theorem.
Show that for integration of an exact two-form over a parametric surface in $R^{3},$ it reduces to Stokes' theorem.

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Subsection 6.1.1 The generalized Stokes' theorem

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So we far we have seen four integral theorems, all related to integration of exact one- and two-forms: the fundamental theorem of calculus, the fundamental theorem of line integrals, Green's theorem, and Stokes' theorem. While at first the theorems look different, you may have noticed that they all take a similar form. In fact, we could write all four integral theorems in the following form:

\int_{M} d ω = \int_{\partial M} ω .

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All that changes is the meaning of

ω

and

M .

More precisely:

If $ω$ is a zero-form (a function) on $U \subseteq R,$ and $M = [a, b] \subset U$ is an oriented interval, then it becomes the fundamental theorem of calculus, see Theorem 5.1.7.
If $ω$ is a zero-form (a function) on $U \subseteq R^{n},$ and $M$ is a parametric curve $α : [a, b] \to R^{n}$ whose image is in $U,$ then it becomes the fundamental theorem of line integrals, see Theorem 5.1.8.
If $ω$ is a one-form on $U \subseteq R^{2},$ and $M \subset U$ is a closed bounded oriented region, then it becomes Green's theorem, see Theorem 5.7.1.
If $ω$ is a one-form on $U \subseteq R^{3},$ and $M$ is a parametric surface $α : D \to R^{3}$ whose image is in $U,$ then it becomes Stokes' theorem, see Theorem 5.8.1.

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In mathematics, when we see something like this, we dig deeper and try to determine whether it is a coincidence or not that all these integral theorems pretty much take the same form. More often than not, such a coincidence is a hint that there is something going on behind the scenes, that there is a unifying principle at play. This is precisely the case here.

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The unifying principle is the mother of all integral theorems, known as the “generalized Stokes' theorem”. It states that the relationship above is very general. The precise statement is the following.

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Theorem 6.1.1. The generalized Stokes' theorem.

Let $M$ be a $k$ -dimensional oriented manifold and $\partial M$ its boundary (which is a $(k - 1)$ -dimensional manifold) with the induced orientation. Let $ω$ be a $(k - 1)$ -form on $M .$ Then

\int_{M} d ω = \int_{\partial M} ω .

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The scope of this theorem is really awe-inspiring, at least in the eye of a mathematician. We will not prove this theorem as it is beyond the scope of the class, but we can try to make sense of it.

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We know what a

k

-form is, at least over open subsets in

R^{n} .

The key object that we have not defined and that appears in the statement of the theorem is the notion of a “manifold”, which is fundamental in differential geometry. So let us say a few words about manifolds.

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Subsection 6.1.2 An informal introduction to manifolds

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The concept of manifold is essential in mathematics and physics to do calculations on complicated geometric spaces. Informally, an

n

-dimensional manifold

M

is a space that “locally looks like

R^{n}

”. What does it mean? It means that for any point

p \in M,

one can find an invertible map that sends an open subset of

M

around

p

to an open subset of

R^{n} .

This map is called a “coordinate chart”; by mapping the open subset of

M

to an open subset of

R^{n},

we are basically defining coordinates on the complicated space

M .

This is the essence of a manifold. The description of most manifolds however requires more than one coordinate charts; we can always map an open subset of

M

to an open subset of

R^{n},

but we cannot generally map the whole space

M

to an open subset of

R^{n},

because the global structure of the space

M

may be quite complicated. The different coordinate charts are then “glued” together in a consistent way, which gives rise to so-called “transition functions”, which are basically changes of coordinates between different charts. To be able to do calculus on manifolds, we usually require that these transition functions, or coordinate changes, are differentiable (or even smooth).

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In the end, the key feature of a manifold is that locally, instead of doing calculations on the space

M

itself, you can use the coordinate chart to do calculations on

R^{n}

instead. How do we do this? We use the pullback! Indeed, our coordinate chart is an invertible map, so we can pullback objects on

M

via the inverse of the coordinate chart to turn them into objects on

R^{n},

where we can do calculus. For instance, using pullback with respect to a coordinate chart, we can define integration of an

n

-form on a region of an

n

-dimensional manifold via integration of an

n

-form over a region in

R^{n},

which is something that we studied in this class. We use once again the fundamental principle of reducing something complicated to something that we already know how to solve, and the pullback is there to help! How neat is this.

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Most of the spaces that we encountered in this class, such as parametric curves, parametric surfaces, etc. are examples of manifolds. But the definition of manifolds is much more general. A key feature of manifolds is that they are defined “intrinsically”. When we talked about parametric curves, or parametric surfaces, we introduced complicated geometry, but the way we did it was by embedding a curve or a surface in a higher-dimensional space

R^{m} .

For manifolds, you do not need to embed them into higher-dimensional spaces to get interesting geometry; the geometry is intrinsic in the definition of a manifold.

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But in the end, you already know many manifolds. Here are a few examples.

The circle is a one-dimensional manifold, with no boundary.
The sphere (i.e. the surface of a ball) is a two-dimensional manifold, with no boundary.
Parametric curves (the way we defined them) are one-dimensional manifolds, possibly with boundary.
Parametric surfaces are two-dimensional manifolds, possibly with boundary.
The graph of a smooth function $f : R^{n} \to R$ is an $n$ -dimensional manifold.
Spacetime, where we live, is a manifold!

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Subsection 6.1.2.1 Back to the generalized Stokes' theorem

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We now understand that an

n

-dimensional manifold

M

is basically a complicated looking space that locally looks like

R^{n} .

If the space is orientable (as we saw for surfaces in

R^{3},

this is not always obvious), we can choose an orientation on

M,

like we did for parametric curves and surfaces. The boundary

\partial M

M

is also a manifold, but one dimension less: it is a

(n - 1)

-dimensional manifold. The chosen orientation on

M

induces an orientation on the boundary

\partial M,

just like we did again for parametric curves and surfaces.

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Even though we haven't defined integration of forms over manifolds, as we mentioned above it can be done via pullback with respect to coordinate charts, and the result is that integration of forms over manifolds is not much different from what we already did in this class. So we can, at least informally, understand the truly beautiful statement of the generalized Stokes' theorem:

\int_{M} d ω = \int_{\partial M} ω .

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To end this section, we summarize in a table how the four integral theorems that we already saw arise as special cases of the generalized Stokes' theorem. We add a fifth integral theorem to the table: the divergence theorem, which is the topic of the next section.

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Table 6.1.2. Integral theorems as special cases of the generalized Stokes' theorem

$M$	$ω$	Integral theorem
Closed interval in $R$	0-form	Fundamental theorem of calculus
Parametric curve in $R^{n}$	0-form	Fundamental theorem of line integrals
Closed bounded region in $R^{2}$	1-form	Green's theorem
Parametric surface in $R^{3}$	1-form	Stokes' theorem
Closed bounded region in $R^{3}$	2-form	Divergence theorem

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Next time someone tells you something about the fundamental theorem of calculus, you can reply: “oh, I know this theorem, it's just a special case of the generalized Stokes' theorem”! :-)