Throughout this lecture, the word differentiable will be used to mean .
Definition 1: A diffeomorphism is a function between open sets of which is differentiable and has a differentiable inverse.
Definition 2: The half-space is the set of points in whose last coordinate is non-negative. A k-dimensional manifold-with-boundary is a subset of such that every point satisfies:
There is a diffeomorphism from an open neighborhood of such that
where the set is either or
The set of which satisfy the condition with is called the boundary of and is denoted If the boundary of is empty, then can be referred to as a manifold.
If , then a coordinate system around is a 1-1 differentiable function for which there is an open set satisfying:
Proposition 1: A subset of is a k-dimensional manifold if and only if every has a coordinate system about
Proof: If has a function as in the definition of manifold, then let be the projection on the first coordinates of and be defined by Then is a coordinate system about . (The condition involving the rank follows by applying the chain rule to where is with being the projection on the first coordinates.)
Conversely, if is a coordinate system about , then define by . If is such that , then is invertible in a neighborhood of . The inverse in this neighborhood is the desired function as in the definition of manifold.
Note: If and are a coordinate systems about , then is a diffeomorphism as the proof showed that is just the first coordinates of the diffeomorphism
In particular, is independent of the choice of coordinate system about This set is called the tangent space of at If is a element of the tangent space at for every in , then there is a unique in the tangent space of at such that We say is a differentiable vector field of if each of the are differentiable vector fields of each . In particular, if is the restriction of a differentiable vector field on some open subset containing , then restricts to a differentiable vector field on
Similarly, a differentiable k-form on is a function which assigns to each an such that is a differentiable k-form on for every coordinate system about . The derivative of is the differentiable k-form on provided by the following result:
Proposition 2: If is a differentiable k-form on , then there is a unique differentiable -form on such that for every coordinate system about , one has
Proof: For each , let be chosen so that . Let
This definition is independent of the choice of coordinate system and is easily shown to be the desired differentiable -form on
Let be a finite dimensional real vector space of dimension and be non-zero. Then is non-zero for every basis of . Thus, the set of (ordered) bases of is partitioned into two sets, such that two bases are in the same set if and only if applied to the bases gives a real number of the same sign. The set to which an (ordered) basis belongs is called its orientation and is denoted Note that two bases being of the same orientation is independent of the choice of The usual orientation of is defined to be
Suppose that for every (where is a k-dimensional manifold), one has chosen an orientation of the tangent space . Then these choices are said to be consistent if and only if for every coordinate system about and every pair , one has if and only if Such a consistent choice is called an orientation of ; a manifold which admits an orientation is said to be orientable.
If the are consistent, then one says that the coordinate system is said to be orientation preserving if for every Clearly, if is a linear transformation with , then exactly one of and is orientation preserving.
Now, let be a -dimensional manifold-with-boundary and Then is a -dimensional subspace of . There are precisely two unit vectors perpendicular to this subspace. Choose a coordinate system about in which 0 maps to and Then the one of the unit vectors of the form with is called the outward unit normal Suppose we have an orientation for . Then choose a basis for such that . Then the define a consistent orientation on called the induced orientation. Note that the orientation induced on from the usual orientation on is the usual orientation if and only if is even.
If is -dimensional manifold contained in which admits an orientation , then one can also define an outward unit normal as the one such that if is a basis of with , then is the usual orientation of
If is a -form on a -dimensional manifold-with-boundary and is a singular in , we can define:
Integrals over -chains are defined in the obvious way. In the special case where , we will always assume that is the restriction to of a coordinate system (where we assume If one has an orientation for , we say that the singular -cube is orientation preserving provided that is.
All the definitions have been set up to guarantee the following result:
Proposition 3: If are two orientation preserving singular -cubes in an oriented -dimensional manifold and is a -form on such that outside of , then
Proof: We have
where and we have used the assumption that is zero outside of It remains to show that
But, if , then we have
since The result now follows by the change of variables formula for integrals.
We can now define integrals. Let be a k-form on an oriented k-dimensional manifold . Choose an open cover of such that for each , there is an orientation preserving singular -cube with Let be a partition of unity for subordinate to this cover. Define
where was chosen so that is zero outside of an compact subset of and is is an orientation preserving singular -cube with Then just as in Chapter 3, the value of this integral is independent of the choice of , , and .
Now suppose we have a -dimensional manifold-with-boundary with orientation . Let be the orientation induced by on . Let be an orientation-preserving -cube in such that lies in and this is the only face which contains any interior points of . Then is orientation preserving if and only is even. In particular, if is a -form on which is zero outside of , we have
Now appears with coefficient in the definition of . So,
This explains the strange choice of signs in the definition of the induced orientation on
Theorem 1: (Stokes' Theorem) Let be a compact oriented -dimensional manifold-with-boundary and be a -form on . Then
where is oriented with the orientation induced from that of
Proof: Begin with two special cases: First assume that there is an orientation preserving -cube in such that outside of Using our earlier Stokes' Theorem, we get
since on But also since on
The second case is where there is an orientation-preserving singular -cube in such that is the only face containing points of and outside of One has:
For the general case, choose an open cover of and a partition of unity subordinate to such that for each , the form is as in one of the two cases already considered. Since is compact, one has a finite sum:
In , the volume can be calculated as the integral of the form . We would like to find a generalization to manifolds of this differential form.
If is a -dimensional manifold, then the usual inner product on induces an inner product on each of the tangent spaces of . (Recall that an inner product of V is a bilinear form such that for all .) With an inner product, one can define an orthonormal basis to be one of the form where where is the Kronecker . Now, if and are both orthonormal bases, then we can write . In particular, one calculates:
which can be expressed as a matrix equation where is the matrix with entries Taking determinants of this means that In particular, if where is a vector space of dimension , then
Proposition 4: is constant for all orthonormal bases of of the same orientation.
Definition 3: Let be an oriented -dimensional manifold in . Then a -form on is called a volume element if for all orientation preserving orthonormal bases
Example: Consider the case of 2-dimensional oriented manifolds in Let be the outward normal at . Then define by
By the definition of the outward normal, is a volume element. Further, if is an orthonormal basis of the same orientation as , one has:
Also, expanding as cofactors of the last row, one gets
On , one can compute for using for some , that
Letting , and , we get:
Proposition 4: Let be an oriented 2-dimensional manifold in and let be the unit outward normal. Then the volume element satisfies:
Further, on , one has:
To calculate a surface area, we need to evaluate for an orientation preserving singular 2-cube . The integrand is
(See Problem 4.9, part e.)
Three separate results will be shown to be special cases of our Stokes' Theorem.
Theorem 2: (Green's Theorem) Let be a compact 2-dimensional manifold-with-boundary. Suppose that are differentiable. Then
Proof: This is just Stoke's Theorem in the case of a 1-form.
Theorem 3: (Divergence Theorem) Let be a compact 3-dimensional manifold-with-boundary and be the unit outward normal on . Let be a differentiable vector field on Then
In terms of , this amounts to:
Proof: Define Then Further, Proposition 4 says that:
So, we see that this is also a special case of Stokes' Theorem
Theorem 4: (Stokes' Theorem) Let be a compact oriented 2-dimensional manifold-with-boundary and be the unit outward normal on determined by the orientation on . Let have the induced orientation. Let be the vector field on with and be a differentiable vector field in an open set containing . Then
In terms of , this amounts to:
Proof: Let on be defined by Again, using Proposition 4, we get:
Since , one has
as one can see by evaluating each equation at . It follows that
So, this is also a special case of our earlier Stokes' Theorem.