CotangentBundleLineIntegrals(GRACLecture5)

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GENERAL RELATIVITY APPRECIATION CLUB

LECTURE 5, LINE INTEGRALS ON MANIFOLDSJASON PAYNE

Contents

1. Dualized Linear Algebra and Manifolds 11.1. The Dual Vector Space 21.2. The Cotangent Bundle 51.3. Covector Fields and Differentials 7

2. Integration of Smooth Covector Fields 102.1. Pullbacks 102.2. Line Integrals 11

2.3. Towards Multiple Integration 14Index 15References 16

1. Dualized Linear Algebra and Manifolds

In the previous lecture we successfully extended the notion of the derivative to the land ofmanifolds, which is surely a great achievement, but it is only half the story. Indeed, elementarycalculus is not merely comprised of differentiation, but a great amount of time is spent coveringintegrationas well. We can immediately see that in trying to extend this to manifolds we run

into precisely the same problem as before: ba

f(x) dx

is nota coordinate-independent expression we have the dx portion to deal with! In general,recall the following theorem from basic (multivariable) calculus:

Theorem(see [3]: Theorem 10.9, page 252). Leth: U Rn be a one-to-one mapping, whereU is an open subset ofRn, such that h = 0 for any p R

n. Furthermore, write (x1, , xn)for the coordinates inU and(y1, , yn) for the coordinates inh(U). Then for any continuousfunctionF : Rn R (which has compact support lying inh(E), but we wont worry so much

about those details here), we haveh(U)

f(y1, , yn) dy1 dyn=

U

f(h(x1, , xn)) | det(h)(x1, , xn)| dx1 dxn.

This theorem is saying that when we make a change of coordiantes h, the integration will bechanged by a factor of the Jacobian. So what can we do to rid ourselves of this coordiantedependence?

Although it may seem like there is nothing we can do, as we have not encountered anythinglikedx thus far, we invite the reader to make the following observation: derivatives and integralsare essentiallydualto each other they, in some sense, cancel each other out:

xi dxj = ij.This is not, by any means, a precise statement; however, the overall conceptual picture shouldremind someone relatively familiar with linear algebra of the construction of a dual vector space,

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General Relativity Appreciation Club Line Integrals on Manifolds

which begins by constructing a basis satisfying something remarkably similar to the above equal-ity. It is with that vague intuition that we hope to complete the following sort of diagram:

differentiation

consider derivations

?? integration

Tp(M) and Fdualize dual ofTp(M) and F

our only hope

With that in mind, let us begin to develop that particular aspect of linear algebra a bit, and seeif it helps us.

1.1. The Dual Vector Space. Apart from the tenuous motivation given above for studyingthese so-called dual vector spaces, let us pause for a moment to reflect on the following observa-tion: so far in the lectures we have encountered a number ofR-vector spaces:

C(M, N), F(M), Tgp (S), Tp(M), F(p)

Since these are showing up so often, it may well behoove us to use the full force of the theoryof vector spaces to aid us in our investigations. With both this, and our hope of recoveringintegration theory, in mind we begin by constructing this dual vector space associated with anarbitrary vector space V.

[Note: Although much of what we will say below makes perfect sense in the context of infinite-dimensinoal vector spaces, all vector spaces considered in these notes will be finite-dimensional.Furthermore, we will set dimR(V) =n as our default dimension.]

Definition (see [2]: page 272). A linear map : V R is called a covector on V. Thecollection of all covectors onVforms a vector space overR, which is denoted byV. This vectorspace is called thedual space to V.

Hopefully this wont appear to be entirely strange and unwarranted we have, in fact, alreadyseen some covectors in action. For example, a derivationX : C(M) R on a manifold Mat a point p is a specific type of covector on C(M). In other words, given an XTp(M), wehave that X(C(M)).

From here we will quickly develop some useful properties of dual vector spaces. To begin with,since this is a vector space, we should identify at least one nice basis to work with, if there areany. Fortunately, this is the case. Choose a basis{E1, , En} for Vand then define covectorsi :V R for each i= 1, , n by

i(Ej) =ij.

For those among the readers who are unfamiliar with linear algebra, the above requirement isenough to specify a linear map. Given any v=

ni=1viEi V, the linearity of

j tells us that

j(v) =j

ni=1

viEi

=

ni=1

vij(Ei) =vj,

so we know how j acts on all ofV. This leads us to the following important proposition.

Proposition (see [2]: Proposition 11.1, page 273). For any vector spaceV, {1, , n} formsa basis forV, called the dual basis. In particular, we have thatdimR(V

) =n.

Proof. First, suppose that f= n

i=1aii V is the zero functional. Then, upon applying it to

each of the vectors Ej we find that

0 =f(Ej) =ni=1

ai(Ej) =ni=1

aiij =aj.

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Hence {1, , n} are linearly independent. Moreover, let fV and define the numbers

fi=f(Ei).

Then for any v=n

j=1ajEj Vwe have that

n

i=1 fii (v) =n

i=1 fin

j=1 aji(Ej) =n

i=1 fin

j=1 ajij=

ni=1

fiai= f

ni=1

aiEi

=f(v);

hence fspan{i}.

This is great, as the dual basis is clearly a realization of the type of structure were hoping touse to construct integrals, but there is more to be said. Indeed, we saw that is wasnt just thevector space Tp(M) that was important for differentiation of smooth maps F : M N, butthe pushforward F : Tp(M) TF(p)(N) was also required. It stands to reason then that we

should not only dualize our notion of a vector space, but also of maps between vector spaces.Such a construction is actually quite natural and easy to define.

Definition(see [2]: page 273). LetV andWbe vector spaces, andf :V W be linear. Wedefine a map f :W V by

f(g) =g f.

f is called the dual mapto for the transpose off.

More explicitly, given an element v V , we have that f(v) W, and so we can plug it intoany covector g W, and this defines a covector f V. That is all the above is saying:

[f(g)](v) =g(f(v)).

This construction, as the second name given to it suggests, is actually very familiar. Those whoknow a little about linear algebra are invited to consider the following example.

Example. LetA: R3 R2 be the linear map represented (with respect to the standard basis)by the following2 3 matrix:

A=

1 3 12 5 4

.

The dual map A should go from(R2) to(R3), so note that these are given by

(R2) ={f : R2 R | f is linear}= {2 1 matrices}={1 2 matrices},

(R3) ={f : R3 R | f is linear}= {3 1 matrices}={1 3 matrices}.

Moreover, since the above proposition implies that vector spaces have the same dimension astheir dual vector spaces, they are isomorphic; henceA can be thought of asA : R2 R3, i.e.a3 2 matrix. Then the definition ofA applied to the basis vectors(1, 0), (0, 1) (R2) yields

A((1, 0)) = (1, 0) A= (1, 0)

1 3 12 5 4

= (1, 3, 1),

A((0, 1)) = (0, 1) A= (0, 1)

1 3 12 5 4

= (2, 5, 4).

On the other hand, writing

A =

a11 a12a21 a22a31 a32

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we have that

A

10

=

a11 a12a21 a22a31 a32

10

=

a11a21a31

and

A 01 = a11 a12a21 a22

a31 a32 01 = a12a22

a32 .

Thus, putting these together reveals that

A =

1 23 51 4

=AT.So, we find that the dual of a linear map corresponds to the transpose of its associated matrixrepresentation!

The dual map also satisifies a few properties that will be familiar to anyone who knows aboutthe transpose of a matrix.

Proposition (see[2]: Proposition 11.4, page 273-274). LetA: V W andB :W Y belinear maps between vector spaces. Then

(i) (B A) =A B.(ii) (idV)

=idV .

Proof. These both follow almost immediately from the definition of the dual map.

(i) Let fY, then we have that

(B A)(f) = (f (B A)) = ((f B) A)

=A(f B) =A(B(f)) = (A B)(f).

(ii) Consider fV. The definition of the dual yields

(idV)(f) =f idV =f=idV (f).

This whole process can, of course, be repeated ad infinitum: V is again a vector space, andso we can consider V := (V). It turns out that (at least in the finite dimensional case thatwe are considering here), this gives us nothing new. To see why, consider the map :V V

defined by

[(v)]() =(v)),i.e. is the map that sends a vector v V to a linear functional (v) on V which takes ina linear functional on V and simply evaluates it at v. (That was certainly a mouthful, so thereader is advised to go back and think about it some more.) It turns out that this map is anisomorphism between V and V:

Proposition (see [2], Proposition 11.8, page 274). For any finite-dimensional vector space V,the map :V V defined above is an isomorphism.

Proof. Note that, from one of the above propositions applied to both V and V, V and V havethe same dimension, and V and V have the same dimension; thus V and V have the same

dimension. For linear maps between vector spaces of the same dimension, a standard result inlinear algebra (see, for example. Theorem 9.5 in [3]) implies that injectivity, surjectivity, andbijectivity are all equivalent; hence we only need to show one of these.

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Suppose that v V and (v) = 0. Writing {E1, , En} and {1, , n} for the bases ofV

and V, respectively, we expand v in terms of the basis as v=n

i=1aiEi and find that

0 = [(v)](j) =j(v) =

ni=1

aij(Ei) =aj;

hence v 0. This means that ker = {0}, and as we saw in the previous set of notes, this

implies that is injective.

Before leaving behind this algebraic digression, we want to leave the reader with one morethought from ([2], page 274): It is important to observe that although V is also isomorphic toV (for the simple reason that any two finite-dimensional vector spaces of the same dimensionare isomorphic), there is no canonical isomorphismV =V. The significance of this statementis that the above map provides an isomorphism between V and V that does not depend onthe choice of basis, but any isomorphismbetween V and V will dependon the choice of basis!So there is definitely something significant differentiatingthese two objects. We leave it to thereader to ponder this a bit more.

Hopefully these new structures will help us because, as Einstein once said:

Figure 1: Truly one of the greats

1.2. The Cotangent Bundle. Now that we have developed all of this algebraic machinery,we may as well begin to put it to use. Since differentiation is what we wish to dualize, andthat comes from understanding Tp(M) and maps between tangent spaces, lets carry out thediscussion in the previous section for these objects.

Definition (see [2]: page 275). Let M be a smooth manifold and p M. The dual space(Tp(M))

of Tp(M), denoted T

p (M), is called the cotangent space at p. The elements ofTp (M) are called the tangent covectors at p.

There is not much else to say about these spaces individually, as we would simply be copyingthe results from the above section; however, it may be useful to look at what tangent covectors

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look like locally. Consider a chart h : U U for some p U Mwith smooth coordinates(x1, , xn). We saw in the previous set of notes that the tangent vectors

x1

p

, ,

xn

p

form a basis for Tp(M). Dualizing this basis, we find a basis for T

p (M) given by

1|p, , n|p,

where

i|p

xj

p

=ij.

Just as in the coordinate computations for tangent vectors, this allows us to take any tangentcovector Tp (M), define the numbers

i=

xi

p

,

and write=

ni=1

ii|p.

Moreover, if we were to choose another set of local coordinates(x1,xn), then the transformationproperty we derived earlier for the standard basis tangent vectors

xi

p

=xj

xi(p)

xj

p

reveals that (using the linearity of)

i = xi p = xj

xi (p)

xj p = xjxi (p) xj p = x

j

xi (p) j.In other words, tangent covectors transform in precisely the same way as the coordinate partialderivatives this is why they were historically refered to as covariant vectors (varying with the

partials), and tangent vectors (whose transformation law has the roles ofXi andXi switched)were calledcontravariant vectors(varyingagainstthe partials). Unfortunately, these names cannow make things confusing for readers who are not careful, as the words covariant and con-travarianthave an entirely different meaning in many modern contexts (perhaps most notablyincategory theory).

Now that we are (hopefully) a little more comfortable with tangent covectors they are simplythe covariant vectors well-known from many elementary subjects in physics let us move backtoward the more general point of view. We have defined cotangent spaces, and so a naturalplace to round out this section is precisely where we went in the theory of the tangent space:lets look at all these cotangent spaces as a single object.

Definition(see [2]: page 276). LetM be a smooth manifold. Then thecotangent bundleofMis defined as the disjoint union

T(M) =pM

Tp (M).

Associated with this bundle is the natural projection map :T(M)Mdefined by

(p, p) =p.

Just as was the case with the tangent bundle, this apparently strange and abstract space isactually rather well behaved:

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Theorem (see [2]: Proposition 11.9, page 276). Let M and T(M) be as above, with M n-dimensional. ThenT(M)is a(2n)-dimensional manifold, and the projection map :T(M)M is smooth.

From here, we continue mimicing our discussion of tangent spaces and consider smooth sectionsof this newvector bundle. Who knows perhaps this is where something incredibly will becomeknown.

Figure 2: Anyone reading these notes should have knownSaganwould show up...

1.3. Covector Fields and Differentials. This is where the really interesting applications ofdual spaces begin to take shape. We begin by continuing the previous program, and mimicingwhat we did with the tangent bundle. First, we have the following crucial definition.

Definition (see[2]: page 277). LetMbe a smooth manifold, andT(M) its cotangent bundle.Then a smooth section ofT(M), i.e. a map : MT(M) such that

=idM,

is called acovector fieldonM. The set of all covector fields onMis denoted byT(M), andit is aC(M)-module.

Just as was the case for vector fields, for any covector field T(M), the condition =idMensures that does pick out an element ofT

p (M)for each p M, and not something else:

p= idM(p) = ( )(p) =(q, q) =q.

Another name for these objects is actually differential 1-forms. So, although Im not makingit too explciit at this point, we have actually begun our discussion of differential forms onmanifolds. There is, however, one distinct difference between these objects and the vector fieldswe encountered in the previous lecture: vector fields can easily be understood geometrically and even visualized as maps that stick a tangent vector on each point of the tangent space.

Figure 3: A vector field, Figure 8.1 in[2], page 175

What is the corresponding geometric understanding of covector fields, if any?

Well, a covector p T

p (M) is a linear functional on Tp(M), and as such it is completelydetermined by two pieces of information: ker p and {v Tp(M) | p(v) = 1}. The first isa codimension-1 subspace of Tp(M), in other words a hyperplane, and the latter is an affine

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hyperplane parallel to its kernel. Thus, a covector field can be though of as assigning a pointp Mto a pair of affine hyperplanes in Tp(M): one through the origin (the kernel), and anotherparallel to it. Note that these vary continuously from point to point. Visually, we have:

Figure 4: A covector field, from [2], page 280

Great. So we should now have a basic and mildly intuitive understanding of the cotangentbundle and covector fields, so lets determine how these objects are actually useful. In order tosave the day, we will appeal to a special case of a covector field that involves derivations.

Definition (see[2]: page 281). LetMbe a smooth manifold andfC

(M). Then we definea covector fielddf onM, called the differential of f, bydf(p) = (p, dfp), wheredfp is definedby

dfp(Xp) =Xp(f).

We immediately get the following proposition:

Proposition (see [2]: Proposition 11.18, page 281). The differential df of a smooth functionf F(M) is a smooth covector field.

As is usual, we can get a handle on what exactly this thing is by looking at it locally. Letp Mandh : UU be a chart atp, with local coordiantes(x1, , xn). We saw earlier that

there is a basis of the cotangent space T

p (M)given by{1

|p, , n

|p}, and so we can write outdfp in terms of this basis, say:

dfp=ni=1

aii|p.

Evaluating this at one of the coordiante partial derivatives, we find that

f

xj(p) =dfp

xj

p

=

ni=1

aii|p

xj

p

=

ni=1

aiij =aj;

thus

dfp =n

i=1f

xi(p)i|p.

In particular, applying this to the coordinate functions xj(x1, , xn) =xj, we find that

dxj|p =ni=1

xj

xi(p)i|p =

ni=1

iji|p =

j |p.

This is interesting, since it reveals that the coordinate covector field j |p is nothing more thanthe differential dxj, and we find that the differential can be written as

dfp=n

i=1f

xi(p)dxi|p.

Passing from tangent covectors to covector fields, this means: df=

ni=1

f

xidxi.

To help solidify this concept, lets work through a simple example.

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Example (see [2]: Example 11.19, page 282). Consider the map f : R2 R defined by

f(x, y) =x2y cos(x).

Applying the above to this function we find that

df=2

i=1f

xi

dxi = f

x

dx+f

y

dy

= (2xy cos(x) x2y sin(x))dx+ (x2 cos(x))dy.

See? How harmless was that? Now that we are a little more comfortable with the notation,we want to point out that this is a REALLY important expression for out purposes; however,let us take a moment to prove a few properties of the differential before discussing what all thismeans.

Proposition (see [2]: Proposition 11.20, page 282). Let M be a smooth manifold and f, g F(M). Then

(i) The operatord isR-linear: d(f+g) = df+ dg for all, R.

(ii) Leibniz rule: d(f g) =f dg+g df.(iii) Quotient rule: On any set whereg = 0, we have that

d

f

g

=

g df f dg

g2 .

(iv) Chain rule: IfIm(f) I, whereI is some interval inR, andh: I R is a smoothfunction, thend(h f) = (h f) df.

(v) Iffconstant, thendf= 0.

Proof. All of these follow from the fact that dfis locally just a sum of partial derivatives, whichsatisfy the above properties; hence d does as well.

With that out of the way, what does this all mean? Well, remember that our lecture begantoday with asking how one can define integration on manifolds, and we found that it was preciselythe dx portion of

f dx which needed to be reimagined without coordinates the expression

fordfderived above shows us that we have, in fact, uncovered what the coordinate-independentversion ofdxshould be: it is a covector field! With this in mind, let us begin out long journey intothe development of integration theory on manifolds. Just as was the case of the derivative, weneed to study maps between structures like this, and so we now apply the dual map constructionto our present situation in order to see what were messing with.

Figure 5: The Milky Way and the Aurora Borealis, because.... space.

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2. Integration of Smooth Covector Fields

2.1. Pullbacks. We begin this discussion by thinking back to what happened while generaliz-ing differentiation: we found that the appropriate understanding of the derivative of a mapF : M N between smooth manifolds at p M was a map F : Tp(M) TF(p)(N).Moreover, we noticed at the beginning of these notes that integration is essentially the dual ofdifferentiation, so we begin with a rather natural definition: lets dualize the pushforward.

Definition(see [2]: page 284). LetF : M N be a smooth map between smooth manifoldsandp M. We can take the pushforward ofF, F : Tp(M)TF(p)(N), and consider its dualmap

(F) :TF(p)(N)T

p (M).

This map, to be denoted simply by F from now on, is called the pullback of F. Explicitly,given TF(p)(N) andXp Tp(M), this map is defined as

(F)(Xp) =(FXp).

One interesting property of this construction is that, unlike the pushforward case in whichonly diffeomorphisms were ensured to pushforward vector fields to vector fields, the pullback ofa arbitrary smooth covector field is alwaysa covector field as well. Indeed, given a smooth mapG: M Nand a smooth covector field on N, we have that a new covector field G canbe defined on M by

(G)p = G(G(p)).

Moreover, the pullback enjoys the following properties.

Proposition (see [2]: Proposition 11.25, page 285). Let G : M N be a smooth map,f F(M), and T(N). Then

(i) Gdf=d(f G).(ii) G(f ) = (f G)G.

Proof. Both of these, as has been the case for nearly everything proposition thus far, followstraight from the definitions.

(i) Given p M, since (Gdf)p is a covector on M, and so lets see what its action is on atangent vector XpTp(M):

(Gdf)p(Xp) =G(dfG(p))(Xp) =dfG(p)(GXp)

= (GXp)(f) =Xp(f G)

=d(f G)p(Xp),

as desired.(ii) Once again, we compute:

[G(f )]p=G((f )G(p)) =G

(f(G(p)G(p))

=f(G(p))G(G(p)) = (f G)(p)(G)p.

Since p was arbitrary, this means that

G(f ) = (f G)G.

Also, we should at least note the following proposition which ensures that this new covectorfield is, in fact, smooth.

Proposition (see [2]: Proposition 11.26, page 285). Let G : M Nbe a smooth map and T(N). ThenG T(M).

All of this makes for a rather abstract and high falutin discussion, so lets bring it back downto earth with a basic example.

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Example (see [2]: Example 11.27, page 286). Consider the map G: R3 R2 defined by

G(x , y , z ) = (x2y, y sin(z)).

We will use(u, v) for the coordinates inR, i.e. u = x2y and v = y sin(z). In addition to this,let T(R2) be defined by

= u dv+v du.

Since this is a covector field on the range ofG, we can pull it back alongG to obtain a covectorfieldG T(R3) onR3. By definition of the pullback, we have that

G=G(u dv+v du) = (u G)d(v G) + (v G)d(u G)

= (x2y)d(y sin(z)) + (y sin(z))d(x2y)

= (x2y) (0 dx+ sin(z) dy+y cos(z) dz) + (y sin(z)) (2xy dx+x2 dy+ 0 dz)

= 2xy2 sin(z) dx+ 2x2y sin(z) dy+x2y2 cos(z) dz.

Ok, that was relatively painless, btu what can we do with the pullback? As it turns out, justas the pushforward allowed us to generalize differentiation, with the pullback we are finally able

to generalize integration somewhat. Alright... (insert witty segue here), therefore

Figure 6: But 7 goes into 28 four times! Uhhh... this is a magic7!

2.2. Line Integrals. With the notion of pullbacks of covector fields out of the way, let us nowtry to construct a theory of integration. We will, as usual, start with the really simple casewe encountered in basic calculus. Let T([a, b]) for some a, b R. Denoting the standardcoordinate on R by t, we know that the dt spans T([a, b]), and so we must have t = f(t) dtfor each t R and some f F([a, b]). This should remind you of the integrands we saw all thetime in elementary calculus, and so by analogy, we define

[a,b] = b

af(t) dt.

As [2] points out, this might seems like nothing more than a trick of notation, i.e. wemight not be saying anything of any import here. This is, in fact, not the case, as the followingproposition demostrates.

Proposition(see [2]: Proposition 11.31, page 288). (Diffeomorphism Invariance of the Integral)Let T([a, b]) and: [c, d][a, b] be an increasing diffeomorphism. Then

[c,d]

=

[a,b]

.

This is important because it says that if we change coordinates, then the smooth covectorfield associated with in terms of the new coordinates yields precisely the same value whenintegrated, i.e. this notion of integration is completely coordinate-independent, and it recapturesthe integration we all know andlove! Now we can move into manifold context once again.

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Definition(see [2]: page 288-289). Just as we considered 1-dimensional domains in the above,here we consider a 1-dimensional submanifold of a smooth manifoldM. What this means is thatwe consider a smooth curve segment: [a, b]M(the word segment here means thatcanbe extended to being defined on some open interval containing[a, b], but this is not a detail thatwill feature prominantly in whats to come). Given T(M), we define the line integralof over to be

= [a,b]

.

More generally, given a piecewise smooth curve segment, we define

=k

i=1

[ai1,ai]

.

Now we can see the power of the pullback in this context. Since the pullbackof a covector fieldalways yields another covector field, we can use the fact that a curve in a manifold M has adomain like the case considered above thus we pullback our given to a covector field on [a, b]and use what we defined above to understand integration along this curve.

From here, let us collect some simple properties of the line integral.Proposition (see [2]: Proposition 11.34, page 289-290). Let M be a smooth manifold and: [a, b]M be a piecewise smooth curve segment. Given, 1, 2 T

(M), we have

(i) For anyc1, c2 R

(c11+c22) =c1

1+c2

2.

(ii) Ifconstant, then

= 0.

(iii) Ifa < b < c, then

=1

+2

,

where1=|[a,b] and2 = |[b,c].(iv) Given a smooth map F :MN, we have

F=

F

.

Exercise 1. Prove the above Proposition. (They all come straight from the definition.)

Without too much effort, one can show that these integrals are actually quite familiar:

Proposition (see [2]: Proposition 11.38, page 291). Given a piecewise smooth curve segment: [a, b]M and T(M), we have

=

ba

(t)((t)) dt,

where the latter integral is simply the Riemann-Lebesgue integral (see [1]).

Time to insert another example to see what this is all about.

Example (see [2]: Example 11.36, page 290). Consider the manifoldM\ {0} and let be thecovector field onMdefined by

= x dx y dx

x2 +y2 .

For the curve: [0, 2]Mdefined by

(t) = (cos(t), sin(t))12
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we can find the pullback of along in the same way we did in the previous example. Writingt for our coordinate on [0, 2] we have that

=(x )d(y ) (y )d(x )

(x )2 + (y )2

=(cos(t))d(sin(t)) (sin(t))d(cos(t))

(cos(t))2

+ (sin(t))2

=(cos(t)) (cos(t) dt) (sin(t)) ( sin(t) dt)

1= (cos2(t) + sin2(t)) dt)

=dt.

Thus, we have that

=

[0,2]

=

20

dt= 2.

All of this is simply meant to convince you that we have really achieved something here. We

hav managed to use this new notion of covector fields and pullbacks in order to define integrationover 1-dimensional submanifolds of a smooth manifold M. Before beginning to tackle the moregeneral problem of multiple integration, we just want to briefly mention two other importantproperties possessed by this new notion of integration on M.

Theorem(see [2]: Proposition 11.37, page 290, and Theorem 11.39, page 291). SupposeM isa smooth manifold and : [a, b]Mbe a piecewise smooth curve segment.

(i) (Parameterization Invariance of the Line Integral) Let T(M). Then for any (for-ward) reparameterization

of, we have

= .(ii) (Fundamental Theorem for Line Integrals) Givenf F(M), we have

df=f((b)) f((a)).

With that out of the way (for proofs, see [2]), we now pose a natural question: what aboutintegration over higher-dimensional submanifolds?

Figure 7: xkcd#1256

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2.3. Towards Multiple Integration. Taking a look back at what we have done, we have seenthat a tangent covector pT

p (M) has the following properties:

p allows us to associate a real number p(Xp) with a tangent vector Xp Tp(M);

p(Xp+X

p) =p(Xp) +p(X

p);

p(Xp) =p(Xp), i.e. p scales in the same way as Xp;

p(0) = 0, i.e. p treats lower-dimensional objects (tangent vectors are, in some sense,one-dimensional, but the zero vector is more like a point) trivially.

The key observation at this point is that these properties are precisely what you would expecta (signed) length measure to do! Indeed, given two one-dimensional objects: their length isthe sum of the lengths of each object; if you stretch or shrink a one-dimensional object, itslength stretchs or shrinks in the same way; and the length of a zero-dimensional object is zero.Perhaps it is not so surprising, then, that this led us to an understanding of integration overone-dimensional submanifolds: dx, in elementary calculus, is a (signed) length measure as well!This suggests that if we want a theory of multiple integrals on M, then we should study anobject that is some sort of (signed) volume measure. Such an object should satisfy entirelyanalogous properties:

p should assign k tangent vectors X1, , Xk TP(M)a real number p(X1, , Xk);

p(X1, , Xi+X

i, , Xk) = p(X1, , Xi, , Xk) + p(X1, , X

i, , Xk);

p(X1, , Xi, , Xk) = p(X1, , Xk), i.e. p scales in the same way as Xi foreach i;

p(X1, , Xk) = 0 ifXi and Xj are linearly dependent for any i and j, i.e. p treatslower-dimensional objects (anym-dimensional subspace withm < k) trivially.

It turns out that the above describes a mathematical object known as an alternating multilinearmap from Tp(M) Tp(M) R (with k-many copies ofTp(M)), and so in lecture 7 wewill pick up with these objects. But, for now, that is all thats going on!

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Index

T(M),7

alternating multilinear map,14

canonical isomorphism between V and V,4Chain rule,9

contravariant vectors,6Cosmos, Ep.13,7cotangent bundle,6cotangent space,5covariant vectors,6covariant/contravariant functors,6covector,2covector field,7

Dazed and Confused,6differential 1-forms,7differential of a smooth function, 8

coordinate representation,8dual basis,2dual map,3dual space,2

Einstein vs. Hawking,9

Fundamental Theorem for Line Integrals,13

Leibniz rulefor a differential, 9

length measure,14line integral

Euclidean case,11

manifold case,12

matrix transpose,3

One of these things,5

Parameterization Invariance of the Line Integral,13pullback,10pushforward,3

quotient rule,9

Rick roll,12Riemann-Lebesgue integral,12

smooth curve segment,12piecewise,12

tangent covectorlocal coordinate representation,6transformation law,6

tangent covectors,5transpose,3

vector bundle,7volume measure,14

What is love?,11Whats Going On?,14

Youre our only hope,2

15


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References

[1] G.B. Folland. Real Analysis: Modern Techniques and Their Applications. John Wiley &Sons, 2nd edition, 1999.

[2] J.M. Lee. Introduction to Smooth Manifolds. Graduate Texts in Mathematics, 218. Springer,2nd edition, 2012.

[3] W. Rudin. Priniciples of Mathematical Analysis. International Series in Pure and Applied

Mathematics. McGraw-Hill, 3rd edition, 1976.[4] N. Straumann. General Relativty. Graduate Texts in Physics. Springer, 2nd edition, 2013.

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http://www.amazon.com/Real-Analysis-Techniques-Applications-Mathematics/dp/0471317160http://www.ebook3000.com/-Introduction-to-Smooth-Manifolds--by-John-M--Lee_175180.htmlhttp://www.ebook3000.com/-Introduction-to-Smooth-Manifolds--by-John-M--Lee_175180.htmlhttp://www.math.boun.edu.tr/instructors/ozturk/eskiders/guz12m331/rud.pdfhttp://www.math.boun.edu.tr/instructors/ozturk/eskiders/guz12m331/rud.pdfhttp://link.springer.com/book/10.1007/978-94-007-5410-2/page/1http://link.springer.com/book/10.1007/978-94-007-5410-2/page/1http://www.math.boun.edu.tr/instructors/ozturk/eskiders/guz12m331/rud.pdfhttp://www.ebook3000.com/-Introduction-to-Smooth-Manifolds--by-John-M--Lee_175180.htmlhttp://www.amazon.com/Real-Analysis-Techniques-Applications-Mathematics/dp/0471317160

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