Appel W. Mathematics for Physics and Physicists

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Exterior algebra 467

that is, when no ambiguity a rises, repeated indices (appearing once as super-

script and once as subs cript) are assumed to be summed in strict ly increasing

order; so if there are k indices, the sum is over





terms instead of n

terms.

For the s pecial case k = n, we deduce the following result:

THEOREM 17.16 The space Λ

∗n

) is of dimension 1.

COROLLARY 17.1 6.1 Any exterior n-form on R

is a multiple of the form “deter-

minant in the canonical basis.”

DEFINITION 17.17 A volume form on R

is any nonzero exterior n-form.

Remark 17.18 We wi ll often denote ω

an exterior k-form, ω

an exterior p-form,

and so on. This supscript notation is only intended to recall the “degree” of the form

which is considered. In particular, it should not be mistaken wit h the exponent in a

power expression.

17.1.d Exterior product

There is a product deﬁned on the various spaces of exterior forms, which

associates a (k+ℓ)-exterior form ω

∧ω

ℓ

to a pair (ω

, ω

ℓ

) in Λ

∗k

(E)×Λ

∗ℓ

(E).

We will ﬁrst give the deﬁnition for the product of two 1-forms, then generalize

it. Note that the notation will be consistent with the prev ious introduction

of basis elements dx

∧···∧ dx

for Λ

∗k

(E) (i.e., th ose will be products of

1-forms).

Exterior product of two 1-forms

DEFINITION 17.19 Let ω, ω

′

be exterior 1-forms. The exterior product ω ∧

′

of ω and ω

′

is deﬁned by

ω ∧ω

′

def

= (ω ⊗ ω

′

−ω

′

⊗ω).

It is easily ch ecked that ω ∧ω

′

is an exterior 2-form.

PROPOSITION 17.20 Let ω and ω

′

be e xterior 1-forms. Then we ha ve

(ω ∧ω

′

)( x, y) =



ω( x) ω

′

( x)

ω( y) ω

′

( y)



and moreover ω∧ω

′

= −ω

′

∧ω. The exterior product is bilinear and antisymmetric.

Similarly, the product of p exterior 1-forms may be deﬁned as follows:

DEFINITION 17.21 Let ω

, . . . , ω

be exterior 1-forms. Then the exterior

product ω

∧ ··· ∧ ω

∈ Λ

∗p

(E) is deﬁned as the multilinear map on E

468 Differen tial forms

such that

∧···∧ ω

( x

, . . . , x

)

def



( x

) . . . ω

( x

)

( x

) . . . ω

( x

)



Exterior product of two arbitrary exterior forms

We now introduce the general deﬁnition of the exterior product of two exte-

rior forms.

DEFINITION 17.22 (Exterior product) Let k, ℓ ∈ N

∗

, ω

∈ Λ

∗k

(E) and ω

ℓ

∈

∗ℓ

(E). The exterior product of ω

and ω

ℓ

is the exterior (k + ℓ)-form

denoted ω

∧ω

ℓ

such that

∧ω

ℓ

( x

, . . . , x

k+ℓ

)

def

σ∈S

k+ℓ

ǫ(σ) ω

( x

σ(1)

, . . . , x

σ(k)

)

·ω

ℓ

( x

σ(k+1)

, . . . , x

σ(k+ℓ)

Example 17.23 For a 2-form ω

and a 1-form ω

, we have

∧ω

(a, b , c) = 2ω

(a, b ) ω

(b, c) ω

(a) + 2ω

(c, a) ω

(b).

The reader is invited, as an exercise, to check the following result:

PROPOSITION 17.24 The exterior product is

• (anti)symmetric: ω

∧ ω

= (−1)

∧ ω

for ω

∈ Λ

∗p

(E) and

∈ Λ

∗q

(E);

• distributive with respect to addition: ω ∧(ω

′

+ ω

′′

) = ω ∧ω

′

+ ω ∧ω

′′

;

• associative: (ω ∧ω

′

) ∧ω

′′

= ω ∧(ω

′

∧ω

′′

Example 17.25 Let ω and ω

′

be exterior 1-forms. Then using Deﬁnition 17.22 we get

ω ∧ ω

′

( x, y) = ω( x) ω

′

( y) −ω

′

( x) ω( y),

i.e. ω ∧ω

′

= ω ⊗ω

′

−ω

′

⊗ω as deﬁned in the preceding paragraph.

PROPOSITION 17.26 Let ω

, . . . , ω

, ω

p+1

, . . . , ω

p+q

be exter ior 1-forms. Th en

we have

(ω

∧···∧ω

) ∧(ω

p+1

∧···∧ω

p+q

) = ω

∧···∧ω

∧ω

p+1

∧···∧ω

p+q

Proof. This formula follows from the deﬁnition of the exterior product of 1-forms

and the compatibility with t h e general deﬁnition.

Differential forms on a vector space 469

DEFINITION 17.27 The e xterior algebra on R

is the vector space

∗

)

def

k=0

∗k

The exterior product is extended to the exterior algebra by distributivity.

17.2

Differential forms on a vector space

17.2.a Deﬁnition

As before, in all this section, E = R

is a real vector space of ﬁnite dimen-

sion n.

DEFINITION 17.28 Let p be an integer or +∞. A differential form of

degree k on RR

of CC

-class, or differential k-form, is a map of C

-class

ω : R

−→ Λ

∗k

x 7−→ ω( x).

For any point x in R

, ω( x) is therefore an exterior k-form on R

What is the precise meaning of “C

-class” here? Any differential k-form

may be written

ω( x) = ω

,...,i

( x) dx

∧···∧dx

where x 7→ ω

,...,i

( x) are real-valued functions on R

. Then x 7→ ω( x) is

of C

-class if and only if the





functions x 7→ ω

,...,i

( x) are.

Before going farther, here are some examples of differential k-forms, with

E = R

• For k = 0: a differential form of degree 0 is none ot her than a function

on E, of C

-class, with values in R.

• For k = 1: let ω

be a dif ferential 1-form. Then there exists an n-tuple

( f

, . . . , f

) of functions of C

-class such that

= f

+ ···+ f

470 Differ en tial forms

• For k = 2: let ω

be a d ifferential 2-form. There exist n(n − 1)/2

functions ( f

i j

)

1¶i< j¶n

(of C

-class) such that

= f

∧dx

+ f

∧dx

+ ···+ f

∧dx

+ ···+ f

n−1,n

n−1

∧dx

1¶i< j¶n

i j

∧dx

= f

i j

∧dx

Note again the use of the convention of partial summation of repeated

indices.

Remark 17.29 Warning: in what follows, as in most books, the point x in space

where the extorior form is e valuated will be omitted from the notation, and we will

often write ω instead of ω( x), when no ambiguity exists. Thus, for a 2-form ω,

ω(ξξ

ξξ

,ξξ

ξξ

) will be the real number obtained by applying the exterior 2-form ω( x) to

the vectors ξξ

ξξ

and ξξ

ξξ

in E.

17.2.b Exterior derivative

DEFINITION 17.30 Let ω = ω

,...,i

( x) dx

∧ ···∧ dx

be a differential k-

form. Since the functions x 7→ ω

,...,i

( x) are differentiable, the differentials

dω

,...,i

( x) are exterior 1-forms. The exterior derivative of the k-form ω is

the differential form of degree k + 1 given by

dω = dω

,...,i

( x) ∧dx

∧···∧ dx

Example 17.31 Let ω = ω

dx + ω

dy + ω

dz be a differential 1-form on R

. Then we have

dω

∂ω

∂ x

dx +

∂ω

∂ y

dy +

∂ω

∂ z

dz,

and similarly for dω

and dω

. It follows that

dω =



∂ω

∂ x

dx +

∂ω

∂ y

dy +

∂ω

∂ z



∧dx +



∂ω

∂ x

dx +

∂ω

∂ y

dy +

∂ω

∂ z



∧dy



∂ω

∂ x

dx +

∂ω

∂ y

dy +

∂ω

∂ z



∧dz



∂ω

∂ x

−

∂ω

∂ y



dx ∧dy +



∂ω

∂ y

−

∂ω

∂ z



dy ∧dz +



∂ω

∂ z

−

∂ω

∂ x



dz ∧dx.

THEOREM 17.32 Let ω be a differential k-form of c lass at least C

. Then ddω ≡

0, in other words we have identically d

≡ 0.

Proof. We have by deﬁnition dω = dω

,...,i

( x) ∧dx

∧··· ∧dx

, with

dω

,...,i

∂ω

,...,i

∂ x

Integration of differential forms 471

It follows that

ddω =

∂

,...,i

∂ x

∧dx

∧··· ∧dx

But since the form ω is of C

-class, Schwarz’s theorem on the commutativity of second

partial derivatives implies t h at

∂

,...,i

∂ x

∂

,...,i

∂ x

and dx

∧dx

= −dx

∧dx

which shows that the preceding expression is identically zero.

17.3

Integration of differential forms

The general theory of integration of d ifferential forms is outside the scope

of this book. Hence the results we present a re only intended to give an

idea of the ﬂavor of this subject. The interested reader can turn for more

details to a book of differential geometry such as Carta n [17], Nakahara [68],

or Arnold [9]. Before discussing the general case, we mention two special

situations: the integration of n-forms on R

and of 1-forms on any R

DEFINITION 17.33 (Integral of a differential n-form) Let Ω be a sufﬁciently

regular domain in R

(so Ω is an n-dimensional “volume”) and let ω be a

differential n-form. Then there exists a function f such that we can wr ite

ω( x) = f ( x) dx

∧···∧dx

and the integral of ω on Ω is deﬁned to be

Ω

def

Ω

f ( x) dx

···dx

Be careful that this special case of integration of forms of degree n on a

space of the s ame dimension is indeed very special.

The next special case is that of differential 1-forms. Before trying to in-

tegrate a 1-form ω on a domain, we must look again at what is the nature

of ω. At a point x ∈ R

, the value ω( x) of ω is an exterior 1-form, that

is, simply a linear form which as sociates a real number to a vector. Where

may we naturally ﬁnd vectors? In differential geometry, vectors are deﬁned in

the same manner that velocity vectors are deﬁned in physics: one chooses a

curve t 7→ γγ

γγ(t) going t hrough x and one looks a t what is t he “velocity” of

the moving point γγ

γγ(t) when it passes at the point x.

So, consider a path γγ

γγ : [0, 1] → R

, t 7→ γγ

γγ(t). Then for any t ∈ [0, 1],

the velocity vector γγ

γγ

′

(t) = dγγ

γγ/dt provides, after being “fed” to ω



γγ

γγ(t)



(the

472 Differential forms

George Stokes (1819—1903), English p hysi cist and mathematician,

discovered the concept of uniform convergence, but his most impor-

tant contributions were in physics, in partic u l ar in ﬂuid mechan-

ics, where he investigated the differential equations introduced by

Navier that now bear their names. The formula linking the circu-

lation of a vector ﬁeld and the ﬂux of its curl is also due to him.

exterior form given by evaluating ω at the point γγ

γγ(t)) a real number, namely,

g(t) = ω



γγ

γγ(t)



γγ

′

(t)



; this real number may be integrated along the path,

and the resulting integral will give the deﬁnition of the integral of ω on the

path γγ

γγ:

DEFINITION 17.34 (Integral of a differential 1-form on a path)

Let γγ

γγ be a path and ω = f

a differential 1-form. T he integral of ω on γγ

γγ

is deﬁned as the integral

γγ

ω =

γγ



γγ

γγ(t)



dt,

where the path is parameterized by

γγ

γγ : [0, 1] −→ R

t 7−→ γγ

γγ(t) =



(t), . . . , x

(t)



Example 17.35 In the plane R

, with ω = f dx + g dy, we obtain

ω =

f dx + g dy =



γ(s)



ds +



γ(s)



ds.

This recovers the notion already seen (in complex analysis) of contour integration.

In the general case, h ow do we integrate a differential k- form? Most impor-

tantly, on what domain should one perform such an integral? Take a 2-form,

for instance. At every point of this integration domain, we need two vectors.

This is easily given if this domain is a smooth surface, for t he t wo vectors can

then be a basis of the tangent plane. A differential 2-form must be integrated along

a 2-dimensional surface.

As an example, we consider a 2-form ω

in the space R

. We denote it

ω = f

dy ∧ dz + f

dz ∧dx + f

dx ∧dy.

(Note that we have changed conventions s omewhat, using dz ∧dx instead of

dx ∧dz; this only amounts to changing the sign of f

.) Let S be an oriented

Integration of differential forms 473

surface. At any point x in S , we can perform the orthogonal projection

of th e basis vectors of R

to the tangent plane of S at x (this is a linear

operation). Hence three vectors u

, u

, and u

are obtained, attach ed to each

point x ∈ S ; they are not linearly independent since t hey belong to the two-

dimensional tangent plane to the surface. Consider then f

dx ∧dy( u

, u

which is a real number. This may be integrated over the whole surface. Note

that dx ∧dy( u

, u

) = n · e

, and therefore we obtain

dy ∧dz + f

dz ∧dx + f

dx ∧dy

dx + f

dy + f

dz =

f · n,

where n is the normal vector to the surface. The ﬁrst integral involves the

differential form ω, whereas the integral on the second line is an integral of

functions.

This is easy to memorize: it sufﬁces to replace dx ∧dy by dz and, similarly

dy ∧dz by dx and dz ∧dx by dy.

To generalize this result, it is necessary to study the behavior of a differen-

tial form during a change of coordinates. This is done in sidebar 6. One shows

that a differential k-form may be integrated on a “surface” of d imension k,

that is, there appears a duality

between

• differential 1-forms and paths (curves);

• differential 2-forms and surfaces;

• differential 3-forms and volumes (of dimension 3) or three-dimensional

surfaces in R

;

• etc.

The following result is then particularly interesting:

THEOREM 17.36 (Stokes) Let Ω be a smooth (k + 1)-dimensional domain with k-

dimensional boundary ∂ Ω (possibly empty), and let ω be a differential k-form on E.

Then we have

∂Ω

ω =

Ω

dω.

This formula is associated with many names, including Newton, Leibniz, Gauss, Green,

Ostrogradski, Stokes, and Poincaré, but it is in general called the “Stokes formula.”

Example 17.37 The boundary of a path γ : [0, 1] → R

is made of only the points b = γ(1)

and a = γ(0) . Let us consider the case n = 1 and let ω be a differential 0-form, that is, a

The following is meant by “duality”: the meeting of a k-form and a k-surface gives a

number, just as the meeting of a linear form and a vector gives a number.

474 Differential forms

function ω = f : R → R. Then the Stokes formula means that we have

d f =

d f



γ(t)



·γ

′

(t) dt =

′

(s) ds = f (b ) − f (a),

which is of course known to the youngest children.

17.4

Poincar

e’s theorem

DEFINITION 17.38 (Exact, closed differential forms) A differential form ω is

closed if dω = 0.

A differential k- form ω is exact if there exists a differential (k − 1)-form

A such that ω = dA.

THEOREM 17.39 Any exact dif ferential form is closed.

Proof. Indeed, if ω is e xact, there exists a form A such that ω = dA, and we then

have dω = ddA = 0.

Note that an exact differential 1-form is none oth er t han a differential (of

a function). Indeed, if ω is an exact differential 1-form, there exists a 0-form,

that is, a function f , such that ω = d f .

THEOREM 17.40 (Schwarz) Let ω be a differential 1-form. If we write ω =

(x) dx

, then the following are equivalent

a) ω is closed (i.e., dω = 0) ;

b) we have

∂ f

∂ x

∂ f

∂ x

for any i, j ∈ {1, 2, 3}.

Proof. Assume ﬁrst that ω is closed. Then dω = 0 = ∂

∧dx

; since dx

∧

= dx

⊗dx

−dx

⊗dx

, this shows that ∂

= ∂

Conversely, ∂

= ∂

obviously implies that dω = 0.

Knowing whether the converse to Theorem 17.39 holds, that is, wheth er

any closed form is exact, turns out to be of paramount importance. The

answer depends in a cr ucial way on the shape and structure of the domain of

deﬁnition of the differential form.

DEFINITION 17.41 (Contractible open set) A contractible open set in R

an open subset which may be continuously deformed to a point. Hence, Ω is

contractible if and only if there exists a continuous map C : Ω ×[0, 1] → Ω

and a point p ∈ Ω such that

Poincaré’s theorem 475

Henri Poincaré (1 854—1912) is one the most universal geniuses

in the whole history of mathematics; he studied — and often

revolutionized — most areas of the mathematics of his time, and

also mechanics, astronomy, and philosophy. He created topology,

used this new tool to investigate differential equations, ﬂirted

with chaos, studied asymptotic expansions, and proved th at the

time series used in astronomy are of this type. He also gave a

major contribution to the three-body problem.

a) C (x, 0) = x for all x ∈ Ω;

b) C (x, 1) = p for all x ∈ Ω.

Example 17.42 A star-shaped open set is contractible. Here, an open set Ω is star-shaped if

and only if, for any x ∈ Ω, the line segment

[0, x]

def

λ x ; λ ∈ [0, 1]

is contained is Ω. In particular, R

is star-shaped and hence contractible. Here we can choose

p = 0 and the map C may be deﬁned by

∀x ∈ R

∀θ ∈ [0, 1] C (x, θ) = (1 −θ) x.

Example 17.43 The surface of a torus is not contractible.

\{0} is not contractible (although it is simply connected as soon as n ¾ 2).

THEOREM 17.44 (Poincar

e) Let Ω ⊂ R

be a contractible open set in R. Then

any closed differential form on Ω is exact.

In one important special case, a more general condition than contractibil-

ity is sufﬁcient to ensure that a closed form is ex act. Indeed, for differential

1-forms, we have

THEOREM 17.45 Let Ω ⊂ R

be a simply connected and connected open set.

Then any closed differential 1-form on Ω is exact.

Remark 17.46 Be careful that this theorem is only valid for differential 1-forms. In

general, it is not sufﬁcient for a closed differential form to be deﬁned on a simply

connected set in order for it to be exact. We will see below (see page 480) a simple

consequence of this for the ﬁeld created by a magnetic monopole.

Notice that a ny contractible subset is simply connected (it sufﬁces to look

at the image of a path γ under th e “contraction” C to see that it also contracts

to a single point), but the converse is not true.

476 Differential forms

Counterexample 17.47 Consider the differential 1-form

ω =

(x − y) dx + (x + y) dy

+ y

in R

. It is easy to check that dω = 0 on R



(0, 0)



, using Schwarz’s theorem, for instance.

However, ω is not exact. One may indeed integrate ω to get

f (x, y) =

log(x

+ y

) −arctan





+ C

but this expression is singular at y = 0 and does not admit any conti nuous continuation

on R



(0, 0)



. This is possible because R



(0, 0)



is not simply connected.

17.5

Relations with vector calculus:

gradient, divergence, curl

17.5.a Differential forms in dimension 3

Consider now the case of differential forms in three-dimensional space R

These are

• the 0-forms, that is, functions;

• the 1-forms, which are linear combinat ions of dx, dy, and dz;

• the 2-forms which are linear combinatons of dx ∧ dy, dy ∧ dz, and

dz ∧dx;

• the forms of degre 3, multiples of the volume form dx ∧ dy ∧dz.

Let ω be a differential 1-form, so that

ω = f

= f

dx + f

dy + f

dz.

Let now γ be a smooth closed path in R

. Then this path bounds an oriented

surface (oriented using the orientation of γ), denoted S . So we have γ = ∂S .

Since the exterior derivative of ω is given by

dω =



∂ f

∂ x

−

∂ f

∂ y



dx∧dy+



∂ f

∂ y

−

∂ f

∂ z



dy∧dz+



∂ f

∂ z

−

∂ f

∂ x



dz∧dx,

Stokes’s formula gives

dω =

ω,