In this post I would like to say more about the generalised Stokes Theorem:
- is an oriented smooth n-manifold (a space whose local neighbourhoods look like n-dimensional Euclidean space)
- is an (n-1)-form with compact support on (further explanation will follow)
- is the boundary of with induced orientation (sign/orientation matter!)
- is the exterior derivative of (further explanation will follow)
This gem of a formula shows a duality between the boundary operator and the exterior derivative, where the two are linked by the integration operation. It relates the algebraic structure of differential forms (integrands) to the topology of manifolds. This duality is understood better by studying homology (of chains) and (de Rham) cohomology.
Apart from this duality, another reason this formula appeals is that through the language of differential forms it generalises many integration formulas:
1. The fundamental theorem of calculus (FTOC):
2. The gradient theorem (FTOC for line integrals):
3. Green’s theorem in the plane:
4. Cauchy’s integral theorem (can be proved from Green’s theorem and Cauchy-Riemann conditions if we assume continuous partial derivatives):
5. The curl theorem (Stokes’s theorem), a more general form of the previous two equations:
6. The divergence theorem:
One only needs to recall the general form and 1-6 all drop out as special cases.
About Differential Forms
Élie Cartan was the genius who recognised that differential forms were the right objects to study when generalising the fundamental theorem of calculus. These can be viewed simply as things that are integrated and combine the ideas of smooth functions and multilinear forms (i.e. many-variable real-valued functions that are linear in each variable).
A 0-form is simply a smooth function on a manifold M. For k > 0 a differential k-form assigns to each point p of M an alternating multilinear map on k tangent vectors in the tangent space at p. Think of the tangent space at p as the vector space of directions in which one could pass through p locally. A map is alternating if it evaluates to 0 when two of its variables are the same. The wedge product of k vectors is such an example and it turns out by linearity that a k-form can be written as a linear combination of wedge products of k 1-forms. Alternating forms generalise the calculation of the signed volume of a parallelepiped spanned by k vectors.
For example the 1-form dx associates to any point p a map which assigns to any tangent vector at p its x-coordinate. A 1-form multiplied by a smooth function is also a 1-form.
Higher k-forms can be formed by taking wedge product of two forms. The 2-form (also written ) evaluated at a point p maps a pair of tangent vectors at p to the signed area of the projection onto the x-y plane of the parallelogram spanned by the tangent vectors.
In general a k-form may be written as
where the multi-index represents a k-tuple of increasing integers from 1 to n. The notation is short for
Here are some examples of differential forms on the manifold . Note that one cannot add a k-form and an l-form if k and l are not equal.
The exterior derivative unifies the grad/div/curl operations of vector calculus and generalises the notion of a differential of a function. In fact the exterior derivative of a 0-form (smooth function) f is defined as its differential:
This is seen as a linear combination of 1-forms and so is a 1-form. With a general differential k-form as given above, its exterior derivative is a (k+1)-form that may be defined as
The differentiation operator d obeys the product rule and commutativity of partial derivatives implies that . This is dual to the result that the boundary of a boundary is 0. The wedge product operation is alternating and therefore skew-symmetric, meaning whenever and are 1-forms.
Differential forms are integrated by mapping the region of integration on the manifold to open sets in (via a parametrisation), then performing the integration there as a multi-dimensional integral on chains (linear combinations of simplices). Intuitively, to find :
(i) divide M into infinitesimal parallelepipeds (which reduce to segments or parallelograms in lower dimensions)
(ii) for each vertex of the subdivision evaluate at the k tangent vectors spanning the parallelepiped to obtain an infinitesimal scalar
(iii) sum the infinitesimal scalars over all infinitesimal parallelepipeds in the entire region of integration to obtain a scalar. Take the limit as the largest infinitesimal parallelepiped tends to 0 in volume.
Example: Green’s Theorem
We will show some of the manipulations of differential forms required in establishing Green’s theorem (3., above) from the generalised Stokes theorem. Here is the 1-form in a subset of the plane and
Substituting this into the generalised Stokes theorem gives Green’s theorem.
Apart from Stokes’s formula, the language of differential forms is useful for making the change of variable formula for integration straightforward. More about that in a future post perhaps!