DIFFERENTIAL FORMS - Line and Surface Integrals; Differential Forms and Stokes Theorem - Advanced Calculus of Several Variables

Advanced Calculus of Several Variables (1973)

Part V. Line and Surface Integrals; Differential Forms and Stokes' Theorem

Chapter 5. DIFFERENTIAL FORMS

We have seen that a differential 1-form ω on Imagen is a mapping which associates with each point Image a linear function ω(x) : ImagenImage, and that each linear function on Imagen is a linear combination of the differentials dx1, . . . , dxn, so

Image

where a1, . . . , an are real-valued functions on Imagen.

A differential k-form α, defined on the set Image is a mapping which associates with each point Image an alternating k-multilinear function α(x) = αx on Imagen. That is,

Image

where λk(Imagen) denotes the set of all alternating k-multilinear functions on Imagen. Since we have seen in Theorem 3.4 that every alternating k-multilinear function on Imagen is a (unique) linear combination of the “multidifferentials” dxi, it follows that α(x) can be written in the form

Image

where as usual [i] denotes summation over all increasing k-tuples i = (i1, . . . , ik) with Image, and each ai is a real-valued function on U. The differential k-form α is called continuous (or Image, etc.) provided that each of the coefficient functions ai is continuous (or Image, etc.).

For example, every differential 2-form α on Image3 is of the form

Image

while every differential 3-form β on Image3 is a scalar (function) multiple of the single multidifferential dx(1, 2, 3),

Image

Similarly, every differential 2-form on Imagen is of the form

Image

A standard and useful alternative notation for multidifferentials is

Image

if i = (i1, . . . , ik); we think of the multidifferential dxi as a product of the differentials Image Recall that, if A is the n × k matrix whose column vectors are a1, . . . , ak, then

Image

where Ai denotes the k × k matrix whose rth row is the irth row of A. If ir = is, then the rth and sth rows of Ai are equal, so

Image

In terms of the product notation of (2) , it follows that

Image

unless the integers i1, . . . , ik are distinct. In particular,

Image

for each i = 1, . . . , n. Similarly,

Image

since the sign of a determinant is changed when two of its rows are interchanged.

The multiplication of differentials extends in a natural way to a multiplication of differential forms. First we define

Image

if i = (i1, . . . . , ik) and j = (j1, . . . , jl). Then, given a differential k-form

Image

and a differential l-form

Image

their product α Image β (sometimes called exterior product) is the differential (k + l)-form defined by

Image

This means simply that the differential forms α and β are multiplied together in a formal term-by-term way, using (5) and distributivity of multiplication over addition. Strictly speaking, the result of this process, the right-hand side of (6) , is not quite a differential form as defined in (1) , because the typical (k + l)-tuple (i, j) = (i1, . . . , ik, j1, . . . , jl) appearing in (6) is not necessarily increasing. However it is clear that, by use of rules (3) and (4) , we can rewrite the result in the form

Image

with the summation being over all increasing (k + l)-tuples. Note that α Image β = 0 if k + l > n.

It will be an instructive exercise for the student to deduce from this definition and the anticommutative property (4) that, if α is a k-form and β is an l-form, then

Image

Example 1 Let α = a1 dx1 + a2 dx2 + a3 dx3 and β = b1 dx1 + b2 dx2 + b3 dx3 be two 1-forms on Image3. Then

Image

using (3) and (4) , respectively in the last two steps. Similarly, consider a 1-form ω = P dx + Q dy + R dz and 2-form α = A dx Image dy + B dx Image dz + C dy Image dz. Applying (3) to delete immediately each multidifferential that contains twice a single differential dx or dy or dz, we obtain

Image

We next define an operation of differentiation for differential k-forms, extending our previous definitions in the case of 0-forms (or functions) on Imagen and 1-forms on Image2. Recall that the differential of the Image function f: ImagenImage is defined by

Image

Given a Image differential k-form Image defined on the open set Image its differential is the (k + 1)-form defined on U by

Image

Note first that the differential operation is clearly additive,

Image

Example 2 If ω = P dx + Q dy + R dz, then

Image

If α = A dy Image dz + B dz Image dx + C dx Image dy, then

Image

If ω is of class Image then, setting α = dω, we obtain

Image

by the equality, under interchange of order of differentiation, of mixed second order partial derivatives of Image functions. The fact, that d() = 0 if ω is a Image differential 1-form in Image3, is an instance of a quite general phenomenon.

Proposition 5.1 If α is a Image differential k-form on an open subset of Imagen, then d() = 0.

PROOF Since d(β + γ) = dβ + dγ, it suffices to verify that d() = 0 if

Image

Then

Image

so

Image

But since dxr Image dxs = −dxs Image dxr, the terms in this latter sum cancel in pairs, just as in the special case considered in Example 2.

Image

There is a Leibniz-type formula for the differential of a product, but with an interesting twist which results from the anticommutativity of the product operation for forms.

Proposition 5.2 If α is a differential k-form and β a differential l-form, both of class Image, then

Image

PROOF By the additivity of the differential operation, it suffices to consider the special case

Image

where a and b are Image functions. Then

Image

the (−1)k coming from the application of formula (7) to interchange the 1-form db and the k-form α in the second term.

Image

Recall (from Section 1 ) that, if ω is a Image differential 1-form on Imagen, and γ: [a, b] → Imagen is a Image path, then the integral of ω over γ is defined by

Image

We now generalize this definition as follows. If α is a Image differential k-form on Imagen, and φ: QImagen is a Image k-dimensional surface patch, then the integral of α over φ is defined by

Image

Note that, since the partial derivatives D1 φ, . . . , Dkφ are vectors in Imagen, the right-hand side of (10) is the “ordinary” integral of a continuous real-valued function on the k-dimensional interval Image

In the special case k = n, the following notational convention is useful. If α = f dx1 Image · · · Image dxk is a Image differential k-form on Imagek, we write

Image

(“ordinary” integral on the right). In other words, ∫Q α is by definition equal to the integral of α over the identity (or inclusion) surface patch Image (see Exercise 5.7) .

The definition in (10) is simply a concise formalization of the result of the following simple and natural procedure. To evaluate the integral

Image

first make the substitutions Image throughout. After multiplying out and collecting coefficients, the final result is a differential k-form β = g du1 Image · · · Image duk on Q. Then

Image

Before proving this in general, let us consider the special case in which α = f dy Image dz is a 2-form on Image3, and φ: QImage3 is a 2-dimensional surface patch. Using uv-coordinates in Image2, we obtain

Image

thus verifying (in this special case) the assertion of the preceding paragraph.

In order to formulate precisely (and then prove) the general assertion, we must define the notion of the pullback φ*(α) = φ*α, of the k-form α on Imagen, under a Image mapping Image This will be a generalization of the pullback defined in Section 2 for differential forms on Image2. We start by defining the pullback of a 0-form (or function) f or differential dxi by

Image

We can then extend the definition to arbitrary k-forms on Imagen by requiring that

Image

Exercise 5.8 gives an important interpretation of this definition of the pullback operation.

Example 3 Let φ be a Image mapping from Image to Image If ω = P dx + Q dy + R dz, then

Image

If α = A dy Image dz, then

Image

In terms of the pullback operation, what we want to prove is that

Image

this being the more precise formulation of Eq. (12) . We will need the following lemma.

Lemma 5.3 Let ω1, . . . , ωk be k differential 1-forms on Imagek, with

Image

in u-coordinates. Then

Image

where A is the k × k matrix (aij).

PROOF Upon multiplying out, we obtain

Image

where the notation {j} signifies summation over all permutations j = (j1, . . . , jk) of (1, . . . , k). If σ(j) denotes the sign of the permutation j, then

Image

so we have

Image

by the standard definition of det A.

Image

Theorem 5.4 If φ : QImagen is a k-dimensional Image surface patch, and α is a differential k-form on Imagen, then

Image

PROOF By the additive property of the pullback, it suffices to consider

Image

Then

Image

by Lemma 5.3 (here Image is the element in the rth row and jth column). Therefore, applying the definitions, we obtain

Image

as desired.

Image

Example 4 Let Image and suppose φ : QImage3 is defined by the equations

Image

We compute the surface integral

Image

in two different ways. First we apply the definition in Eq. (10) . Since

Image

we see that

Image

Therefore

Image

Second, we apply Theorem 5.4 . The pullback φ*α is simply the result of substituting

Image

into α. So

Image

Therefore Theorem 5.4 gives

Image

Of course the final computation is the same in either case. The point is that Theorem 5.4 enables us to proceed by formal substitution, making use of the equations which define the mapping φ, instead of referring to the original definition of ∫φ α.

Theorem 5.4 is the k-dimensional generalization of Lemma 2.2(b) , which played an important role in the proof of Green‘s theorem in Section 2. Theorem 5.4 will play a similar role in the proof of Stokes’ theorem in Section 6 . We will also need the k-dimensional generalization of part (c) of Lemma 2.2 —the fact that the differential operation d commutes with pullbacks.

Theorem 5.5 If φ : ImagemImagen is a Image mapping and α is a Image differential k-form on Imagen, then

Image

PROOF The proof will be by induction on k. When k = 0, α = f, a Image function on Imagen, and φ*f = f Image φ, so

Image

But

Image

which is the same thing.

Supposing inductively that the result holds for (k − 1)-forms, consider the k-form

Image

where Image . Then

Image

by Proposition 5.2. Therefore

Image

since φ*() = d(φ*β) by the inductive assumption. Since Image by (13) , we now have

Image

using Propositions 5.1 and 5.2.

Image

Our treatment of differential forms in this section has thus far been rather abstract and algebraic. As an antidote to this absence of geometry, the remainder of the section is devoted to a discussion of the “surface area form” of an oriented smooth (Image) k-manifold in Imagen. This will provide an example of an important differential form that appears in a natural geometric setting.

First we recall the basic definitions from Section 4 of Chapter III . A coordinate patch on the smooth k-manifold Image is a one-to-one Image mapping φ : U → M, where U is an open subset of Imagek, such that u has rank k for each Image this implies that [det(φ′(u)tφ′(u)]1/2 ≠ 0. An atlas for M is a collection {Imagei} of coordinate patches, the union of whose images covers M. An orientation for M is an atlas {Imagei} such that the “change of coordinates” mapping, corresponding to any two of these coordinate patches Imagei and Imagej whose images Imagei(Ui) and Imagej(Uj) overlap, has a positive Jacobian determinant. That is, if

Image

then det Tij > 0 wherever Tij is defined (see Fig. 5.29) . The pair (M, {Imagei}, is then called an oriented manifold. Finally, the coordinate patch φ : U → M is called orientation-preserving if it overlaps positively (in the above sense) with each of the Imagei, and orientation-reversing if it overlaps negatively with each of the Imagei (that is, the appropriate Jacobian determinants are negative at each point).

The surface area form of the oriented k-dimensional manifold Image is the differential k-form

Image

Image

Figure 5.29

whose coefficient functions ni are defined on M as follows. Given i = (i1, . . . , ik) and Image choose an orientation-preserving coordinate patch φ : U → M such that Image. Then

Image

where

Image

Example 5 Let M = S2, the unit sphere in Image3. We use the spherical coordinates surface patch defined as usual by

Image

Here D = sin φ (see Example 3 in Section 4) . Hence

Image

Thus the area form of S2 is

Image

Of course we must prove that ni is well-defined. So let Image : V → M be a second orientation-preserving coordinate patch with Image If

Image

then φ = Image Image T on Image, so an application of the chain rule gives

Image

Therefore

Image

because det T′(u) > 0. Thus the two orientation-preserving coordinate patches φ and Image provide the same definition of ni(x).

The following theorem tells why dA is called the “surface area form” of M.

Theorem 5.6 Let M be an oriented k-manifold in Imagen with surface area form dA. If φ : Q → M is the restriction, to the k-dimensional interval Image, of an orientation-preserving coordinate patch, then

Image

PROOF The proof is simply a computation. Using the definition of dA, of the area a(φ), and of the integral of a differential form, we obtain

Image

Image

Recall that a paving of the compact smooth k-manifold M is a finite collection Image = {A1, . . . , Ar} of nonoverlapping k-cells such that Image. If M is oriented, then the paving Image is called oriented provided that each of the k-cells Ai has a parametrization φi : QiAi which extends to an orientation-preserving coordinate patch for M (defined on a neighborhood of Image. Since the k-dimensional area of M is defined by

Image

we see that Theorem 5.6 gives

Image

Given a continuous differential k-form α whose domain of definition contains the oriented compact smooth k-manifold M, the integral of α on M is defined by

Image

where φ1, . . . , φr are parametrizations of the k-cells of an oriented paving Image of M (as above). So Eq. (17) becomes the pleasant formula

Image

The proof that the integral ∫M α is well defined is similar to the proof in Section 4 that a(M) is well defined. The following lemma will play the role here that Theorem 4.1 played there.

Lemma 5.7 Let M be an oriented compact smooth k-manifold in Imagen and α a continuous differential k-form defined on M. Let φ : U → M and Image : V → M be two coordinate patches on M with φ(U) = Image(V), and write T = Image−1 Image φ : U → V. Suppose X and Y are contented subsets of U and V, respectively, with T(X) = Y. Finally let Image; = φ Image X and Image = Image Image Y. Then

Image

if φ and Image are either both orientation-preserving or both orientation-reversing, while

Image

if one is orientation-preserving and the other is orientation-reversing.

PROOF By additivity we may assume that α = a dxi. Since φ = Image Image T (Fig. 5.30) , an application of the chain rule gives

Image

Therefore

Image

On the other hand, the change of variables theorem gives

Image

Image

Figure 5.30

Since det T′ > 0 either if φ and Image are both orientation-preserving or if both are orientation-reversing, while det T′ > 0 otherwise, the conclusion of the lemma follows immediately from a comparison of formulas (19) and (20) .

Image

Now let Image = {A1, . . . , Ar} and Image = {B1, . . . , Bs} be two oriented pavings of M. Let φi and Imagej be orientation-preserving parametrizations of Ai and Bj, respectively. Let

Image

If φij = φiImage Xij and Imageij = ImagejImageYij, then it follows from Lemma 5.7 that

Image

for any k-form α defined on M. Therefore

Image

so the integral ∫M α is indeed well defined by (18) .

Integrals of differential forms on manifolds have a number of physical applications. For example, if the 2-dimensional manifold Image is thought of as a lamina with density function ρ, then its mass is given by the integral

Image

If M is a closed surface in Image3 with unit outer normal vector field N, and F is the velocity vector field of a moving fluid in Image3., then the “flux” integral

Image

measures the rate at which the fluid is leaving the region bounded by M. We will discuss such applications as these in Section 7 .

Example 6 Let T be the “flat torus” Image, which is the image in Image4 of the surface patch F : Q = [0, 2π]2Image4 defined by

Image

The surface area form of T is

Image

(see Exercise 5.11) . If Q is subdivided into squares Q1, Q2, Q3, Q4 as indicated in Fig. 5.31 , and Ai = φ(Qi), then {A1, A2, A3, A4} is a paving of T, so

Image

Image

Figure 5.31

Now

Image

so

Image

Therefore

Image

Consequently

Image

Exercises

5.1Compute the differentials of the following Compute the differentials of the following Compute the differentials of the following differential forms.

(a)Compute the differentials of the following Image

(b)rnα, where r = [x12 + · · · + xn2]1/2.

(c)Image, where (x1, . . . , xn, y1, . . . , yn) are coordinates in Image2n

5.2If F : ImagenImagen is a Image mapping, show that

Image

5.3If Image is a differential 2-form on Imagen, show that

Image

5.4The function f is called an integrating factor for the 1-form ω if f(x) ≠ 0 for all x and d() = 0. If the 1-form ω has an integrating factor, show that ω Image = 0.

5.5(a)If = 0 and = 0, show that d(α Image β) = 0.

(b)The differential form β is called exact if there exists a differential form γ such that dγ = β. If = 0 and β is exact, prove that α Image β is exact.

5.6Verify formula (7) in this section.

5.7If φ : QImagek is the identity (or inclusion) mapping on the k-dimensional interval Image, and α = f dx1 Image · · · Image dxk, show that

Image

5.8Let φ : ImagemImagen be a Image mapping. If α is a k-form on Imagen, prove that

Image

This fact, that the value of φ*α on the vectors v1, . . . , vk is equal to the value of α on their images under the induced linear mapping , is often taken as the definition of the pullback φ*α.

5.9Let C be a smooth curve (or 1-manifold) in Imagen, with pathlength form ds. If φ : U → C is a coordinate patch defined for Image, show that

Image

5.10Let M be a smooth 2-manifold in Imagen, with surface area form dA. If φ : U → M is a coordinate patch defined on the open set Image, show that

Image

with E, G, F defined as in Example 5 of Section 4 .

5.11Deduce, from the definition of the surface area form, the area form of the flat torus used in Example 6.

5.12Let Image be an oriented smooth k-manifold with area form dA1, and Image an oriented smooth l-manifold with area form dA2. Regarding dA1 as a form on Imagem + n which involves only the variables x1, . . . , xm, and dA2 as a form in the variables xm + 1, . . . , xm + n, show that

Image

is the surface area form of the oriented (k + l)-manifold Image. Use this result to obtain the area form of the flat torus in Image4.

5.13Let M be an oriented smooth (n − 1)-dimensional manifold in Image n. Define a unit vector field N : MImagen on M as follows. Given Image, choose an orientation-preserving coordinate patch φ: U → M with x = φ(u). Then let the ith component ni(x) of N(x) be given by

Image

(a)Show that N is orthogonal to each of the vectors ∂φ/∂u1, . . . , ∂φ/∂un−1, so N is a unit normal vector field on M.

(b)Conclude that the surface area form of M is

Image

(c)In particular, conclude (without the use of any coordinate system) that the surface area form of the unit sphere Image is

Image

5.14(a)If F : ImagelImagem and G : ImagemImagen are Image mappings, show that

Image

That is, if α is a k-form on Imagen and H = G Image F then H*α = F*(G*α).

(b)Use Theorem 5.4 to deduce from (a) that, if φ is a k-dimensional surface patch in Imagem, F : ImagemImagen is a Image mapping, and α is a k-form on Imagen, then

Image

5.15Let α be a differential k-form on Imagen. If

Image

prove that

Image

where A = (aij).