We now turn to the study of hyper-surfaces in . A hyper-surface is something like a curve or a surface in or their higher dimensional analogues. A differential form is the something we can integrate over a hyper-surface.

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For example, a parameterized curve in is a map , we will call this a curve if is . Given such a curve with and a function we can form the integral of “ along :”

We will call the expression

a *differential one form*. We will generalize this notion to make *differential -forms *and integrate these over *hypersurfaces of dimension .*

** Hypersurfaces **

Definition 1Aparameterized -surfacein is a map where is open. If with an open set, we say thatis a parameterized -surface in .

Remark 2We will call a -surface for short. In principle we should define a -surface as an equivalence class of parameterized -surface where we make if with a one-to-one map with at all points of it’s domain. Think of a curve, say where . This is the closed unit circle in . As a parametrized curve this is distinct from defined on the domain , but we see that the difference is simply one ofreparameterizingthe domain. We should keep this in mind, but for the sake of simplicity we will refer to parameterized -surfaces simply as -surfaces.

** Differential forms **

Definition 3Let be an open set and an integer. Adifferential -formis a function from to the space of alternating -forms on . If with a non-negative integer we say that is a –formor that is ofclass(for we also say that is acontinuous-form).

Remark 4Since by definition, a differential -form on is just a real valued function on .

We will develop some special notation for -forms in a bit. For the moment we will denote the value of at a point by . Note that for each is an alternating -form, which in particular is a *function* taking as an argument -vectors in . We will denote the action of on the vectors by

Remark 5is if and only if for each the map is .

Lemma 6Suppose is a -form on and is a parameterized -surface. Thenis a -form on , which is continuous if is.

*Proof:* That is a differential -form follows easily from the fact that is. The continuity follows easily from the continuity of and .

Definition 7Let be a continuous -form, let be a parameterized -surface in and let be the standard basis of . Then the integral of over is

Remark 81) For the most part, it will suffice to consider only in case has compact support in . (Here thesupportof a -form is the closure of the set on which it is not identically zero.) In that case the integral on the right hand side of (1) is well defined, since we may continuously extend to all of by taking it to be zero off of and set2) Sometimes it is useful to consider a -surface defined on a -cell in . If extends continuously to the boundary points of then the integral on the right hand side of (1) is well defined as an iterated integral. 3) For a -form in we will simply write

with the standard basis and a -cell. This amounts to identifying the -cell with the -surface given by the identity map.

Why is (1) a good definition? The answer comes from considering change of variables. Indeed, suppose and are parameterized surfaces in with where is a one-to-one map with everywhere. Then

If the domain of is connected then we either have or for each . Thus, by the change of variables formula,

with the sign corresponding to the sign of .

What does this sign mean? Remember that for the fundamental theorem of calculus in it was useful to define for , thus making the integral into an *oriented integral* that depends on the orientation of the interval. In the same way, a -surface in has two possible orientations and switching orientations results in a minus sign. You may have encountered this in vector calculus when integrating over surfaces in — there are two choices of unit normal vectors and the choice determines the sign of integrals like .

** Wedge product and elementary forms **

Definition 9Given a differential -form and a differential -form we define a from by

Remark 10Note that given a -form and a -form a on an open set . Given a parameterized -surface we have

Remark 11The “wedge” product of a zero form and a -form is the -formWe will typically denote this product as just .

Exercise 12Let be a -form , let be a -form and let be an -form. Show that and that

Exercise 13Let and be -forms and let be an -form. Show that

To proceed it is useful to introduce a notation for forms. First we define the *elementary forms* on by

so if and if . The for each we define the *elementary forms* to be the following

where — this makes sense without parentheses by exercise 12. For instance

Note that if for any then the resulting form is zero and more generally that

for any permutation of . Hence, up to sign, there are only elementary forms, given by

where .

Let us introduce a compact notation for these elementary -forms. Given we can write its elements in increasing order where . Let

Note that given of size ,

with is the unique permutation of that puts in increasing order.

Theorem 14Let be a differential -form on an open set then to each subset of size there is a function on such thatFurthermore is if and only if the functions for each .

*Proof:* Given of size with let

Clearly if is then is also. Furthermore

Now for each we have

where Note also that

if . Thus

as claimed.

Since is constant as a function of , it is so that if is for each then is too.

Definition 15Let be a parameterized -surface in . Let points in be denoted and points in be denoted , so . Given and define theJacobianGiven and let denote the correspnding Jacobian with ‘s and ‘s in increasing order.

Proposition 16Let be a parameterized -surface in and a -form defined on a neighborhood of . Then

*Proof:* Let denote the standard basis of and the standard basis of . Then

Thus given ,

where in the second to last line Together with the previous theorem this proves the result.

A particular case of this result is when , so for we get the explicit expression

** Exterior derivative **

Let be an open set in . If we define

So is a continuous -form on . Similarly if is a -form we define

So is a continuous -form on .

Example 17Let be a -form. Then

Example 18Let be a -surface in , that is a curve in . Suppose then

Lemma 19Let be a -form on an open set . Then .

*Proof:* Let . So

Lemma 20Let be a j-form and let be a -form, both defined on an open set . Then

Exercise 21Prove Lemma 20. (Hint: it is essentially the product rule.)

Note that the notation for the elementary one forms is consistent with the exterior derivative: *is indeed the exterior derivative of the function .*

Theorem 22Let be a -form on an open set and let be a parameterized -surface in . Then

*Proof:* First consider a zero form, namely a function . Then and we see that

Specializing to a coordinate function we see that

where denotes the -th coordinate function of . Since (see remark 10) we see that

for any elementary -form . In particular, it follows from the product rule (Lemma 20) that

Putting equations (3) and (4) together we find for a general -form that

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