DIFFERENTIAL FORMS - Line and Surface Integrals; Differential Forms and Stokes Theorem

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 ⁿ is a mapping which associates with each point a linear function ω(x) : ⁿ → , and that each linear function on ⁿ is a linear combination of the differentials dx₁, . . . , dx_n, so

where a₁, . . . , a_n are real-valued functions on ⁿ.

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

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

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

For example, every differential 2-form α on ³ is of the form

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

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

A standard and useful alternative notation for multidifferentials is

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

where A_i denotes the k × k matrix whose rth row is the i_rth row of A. If i_r = i_s, then the rth and sth rows of A_i are equal, so

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

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

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

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

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

and a differential l-form

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

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, . . . , i_k, j₁, . . . , j_l) 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

with the summation being over all increasing (k + l)-tuples. Note that α 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

Example 1 Let α = a₁ dx₁ + a₂ dx₂ + a₃ dx₃ and β = b₁ dx₁ + b₂ dx₂ + b₃ dx₃ be two 1-forms on ³. Then

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 dy + B dx dz + C dy dz. Applying (3) to delete immediately each multidifferential that contains twice a single differential dx or dy or dz, we obtain

We next define an operation of differentiation for differential k-forms, extending our previous definitions in the case of 0-forms (or functions) on ⁿ and 1-forms on ². Recall that the differential of the function f: ⁿ → is defined by

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

Note first that the differential operation is clearly additive,

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

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

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

by the equality, under interchange of order of differentiation, of mixed second order partial derivatives of functions. The fact, that d(dω) = 0 if ω is a differential 1-form in ³, is an instance of a quite general phenomenon.

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

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

Then

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

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 , then

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

where a and b are functions. Then

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.

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

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

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

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

(“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 (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

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

Before proving this in general, let us consider the special case in which α = f dy dz is a 2-form on ³, and φ: Q → ³ is a 2-dimensional surface patch. Using uv-coordinates in ², we obtain

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 ⁿ, under a mapping This will be a generalization of the pullback defined in Section 2 for differential forms on ². We start by defining the pullback of a 0-form (or function) f or differential dx_i by

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

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

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

If α = A dy dz, then

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

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

Lemma 5.3 Let ω₁, . . . , ω_k be k differential 1-forms on ^k, with

in u-coordinates. Then

where A is the k × k matrix (a_ij).

PROOF Upon multiplying out, we obtain

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

so we have

by the standard definition of det A.

Theorem 5.4 If φ : Q → ⁿ is a k-dimensional surface patch, and α is a differential k-form on ⁿ, then

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

Then

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

as desired.

Example 4 Let and suppose φ : Q → ³ is defined by the equations

We compute the surface integral

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

we see that

Therefore

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

into α. So

Therefore Theorem 5.4 gives

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 : ^m → ⁿ is a mapping and α is a differential k-form on ⁿ, then

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

But

which is the same thing.

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

where . Then

by Proposition 5.2. Therefore

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

using Propositions 5.1 and 5.2.

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 () k-manifold in ⁿ. 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 is a one-to-one mapping φ : U → M, where U is an open subset of ^k, such that dφ_u has rank k for each this implies that [det(φ′(u)^tφ′(u)]^1/2 ≠ 0. An atlas for M is a collection {_i} of coordinate patches, the union of whose images covers M. An orientation for M is an atlas {_i} such that the “change of coordinates” mapping, corresponding to any two of these coordinate patches _i and _j whose images _i(U_i) and _j(U_j) overlap, has a positive Jacobian determinant. That is, if

then det T′_ij > 0 wherever T_ij is defined (see Fig. 5.29) . The pair (M, {_i}, 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 _i, and orientation-reversing if it overlaps negatively with each of the _i (that is, the appropriate Jacobian determinants are negative at each point).

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

Figure 5.29

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

where

Example 5 Let M = S², the unit sphere in ³. We use the spherical coordinates surface patch defined as usual by

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

Thus the area form of S² is

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

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

Therefore

because det T′(u) > 0. Thus the two orientation-preserving coordinate patches φ and provide the same definition of n_i(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 ⁿ with surface area form dA. If φ : Q → M is the restriction, to the k-dimensional interval , of an orientation-preserving coordinate patch, then

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

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

we see that Theorem 5.6 gives

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

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

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 ⁿ and α a continuous differential k-form defined on M. Let φ : U → M and : V → M be two coordinate patches on M with φ(U) = (V), and write T = ⁻¹ φ : U → V. Suppose X and Y are contented subsets of U and V, respectively, with T(X) = Y. Finally let ; = φ X and = Y. Then

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

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

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

Therefore

On the other hand, the change of variables theorem gives

Figure 5.30

Since det T′ > 0 either if φ and 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) .

Now let = {A₁, . . . , A_r} and = {B₁, . . . , B_s} be two oriented pavings of M. Let φ_i and _j be orientation-preserving parametrizations of A_i and B_j, respectively. Let

If φ_ij = φ_i X_ij and _ij = _jY_ij, then it follows from Lemma 5.7 that

for any k-form α defined on M. Therefore

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 is thought of as a lamina with density function ρ, then its mass is given by the integral

If M is a closed surface in ³ with unit outer normal vector field N, and F is the velocity vector field of a moving fluid in ³., then the “flux” integral

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” , which is the image in ⁴ of the surface patch F : Q = [0, 2π]² → ⁴ defined by

The surface area form of T is

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

Figure 5.31

Now

Therefore

Consequently

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

(b)r⁻ⁿα, where r = [x₁² + · · · + x_n²]^1/2.

(c), where (x₁, . . . , x_n, y₁, . . . , y_n) are coordinates in ²ⁿ

5.2If F : ⁿ → ⁿ is a mapping, show that

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

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

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

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

5.6Verify formula (7) in this section.

5.7If φ : Q → ^k is the identity (or inclusion) mapping on the k-dimensional interval , and α = f dx₁ · · · dx_k, show that

5.8Let φ : ^m → ⁿ be a mapping. If α is a k-form on ⁿ, prove that

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

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

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

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 be an oriented smooth k-manifold with area form dA₁, and an oriented smooth l-manifold with area form dA₂. Regarding dA₁ as a form on ^{m + n} which involves only the variables x₁, . . . , x_m, and dA₂ as a form in the variables x_m_{+ 1}, . . . , x_{m + n}, show that

is the surface area form of the oriented (k + l)-manifold . Use this result to obtain the area form of the flat torus in ⁴.

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