2.4 Vector Fields 15 Proof.To prove injectivity.suppose D=0for vT(R").Applying D to the coordinate functionx gives 0=n=∑品，=∑W= Hence.v=0 and is injective. Toprove surjectivity,let Dbeaderivation of at pandlet(f.V)be arepresentative of a g Making V smaller if nece that l ped.Bv T aylor's th C functionsg)in a neighborhood of p such that fa=fn+∑d-pa.8)=部o Applying Dtoboth sides and noting that Df(p)=0and D(p)=0by Lemma2. we get by the Leibniz rule Df(x)=(Dxi)gi(p)+(pi-p)Dgi(x) =or影p This proves that D=D.forv=(Dx!.Dx"). This theorem shows that one may identify the tangent vectors at p with the derivations at p.Under the identification T(R")Dp(R").the standard basis for)corresponds to the o=2需hiso t angent vector =E副 The vector space Dp(R")of derivations at p,although not as geometric as arrows, turns out to be more suitable for generalization to manifolds. 2.4 Vector Fields Avector field on an open subset U of is a function that assigns to each point ngent vector na taneent vector i (Sincea lp).the vector n U a ta xp=∑dpp We say that the vector field X if the coefficient functions are all c on U
2.4 Vector Fields 15 Proof. To prove injectivity, suppose Dv = 0 for v ∈ Tp(Rn). Applying Dv to the coordinate function xj gives 0 = Dv(xj ) = i vi ∂ ∂xi p xj = i vi δ j i = vj . Hence, v = 0 and φ is injective. To prove surjectivity, let D be a derivation of at p and let(f, V ) be a representative of a germ in C∞ p . Making V smaller if necessary, we may assume that V is an open ball, hence star-shaped. By Taylor’s theorem with remainder (Lemma 1.4) there are C∞ functions gi(x) in a neighborhood of p such that f (x) = f (p) +(xi − pi )gi(x), gi(p) = ∂f ∂xi (p). ApplyingD to both sides and noting thatD(f (p)) = 0 andD(pi ) = 0 by Lemma 2.2, we get by the Leibniz rule Df (x) = (Dxi )gi(p) +(pi − pi )Dgi(x) = (Dxi ) ∂f ∂xi (p). This proves that D = Dv for v = Dx1,. ,Dxn. This theorem shows that one may identify the tangent vectors at p with the derivations at p. Under the identification Tp(Rn) Dp(Rn), the standard basis {e1,.,en} for Tp(Rn) corresponds to the set {∂/∂x1|p, . . . , ∂/∂xn|p} of partial derivatives. From now on, we will make this identification and write a tangent vector v = v1,.,vn = vi ei as v = vi ∂ ∂xi p . The vector space Dp(Rn) of derivations at p, although not as geometric as arrows, turns out to be more suitable for generalization to manifolds. 2.4 Vector Fields A vector field X on an open subset U of Rn is a function that assigns to each point p in U a tangent vector Xp in Tp(Rn). Since Tp(Rn) has basis {∂/∂xi |p}, the vector Xp is a linear combination Xp = ai (p) ∂ ∂xi p , p ∈ U. We say that the vector field X is C∞ on U if the coefficient functions ai are all C∞ on U.
16 2 Tangent Vectors in R"as Derivations Example 2.4.On R2-(0).let p =(x.y).Then V2+反+ X三 2+z is the vector field of Figure 2.3. Fig.2.3.A vector field on R2-(). One can identify vector fields on U with column vectors of c functions on U: 「a7 X=∑+品 ←→ The ring of C functions on U is commonly denoted C(U)or F(U).Since 8Sw R.but also a module over the ring C(U).We recall the efini Definition 2.5.If R is a commutative ring with identity,then an R-module is a set A with two operations.addition and scalar multiplication,such that (1)under addition,A is an abelian group: (2)forr.s∈R and a.b∈A. ((closure)ra∈A; (ii)(identity)if I is the multiplicative identity in R.then la=a: (iii)(associativity)(rs)a =r(sa); (iv)(distributivity)(r+s)a ra+sa.r(a+b)=ra+rb. If R is a field,then an R-module is precisely a vector space over R.In this sense, a module generalizes a vector space by allowing scalars in a ring rather than a field
16 2 Tangent Vectors in Rn as Derivations Example 2.4. On R2 − {0}, let p = (x, y). Then X = −y x2 + y2 ∂ ∂x + x x2 + y2 ∂ ∂y is the vector field of Figure 2.3. Fig. 2.3. A vector field on R2 − {0}. One can identify vector fields on U with column vectors of C∞ functions on U: X = ai ∂ ∂xi ←→ ⎡ ⎢ ⎣ a1 . . . an ⎤ ⎥ ⎦ . The ring of C∞ functions on U is commonly denoted C∞(U ) or F(U ). Since one can multiply a C∞ vector field by a C∞ function and still get a C∞ vector field, the set of all C∞ vector fields on U, denoted X(U ), is not only a vector space over R, but also a module over the ring C∞(U ). We recall the definition of a module. Definition 2.5. If R is a commutative ring with identity, then an R-module is a set A with two operations, addition and scalar multiplication, such that (1) under addition, A is an abelian group; (2) for r, s ∈ R and a, b ∈ A, (i) (closure) ra ∈ A; (ii) (identity) if 1 is the multiplicative identity in R, then 1a = a; (iii) (associativity) (rs)a = r(sa); (iv) (distributivity) (r + s)a = ra + sa, r(a + b) = ra + rb. If R is a field, then an R-module is precisely a vector space over R. In this sense, a module generalizes a vector space by allowing scalars in a ring rather than a field.
2.5 Vector Fields as Derivations 17 2.5 Vector Fields as Derivations If X is a c vector field on an open subset U of R"and f is a C function on U, we define a new function xf on U by (Xf)(p)=Xpf for any peU Writing X =aa/axi,we get Xnp=∑dppn or xr=∑影 which shows that Xf is a c function on U.Thus,a c vector field X gives rise to an R-linear map Co(U)-C(U) f→Xf. rule (Leibniz rule): x(f)=(X)g+fXg. Proof.At each point pU,the vector Xp satisfies the Leibniz rule: Xp(fg)=(Xpf)g(p)+f(p)Xpg. As p varies over U,this becomes an equality of functions: X(fg)=(xf)g+fXg. If A is an algebra over a field K.aderivation of A is a K-linear map D:A-A such that D(ab)=(Da)b+aDb for all a,. The set of all derivations of A is closed under addition and scalar multiplication and s a vector space oted Der(A).As setives rise to a derivation of the alge We therefore have a map p:x(U)-→Der(C(U). X→(f→Xf). Just as the tangent vectors at a point p can be identified with the point-derivations of Co,so the vector fields on an open set U can be identified with the derivations of the algebra CU),i.e.,the mapo is an isomorphism of vector spaces.The injectivity of is easy to establish,but the surjectivity of takes some work(see Problem 19.11). Note that a derivation at p is not a derivation of the algebra C.A derivation at p is a map from Co to R,while a derivation of the algebra Co is a map from Co to Cpo
2.5 Vector Fields as Derivations 17 2.5 Vector Fields as Derivations If X is a C∞ vector field on an open subset U of Rn and f is a C∞ function on U, we define a new function Xf on U by (Xf )(p) = Xpf for any p ∈ U. Writing X = ai ∂/∂xi , we get (Xf )(p) = ai (p) ∂f ∂xi (p), or Xf = ai ∂f ∂xi , which shows that Xf is a C∞ function on U. Thus, a C∞ vector field X gives rise to an R-linear map C∞(U ) −→ C∞(U ) f → Xf. Proposition 2.6 (Leibniz rule for a vector field). If X is a C∞ vector field and f and g are C∞ functions on an open subset U of Rn, then X(fg) satisfies the product rule (Leibniz rule): X(fg) = (Xf )g + f Xg. Proof. At each point p ∈ U, the vector Xp satisfies the Leibniz rule: Xp(fg) = (Xpf )g(p) + f (p)Xpg. As p varies over U, this becomes an equality of functions: X(fg) = (Xf )g + f Xg. If A is an algebra over a field K, a derivation of A is a K-linear map D: A −→ A such that D(ab) = (Da)b + aDb for all a, b ∈ A. The set of all derivations of A is closed under addition and scalar multiplication and forms a vector space, denoted Der(A). As noted above, a C∞ vector field on an open set U gives rise to a derivation of the algebra C∞(U ). We therefore have a map ϕ : X(U ) −→ Der(C∞(U )), X → (f → Xf ). Just as the tangent vectors at a point p can be identified with the point-derivations of C∞ p , so the vector fields on an open set U can be identified with the derivations of the algebra C∞(U ), i.e., the map ϕ is an isomorphism of vector spaces. The injectivity of ϕ is easy to establish, but the surjectivity of ϕ takes some work (see Problem 19.11). Note that a derivation at p is not a derivation of the algebra C∞ p . A derivation at p is a map from C∞ p to R, while a derivation of the algebra C∞ p is a map from C∞ p to C∞ p .
18 2 Tangent Vectors in R as Derivations Problems 2.1.Vector fields Let X be the vector field x a/ax +ya/ay and f(x.y.)the function x2+y2+2 2.2.Algebra structure on C Define carefully addition,multiplication,and scalar multiplication in C.Prove that addition in Cis commutative. 2.3.Vector space structure on derivations at a point Let Dand D'be derivations at p inR",andcR.Prove that (a)the sum D+D'is a derivation at p. (b)the s scalar multiple cD is a derivat on at p. 2.4.Product of derivations Let A be an algebra over a field K.If Di and D are derivations of A.show that D1。D2((D1orD2=.以butD1。D2-D2。Dl isawaysa derivation of A
18 2 Tangent Vectors in Rn as Derivations Problems 2.1. Vector fields Let X be the vector field x ∂/∂x + y ∂/∂y and f (x, y, z) the function x2 + y2 + z2 on R3. Compute Xf . 2.2. Algebra structure on C∞ p Define carefully addition, multiplication, and scalar multiplication in C∞ p . Prove that addition in C∞ p is commutative. 2.3. Vector space structure on derivations at a point Let D and D be derivations at p in Rn, and c ∈ R. Prove that (a) the sum D + D is a derivation at p. (b) the scalar multiple cD is a derivation at p. 2.4. Product of derivations Let A be an algebra over a field K. If D1 and D2 are derivations of A, show that D1 ◦ D2 is not necessarily a derivation (it is if D1 or D2 = 0), but D1 ◦ D2−D2 ◦ D1 is always a derivation of A
Alternating k-Linear Functions pureyveeo is to devclop the propertics of altemating k-linear er applicati on to the nt space at a poin of a manifold. 3.1 Dual Space If V and W are real vector spaces,we denote by Hom(V.W)the vector space of all linear maps f:VW.Define the dual space V+to be the vector space of all real-valued linear functions on V: V*=Hom(V,R). The elements of V*are called covectors or 1-covectors on V. In the rest of this section,assume V to be a finite-dimensional vector space.Let fe1.en)be a basis for V.Then every v in V is uniquely a linear combination v=ve;with vi E R.Let a:V-R be the linear function that picks out the ith coordinate.a(v)=v.Note that a is characterized by de=8的=0ifi≠j 1 ifi=j: Proposition 3.1.The functions a.a"form a basis for V". Proof.We first prove that a,.a"span V.If=vei in V,ther fu)=∑vfe)=∑f(e)a(w) Hence. f=∑fe)a, which shows that a,.a"span V*
3 Alternating k-Linear Functions This chapter is purely algebraic. Its purpose is to develop the properties of alternating k-linear functions on a vector space for later application to the tangent space at a point of a manifold. 3.1 Dual Space If V and W are real vector spaces, we denote by Hom(V , W ) the vector space of all linear maps f : V −→ W. Define the dual space V ∗ to be the vector space of all real-valued linear functions on V : V ∗ = Hom(V , R). The elements of V ∗ are called covectors or 1-covectors on V . In the rest of this section, assume V to be a finite-dimensional vector space. Let {e1,.,en} be a basis for V . Then every v in V is uniquely a linear combination v = vi ei with vi ∈ R. Let αi : V −→ R be the linear function that picks out the ith coordinate, αi (v) = vi . Note that αi is characterized by αi (ej ) = δi j = 1 if i = j ; 0 if i = j. Proposition 3.1. The functions α1,.,αn form a basis for V ∗. Proof. We first prove that α1,.,αn span V ∗. If f ∈ V ∗ and v = vi ei in V , then f (v) = vi f (ei) = f (ei)αi (v). Hence, f = f (ei)αi , which shows that α1,.,αn span V ∗.