Differential forms are important concepts in differential geometry and mathematical physics. For example, they can be used to express Maxwell’s equations (see Some Basics of (Quantum) Electrodynamics) in a very elegant form. In this post, however, we will introduce these mathematical objects as generalizing certain aspects of integral calculus (see An Intuitive Introduction to Calculus), allowing us to perform integration over surfaces, volumes, or their higher-dimensional analogues.
We recall from An Intuitive Introduction to Calculus the statement of the fundamental theorem of calculus:
Regarding the left hand side of this equation, we usually we say that we integrate over the interval from to ; we may therefore also write it more suggestively as
We note that and form the boundary of the interval . We denote the boundary of some “shape” by . Therefore, in this case, .
Next we are going to perform some manipulations on the notation, which, while we will not thoroughly justify in this post, are meant to be suggestive and provide intuition for the discussion on differential forms. First we need the notion of orientation. We can imagine, for example, an “arrow” pointing from to ; this would determine one orientation. Another would be determined by an “arrow” pointing from to . This is important because we need a notion of integration “from to ” or “from to “, and the two are not the same. In fact,
i.e. there is a change of sign if we “flip” the orientation. Although an interval such as is one-dimensional, the notion of orientation continues to make sense in higher dimension. If we have a surface, for example, we may consider going “clockwise” or “counterclockwise” around the surface. Alternatively we may consider an “arrow” indicating which “side” of the surface we are on. For three dimensions or higher it is harder to visualize, but we will be able to make this notion more concrete later on with differential forms.
Given the notion of orientation, let us now denote the boundary of the interval , taken with orientation, for instance, “from to “, by .
Let us now write
and then we can write the fundamental theorem of calculus as
Then we consider the idea of “integration over points”, by which we refer to simply evaluating the function at those points, with the orientation taken into account, such that we have
Recalling that , this now gives us the following expression for the fundamental theorem of calculus:
Things may still be confusing to the reader at this point – for instance, that integral on the right hand side looks rather weird – we will hopefully make things more concrete shortly. For now, the rough idea that we want to keep in mind is the following:
The integral of a “differential” of some function over some shape is equal to the integral of the function over the boundary of the shape.
In one dimension, this is of course the fundamental theorem of calculus as we have stated it earlier. For two dimensions, there is a famous theorem called Green’s theorem. In three dimensions, there are two manifestations of this idea, known as Stokes’ theorem and the divergence theorem. The more “concrete” version of this statement, which we want to discuss in this post, is the following:
The integral of the exterior derivative of a differential form over a manifold with boundary is equal to the integral of the differential form over the boundary.
We now discuss what these differential forms are. Instead of the formal definitions, we will start with special cases, develop intuition with examples, and attempt to generalize. The more formal definitions will be left to the references. We will start with the so-called -forms, which are “linear combinations” of the “differentials”.
We can think of these “differentials” as merely symbols for now, or perhaps consider them analogous to “infinitesimal quantities” in calculus. In differential geometry, however, they are actually “dual” to vectors, mapping vectors to numbers in the same way that row matrices map column matrices to the numbers which serve as their scalars (see Matrices) of the coordinates, with coefficients which are functions:
From now on, to generalize, instead of the coordinates , , and we will use , , , and so on. We will write exponents as , to hopefully avoid confusion.
From these -forms we can form -forms by taking the wedge product. In ordinary multivariable calculus, the following expression
represents an “infinitesimal area”, and so for example the integral
gives us the area of a square with vertices at , , , and . The wedge product expresses this same idea (in fact the wedge product is often also called the area form, mirroring the idea expressed by earlier), except that we want to include the concept of orientation that we discussed earlier. Therefore, in order to express this idea of orientation, we require the wedge product to satisfy the following property called antisymmetry:
Note that antisymmetry implies the following relation:
In other words, the wedge product of such a differential form with itself is equal to zero.
We can also form -forms, -forms, etc. using the wedge product. The collection of all these -forms, for every , is the algebra of differential forms. This means that we can add, subtract, and form wedge products of differential forms. Ordinary functions themselves form the -forms.
We can also take what is called the exterior derivative of differential forms. If, for example, we have a differential form given by the following expression,
then the exterior derivative of , written , is given by
We note that the exterior derivative of a -form is an -form. We also note that the exterior derivative of an exterior derivative is always zero, i.e. for any differential form . A differential form which is the exterior derivative of some other differential form is called exact. A differential form whose exterior derivative is zero is called closed. The statement can also be expressed as follows:
All exact forms are closed.
However, not all closed forms are exact. This is reminiscent of the discussion in Homology and Cohomology, and in fact the study of closed forms which are not exact leads to the theory of de Rham cohomology, which is a very important part of modern mathematics and mathematical physics.
Given the idea of the exterior derivative, the general form of the fundamental theorem of calculus is now given by the generalized Stokes’ theorem (sometimes simply called the Stokes’ theorem; historically however, as alluded to earlier, the original Stokes’ theorem only refers to a special case in three dimensions):
This is the idea we alluded to earlier, relating the integral of a differential form (which includes functions as -forms) over some “shape” to the integral of the exterior derivative of the differential form over the boundary of that “shape”.
There is much more to the theory of differential forms than we have discussed here. For example, although we have referred to these “shapes” as manifolds with boundary, more generally they are “chains” (see also Homology and Cohomology – the similarities are not coincidental!). There are restrictions on these chains in order for the integral to give a function; for example, an -form must be integrated over an -dimensional chain (or simply -chain) to give a function, otherwise they will give some other differential form. An -form integrated over an -chain gives an form. Also, more rigorously the concept of integration on more complicated spaces involves the notion of “pullback”. We will leave these concepts to the references for now, contenting ourselves with the discussion of the wedge product and exterior derivative in this post. The application of differential forms to physics is discussed in the very readable book Gauge Fields, Knots and Geometry by John Baez and Javier P. Muniain.
Calculus on Manifolds by Michael Spivak
Gauge Fields, Knots and Gravity by John Baez and Javier P. Muniain
Geometry, Topology, and Physics by Mikio Nakahara