ε i (E j )=δj i = 0, if i j, form a basis for V, called the dual basis to (E i ). Therefore, dim V =dim V.

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1 Covectors Definition. Let V be a finite-dimensional vector sace. A covector on V is real-valued linear functional on V, that is, a linear ma ω : V R. The sace of all covectors on V is itself a real vector sace under the obvious oerations of airwise addition and scalar multilication. It is denoted by V and called the dual sace to V. Proosition 6.1. Let V be a finite-dimensional vector sace. If (E 1,,E n ) is any basis for V, then the covectors (ε 1,,ε n ) defined by { 1, if i = j, ε i (E j )=δj i = 0, if i j, form a basis for V, called the dual basis to (E i ). Therefore, dim V =dim V. Examle. If(e i ) denote the standard basis for R n, we denote the dual basus by (e 1,,e n ) (note the uer indices), and call it the standard dual basis. The basis vectors are the linear functions from R n to R given by e j (v) =e j (v 1,,v n )=v j. In other words, e j is just the linear functional that icks out the jth comonent of a vector. In matrix notation, a linear ma from R n to R is reresented by a 1 n matrix; i.e. a row matrix. The basis covectors can therefore also be thought of as the linear functionals reresented by the row matrix e 1 =(1 0 0),,e n =(0 0 1). If (E i ) is a basis for V and (ε j ) is its dual basis, then for any matrix X = X i E i V,wehave ε j (X) =X i ε j (E i )=X i δ j i = Xj. Thus as in the case of R n, ε j icks out the jth comonent of a vector w.r.t. the basis (E i ). More generally, Proosition 6.1 shows that we can exress an arbitrary covector ω V in terms of the dual basis as (6.1) ω = ω j ε j, where the comonents ω j are determined by (6.2) ω j = ω(e j ). We will write basis covectors with uer indices, and comonents of a covector with lower indices, because this hels to ensure that mathematically meaningful exressions such as (6.1) will always follow our index conventions: Any index that is to be summed over in a given term aears exactly twice, once as a subscrit and once as a suerscrit. Tyeset by AMS-TEX 1

2 2 Proosition 6.2. The dual ma satisfies the following roerties. (a) (A B) = B A. (b) (Id V ) ; V V is the identity ma of V. Aart from the fact that dim V =dim V, the second imortant fact is the following characterization of the second dual sace V =(V ). For each vector sace V there is a natural basis-indeendent ma ξ : V V, defined as follows. For each vector X V, define a linear functional ξ(x) :V R by ξ(x)(ω) =ω(x), ω V. Proosition 6.4. For a finite-dimensional vector sace V, the ma ξ : N V is an isomorhism. Proof. Since V and V have the same dimension, it suffices to claim: ξ is injective. Indeed, suose X V is not zero. Extend X to a basis (E 1,E 2,,E n ) for V, X = E 1 and let (ε 1,,ε n ) denote the dual basis for V. Then ξ(x)(ε 1 )=ε 1 (X) =ε 1 (E 1 )=1 0, so ξ(x) 0. The receeding roosition shows that when V is finite-dimensional, we can unambiguously identify V with V itself, because the ma ξ is canonically defined, without reference to any basis. It is imortant to observe that although V is also isomorhic to V, there is no canonical isomorhism V = V. Because of Proosition 6.4, the real number ω(x) obtained by alying a covector ω to a vector X is sometimes denoted by either of the more symmetric-looking notations ω, X, X, ω ; both exressions can be thought of either as the action of the covector ω V on the vector X V, or as the action of the covector ξ(x) V on the element ω V. Whenever one of the arguments is a vector and the other a covector, the notation ω, X is always to be interreted as the actual airing between vectors and covectors, not as an inner roduct.

3 Tangent Covectors on Manifolds Definition. Let M be a smooth manifold. For each M, we define the cotagent sace at, denoted by T M, to be the dual sace to T M: T M =(T M). Elements of T M are called tangent covectors at. If (x i ) are smooth local coordinates on an oen subset U M, then U, the coordinate basis ( x ) give rise to a dual basis for T i M, which we denote for the moment by (λ i ). Any covector ω T M can thus be written uniquely as ω = ω iλ i, where ( ω i = ω x i ). 3 Let ( x j ) be another set of smooth local coordinates whose domain contains, and let ( λ i ) denote the basis for T M dual to ( x ). j We can comute the comonents of the same covector ω w.r.t. the new coordinate system as follows. First recall that the coordinate vector fields transform as follows: (6.4) x i = xj x i () x j. Writing ω in both systems as ω = ω i λ i = ω j λ j, we can use (6.4) to comute the comonents ω i in terms of ω j : ( ) ( (6.5) ω i = ω x j x i = ω x i () ) x j = xj x i () ω j. In the early days of smooth manifold theory, before most of the abstract coordinate-free definition we are using were develoed, mathematicians tended to think of a tangent vector at a oint as an assignment of an n-tules (X 1,,X n ) and ( X 1,, X n ) assigned to two different systems (x i ) and ( x j ) were related by the transformation law: (6.6) Xj = xj x i ()Xi.

4 4 Similarly, a tangent covector was thought of as n-tule (ω 1,,ω n ) that transform, by virtue of (6.5), according to the following slightly different rule: (6.7) ω i = xj x i () ω j. Thus it becomes customary to call the tangent covectors covariant vectors because their comonents transform in the same way as ( vary with ) the coordinate artial derivatives, with the Jacobian matrix ( xj x ) multilying the objects i associated with the new coordinates ( x j ) to obtain those associated with the old coordinates (x i ). Analogously, tangent vectors were called contravariant vectors, because their comonents transform in the oosite way.

5 5 The Cotangent Bundle Definition. The disjoint union T M = T M M is called the cotangent bundle of M. It has a natural rojection ma π : T M M sending ω T M to M. As above, given any smooth local coordinates (x i )onu M, for each U we denote the basis for T M dual to x by (λ i ). i This defines n mas λ 1,,λ n : U T M, called coordinate covector fields. Proosition 6.5. Let M be a smooth manifold and let T M be its cotangent bundle. With the standard rojection ma and the natural vector sace structure on each fiber, T M has a unique smooth manifold structure making it into a rankn vector bundle over M for which all coordinate covector fields are smooth local sections. Proof. Given a smooth chart (U, ϕ) onm, with coordinate functions (x i ), define Φ:π 1 (U) U R n by Φ(ξ i λ i )=(, (ξ 1,,ξ n )), where λ i is the ith coordinate covector field associated with (x i ). Suose (Ũ, ϕ) is another smooth chart on M, with coordinate functions ( xi ), and let Φ :π 1 (Ũ) Ũ Rn be defined analogously. On π 1 (U V ), it follows from (6.5) that Φ Φ 1 (, ( ξ 1,, ξ n )) = ( ( )) x j, x 1 () ξ j,, xj x n () ξ j. The GL(n, R)-valued function ( xj x )() is smooth, so it follows that T M has a i smooth structure making it into a smooth vector bundle for which the ma Φ are smooth local trivilizations. As in the case of the tangent bundle, smooth local coordinates for M yield smooth local coordinates for its cotangent bundle. If (x i ) are smooth coordinates on an oen set U M, then the ma π 1 (U) to R 2n given by ξ i λ i i (x 1 (),,x n (),ξ 1,,ξ n ) is a smooth coordinate chart for T M. We will call (x i,ξ i ) the standard coordinates for T M associated with (x i ).

6 6 Definition. A section of T M is called a covector field or a (differential) 1- form. In any smooth local coordinates on an oen set U M, a covector field ω can be written in terms of the coordinate covector fields (λ i )asω = ω i λ i for n functions ω i : U R called the comonent functions of ω. They are characterized by ( ω i () =ω x i ). Lemma 6.6 (Smoothness Criteria for Covector Fields). Let M be a smooth manifold, and let ω : M T M be a rough section. (1) If ω = ω i λ i is the coordinate reresentation for ω in any smooth chart (U, x i ) for M, then ω is smooth iff its comonents functions are smooth. (2) ω is smooth iff for every vector field X on an oen subset U M, the function ω, X : U R defined by is smooth. ω, X () = ω,x = ω (X ) Definition. We denote the real vector sace of all smooth covector fields on M by T (M). As smooth sections of a vector bundle, elements of T (M) can be multilied by smooth real-valued function: If f C (M) and ω T (M), the covector field fω is defined by (6.8) (fω) = f()ω. Like the sace of smooth vector fields, T (M) is a module over C (M). Geometrically, we think of a vector field on M as a rule that attaches an arrow to each oint of M. What kind of geometric icture can we form of a covector field? The key idea is that a nonzero linear functional ω T M is comletely determined by two ieces of data: (1) its kernel, which is a codimension-1 linear subsace of T M (a hyerlane), and (2) the set of vectors X for which ω (X) = 1, which is an affine hyerlane arallel to the kernel. Thus you can visualize a covector field as defining a air of affine hyerlanes in each tangent sace, one through the origin and another arallel to it, and varying continuously from oint to oint. At oints where the covector field takes on the value zero, one of the hyerlanes goes off to infinity.

7 The Differential of a Function In elementary calculus, the gradient of a smooth real-valued function f on R n is defined as the vactor field whose comonents are the artial derivatives of f. In our notation, this would read grad f = n i=1 f x i x i. Unfortunately, in this form, the gradient does not make coordinate indeendent sense. Examle. Let f(x, y) =x 2 on R 2, and let X be the vctor field X = grad f =2x x. Comute the coordinate exression of X in olar coordinates of X in olar coordinates (on some oen set on which they are defined) and show that grad f is not equal to f r r + f θ θ. Although the first artial derivatives of a smooth function cannot be interreted in a coordinate-indeendent way as the comonents of a vector field, it turns out that they can be interreted as the comonents of a covector field. This is the most imortant alication of covector fields. Definition. Let f be a smooth real-valued function on a smooth manifold M. We define a covector field df, called the differential of f, by df (X )=X f, X T M. 7 Lemma 6.7. The differential of a smooth function is a smooth covector field. Proof. (1) It is straightforward to verify that at each oint M, df (X ) deends linearly on X, so that df is indeed a covector at. (2) To see that df is smooth, we use Lemma 6.6 (b). For any smooth vector field X on an oen subset U M, the function df, X is smooth because it is equal to Xf.

8 8 To see what df looks like more concretely, we need to comute its coordinate reresentations. Let (x i ) be smooth coordinates on an oen subset U M, and let (λ i ) be the corresonding coordinate coframe on U. Writing df in coordinates as df = A i ()λ i for some functions A i : U R, the definition of df imlies ( ) A i () =df x i = x i f = f x i (). This yields the following formula for the coordinate reresentation of df : (6.9) df = f x i ()λi. Thus the comonent functions of df in any smooth coordinate chart are the artial derivatives of f w.r.t. those coordinates. Because of this, we can think of df as an anologue of the classical gradient, reinterreted in a way that makes coordinate-indeendent sense on a manifold. If we aly (6.9) to the secial case in which f is one of the coordinate functions x j : U R, we obtain dx j = xj x i λi = δ j i λi = λ j. In other words, the coordinate covector field λ j is dx j. Therefore, (6.9) can be rewritten as df = f x i ()dxi, or as an equation between covector fields instead of covectors: (6.10) df = f x i dxi. In articular, in the 1-dimensional case, this reduces to df = df dx dx. Examle 6.8. If f(x, y) =x 2 y cos x on R 2, then df = (x2 y cos x) x dx + (x2 y cos x) dy y =(2xy cos x x 2 y sin x)dx +(x 2 cos x)dy.

9 It is imortant to observe that for a smooth real-valued function f : M R, we have now defined two different kinds of derivatives of f at a oint M. (1) The ushforward f is defined as a linear ma from T M to T f() R. (2) The differential df as a covector at ; i.e., a linear ma from T M to R. These are really the same at, once we take into account the canonical identification between R and its tangent sace at any oint; one easy way to see this is to note that both are reresented in coordinates by the row matrix whose comonents are the artial derivatives of f. Proosition 6.11 (Derivative of a Function Along a Curve). Suose M is a smoth manifold, γ : M R is a smooth curve, and f : M R is a smooth function. Then the derivative of the real-valued function f γ : R R is given by (f γ) (t) =df γ(t) (γ (t)). 9 Proof. Directly from the definition, for any t 0 J, df γ(t0)(γ (t 0 )) = γ (t 0 )f = ( d ) γ f dt t0 = d (f γ) dt t0 =(f γ) (t 0 ). If γ is smooth curve in M, we have two different meanings for the exression (f γ) (t). (1) f γ can be interreted as a smooth curve in R. Thus (f γ) (t) is its tangent vector at the oint f γ(t), an element of the tangent sace T f γ(t) R. This tangent vector is equal to f (γ (t)). (2) f γ can also be considered simly as a real-valued function of one real variable, and then (f γ) (t) is just its ordinary derivative. Proosition 6.11 shows that this derivative is equal to the real number df γ(t) (γ (t)).

10 10 Pullbacks Let F : M N be a smooth ma and M be arbitrary. The ushroward ma F : T M T F () N yields a dual linear ma F : TF () N T M, which is characterized by (F ω)(x) =ω(f X), ω T F () N, X T M. When we introduced the ushforward ma, we made a oint of noting that vector fields do not ushforward to vector fields, excet in the secial case of a diffeomorhism. The surrising thing about ullbacks is that smooth vector fields always ull back to smooth covector fields. Definition. Given a smoth ma G : M N and a smooth covector field ω on N, define a covector field G ω on M by (6.12) (G ω) = G (ω G() ). Observe that there is no ambiguity about what oint to ull back from, in contrast to the vector field case. Lemma Let G : M N be a smooth ma. Suose f C (N) and ω T (N). Then (6.13) (6.14) G df =d(f G); G (fω)=(f G)G ω. Proof. To rove (6.13), we let X T M be arbitrary and comute (G df ) (X )=(G (df G() ))(X ) (by (6.12)) =df G() (G X ) (by definition of G ) =(G X )f (by definition of df ) =X (f G) (by definition of G ) =d(f G) (X ) (by definition of d(f G)). Similarly, for (6.14) we comute (G (fω)) =(G (fω) G() ) (by (6.12)) =G (f(g())ω G() ) (by (6.8)) =f(g())g (ω G() ) (by linearity of G ) =f(g())(g ω) (by (6.12)) =((f G)G ω) (by (6.8)).

11 Proosition Suose G : M N is smooth, and let ω be a smooth covector field on N. Then G ω is a smooth vector field on M. Proof. Let M be arbitrary, and choose smooth coordinates (x i ) for M near and (y j ) for N near G(). Writing ω in coordinates as ω = ω j dy j for smooth functions ω j defined near G() and using Lemma 6.12 twice, we have the following comutation in a nbhd of : which is smooth. G ω = G (ω j dy j )=(ω j G)G dy j =(ω j G)d(y j G), In the couse of the receeding roof we derived the following formula for the ullback of a covector field w.r.t. smooth coordinates (x i ) on the domain and (y j ) on the range: (6.15) G ω = G (ω j dy j )=(ω j G)d(y j G) =(ω j G)dG j, where G j is the jth comonent function of G in these coordinates. In other words, to comute G ω, all we need to do is to substitute the comonent functions of G for the coordinate functions of N everywhere that aear in ω. Examle. Let G : R 3 R 2 be the ma given by and let ω T (R 2 ) be the covector field (u, v) =G(x, y, z) =(x 2 y, y sin z), ω = udv+ v du. According to (6.15), the ullback G ω is given by G ω =(u G)d(v G)+(v G)d(u G) =(x 2 )d(y sin z)+(ysin z)d(x 2 y) =x 2 y(sin zdy+ y cos zdz)+ysin z(2xy dx + x 2 dy) =2x 2 y sin zdx +2x 2 y sin zdy + x 2 y 2 cos zdz. Examle. Let (r, θ) be olar coordinates on the half-lane H = {(x, y) :x>0}. We can think of the change of coordinates (x, y) =(r cos θ, r sin θ) as the coordinate exression for the identity ma on H, but using (r, θ) as coordinates for the domain and (x, y) for the range. Then the ullback formula (6.15) tells us what we can comute the olar coordinate exression for a covector field simly by substituting x = r cos θ, y = sin θ and exanding. For examle, xdy ydx=id (xdy ydx) =r cos θd(r sin θ) r sin θd(r cos θ) =r cos θ(sin θdr+ r cos θdθ) r sin θ(cos θdr rsin θdθ) =(r cos θ sin θ r sin θ cos θ)dr +(r 2 cos 2 θ + r 2 sin 2 θ)dθ =r 2 dθ. 11