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:

- has rank for each
- is continuous.

**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:

But then,

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

where

(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.