10 1 Smooth Functions on a Euclidean Space 1.4.A diffeomorphism of an open ball with R" (a)Show that the function h:(-/2./2)[0,o). h(x)= Je-secx forx∈(0，T/2) 0 forx≤0， is c on(-/2./2).strictly increasing on [0./2).and satisfies h()=0 for allk.(Hint:Let f(x)be the function of Example 1.3 and let g(x)=secx Then h(x)= ()Use th of f()) (Detine the mapby forx≠0. F(x)= x衣 0 forx=0. Show that F:B(0,π/2)-→R"is a diffeomorphism 1.5.*Taylor's theorem with remainder to order 2 Prove that if f:R2-R is c,then there exist C functions fu,fi2.f22 on R2 such that c=f0.o0+00r+0.0w dx +xfi(x.y)+xyfi2(x,y)+y2f22(x.y). with a ularity gt,4)= ,for t≠0 10 for t =0. Prove that g(t.)isc for (t.u)R2.(Hint:Apply Problem 1.5.) 1.7.Bijective Co maps Define R一Rb meagfc necessan
10 1 Smooth Functions on a Euclidean Space 1.4. A diffeomorphism of an open ball with Rn (a) Show that the function h: (−π/2,π/2) −→ [0,∞), h(x) = e−1/x sec x for x ∈ (0,π/2), 0 for x ≤ 0, is C∞ on (−π/2,π/2), strictly increasing on [0,π/2), and satisfies h(k) = 0 for all k ≥ 0. (Hint: Let f (x) be the function of Example 1.3 and let g(x) = sec x. Then h(x) = f (x)g(x). Use the properties of f (x).) (b) Define the map F : B(0,π/2) ⊂ Rn −→ Rn by F (x) = ⎧ ⎨ ⎩ h(|x|) x |x| for x = 0, 0 for x = 0. Show that F : B(0,π/2) −→ Rn is a diffeomorphism. 1.5.* Taylor’s theorem with remainder to order 2 Prove that if f : R2 −→ R is C∞, then there exist C∞ functions f11, f12, f22 on R2 such that f (x, y) = f (0, 0) + ∂f ∂x (0, 0)x + ∂f ∂y (0, 0)y + x2f11(x, y) + xyf12(x, y) + y2f22(x, y). 1.6.* A function with a removable singularity Let f : R2 −→ R be a C∞ function with f (0, 0) = 0. Define g(t, u) = f (t,tu) t for t = 0; 0 for t = 0. Prove that g(t, u) is C∞ for (t, u) ∈ R2. (Hint: Apply Problem 1.5.) 1.7. Bijective C∞ maps Define f : R −→ R by f (x) = x3. Show that f is a bijective C∞ map, but that f −1 is not C∞. (In complex analysis a bijective holomorphic map f : C −→ C necessarily has a holomorphic inverse.)
2 Tangent Vectors in R"as Derivations In elementary calculus we normally represent a vector at a point p in R3algebraically as a column of numbers 周 v= or geometrically as an arrow emanating from p(Figure 2.1). Fig.2.1.A vector vat p A vector at p is tangent to a surface at p if it lies in the tangent plane at p (Figure 2.2),which is the limiting position of the secant planes through p.Intuitively, the tangent plane to a surface at p is the plane in R3 that just"touches"the surface at p. Fig.2.2.A tangent vector vto a surface at p
2 Tangent Vectors in Rn as Derivations In elementary calculus we normally represent a vector at a point p in R3 algebraically as a column of numbers v = ⎡ ⎣ v1 v2 v3 ⎤ ⎦ or geometrically as an arrow emanating from p (Figure 2.1). p v Fig. 2.1. A vector v at p. A vector at p is tangent to a surface at p if it lies in the tangent plane at p (Figure 2.2), which is the limiting position of the secant planes through p. Intuitively, the tangent plane to a surface at p is the plane in R3 that just “touches’’ the surface at p. p v Fig. 2.2. A tangent vector v to a surface at p
12 2 Tangent Vectors in R as Derivations Such a definition of a tangent vector to a surface presupposes that the surface is embedded in a Euclidean space,and so would not apply to the projective plane,which does not sit inside an R"in any natural way. Our goal in this chapter is to find acha acterization of a tangent vector in"that would generalize to manifolds 2.1 The Directional Derivative In calculus we visualize the tangent space Tp(R")at p in R"as the vector space of all arrows emanating from p.By the correspondence between arrows and column vectors,this space can be identified with the vector space R".To distinguish between ntoriea point insp)and a vector vin the or We usually denote the standard basis for R"or T(R")by (e1.n).Then v= >vei.We sometimes drop the parentheses and write TpR"for Tp(R").Elements of Tp(R")are called tangent vectors (or simply vectors)at p in R". The line through a point p=(p!,.p")with direction v=(vi.vn)inR" has parametrization c0=(p+1,p”+) Its ith component ci(r)is pi+t.If f is c in a neighborhood of p inR"and v is a tangent vector at p,the directional derivative of f in the direction v at p is defined to be D.f=mfco-f卫_ d f(c(r)). By the chain rule. Df=刀 (2.1) e at p.since v is a vector at p.So Df is a number.not a function.We write n=∑l for the operator that sends a function f to the number Def.To simplify the notation we often omit the subscript p if it is clear from the context
12 2 Tangent Vectors in Rn as Derivations Such a definition of a tangent vector to a surface presupposes that the surface is embedded in a Euclidean space, and so would not apply to the projective plane, which does not sit inside an Rn in any natural way. Our goal in this chapter is to find a characterization of a tangent vector in Rn that would generalize to manifolds. 2.1 The Directional Derivative In calculus we visualize the tangent space Tp(Rn) at p in Rn as the vector space of all arrows emanating from p. By the correspondence between arrows and column vectors, this space can be identified with the vector space Rn. To distinguish between points and vectors, we write a point in Rn as p = (p1,.,pn) and a vector v in the tangent space Tp(Rn) as v = ⎡ ⎢ ⎣ v1 . . . vn ⎤ ⎥ ⎦ or v1,.,vn. We usually denote the standard basis for Rn or Tp(Rn ) by {e1,.,en}. Then v = vi ei. We sometimes drop the parentheses and write TpRn for Tp(Rn). Elements of Tp(Rn) are called tangent vectors (or simply vectors) at p in Rn. The line through a point p = (p1,.,pn) with direction v = v1,.,vn in Rn has parametrization c(t) = (p1 + tv1,.,pn + tvn). Its ith component ci (t) is pi + tvi . If f is C∞ in a neighborhood of p in Rn and v is a tangent vector at p, the directional derivative of f in the direction v at p is defined to be Dvf = lim t−→0 f (c(t)) − f (p) t = d dt t=0 f (c(t)). By the chain rule, Dvf = n i=1 dci dt (0) ∂f ∂xi (p) = n i=1 vi ∂f ∂xi (p). (2.1) In the notation Dvf , it is understood that the partial derivatives are to be evaluated at p, since v is a vector at p. So Dvf is a number, not a function. We write Dv = vi ∂ ∂xi p for the operator that sends a function f to the number Dvf . To simplify the notation we often omit the subscript p if it is clear from the context.
2.2 Germs of Functions 13 2.2 Germs of Functions A relation on a set is a subset R of Sx S.Given x.y in S,we write x ~y if and only if (x,y)ER.The relation is an equivalence relation if it satisfies the following three properties: (i)reflexive:x~x for allx S. (i）symme fx ~y,th (ii)transitive:ifx~y and y~z,thenxz. As long as two functions agree on some neighborhood of a point p.they will have the same directional derivatives at p.This suggests that we introduce an equivalence relation on the Co functions defined in some neighborhood of p.Consider the set of all pairs (f.U).where U is aneighborhood of pand f:URisaCoo function.We say that (f.U)isequi p such th This an open set w cunv ainin clearly an equi alence 二 called the germ of f at p.We write Co(R")or simply Co if there is no possibility of confusion,for the set of all germs of c functions on at p. Example 2.1.The functions f(x)=1-x with domainR-(1)and gx)=1+x+x2+x3+. with domain the open interval (-1.1)have the same germ at any point pin the open interval (-1.1). An algebra over a field K is a vector space A over K with a multiplication map u:AXA-→A, usually written(a,b)=a×b,such that for all a,b,c∈A and r∈K, ()①(associativity)(a×b)×c=a×(b×c. (Gi(distributivity)(a+b)×c=a×c+b×canda×(b+c)=a×b+a×c, (Gi(homogeneity)r(a×b)=(ra)×b=a×(rb). Equivalently,an algebra over a field K is a ring A which is also a vector space over K such that the ring multiplication satisfies the homogeneity condition(iii).Thus,an algebra has three operations:the addition and multiplication of a ring and the scalar multiplication of a vector space.Usually we omit the multiplication sign and write ab instead of a×b. Addition and multiplication of functions induce corresponding operations on C making it into an algebra over R(Problem 2.2)
2.2 Germs of Functions 13 2.2 Germs of Functions A relation on a set S is a subset R of S × S. Given x, y in S, we write x ∼ y if and only if (x, y) ∈ R. The relation is an equivalence relation if it satisfies the following three properties: (i) reflexive: x ∼ x for all x ∈ S. (ii) symmetric: if x ∼ y, then y ∼ x. (iii) transitive: if x ∼ y and y ∼ z, then x ∼ z. As long as two functions agree on some neighborhood of a point p, they will have the same directional derivatives at p. This suggests that we introduce an equivalence relation on the C∞ functions defined in some neighborhood of p. Consider the set of all pairs(f, U ), where U is a neighborhood ofp and f : U −→ R is aC∞ function. We say that (f, U ) is equivalent to (g, V ) if there is an open set W ⊂ U ∩ V containing p such that f = g when restricted to W. This is clearly an equivalence relation because it is reflexive, symmetric, and transitive. The equivalence class of (f, U ) is called the germ of f at p. We write C∞ p (Rn) or simply C∞ p if there is no possibility of confusion, for the set of all germs of C∞ functions on Rn at p. Example 2.1. The functions f (x) = 1 1 − x with domain R − {1} and g(x) = 1 + x + x2 + x3 +··· with domain the open interval (−1, 1) have the same germ at any point p in the open interval (−1, 1). An algebra over a field K is a vector space A over K with a multiplication map µ: A × A −→ A, usually written µ(a, b) = a × b, such that for all a, b, c ∈ A and r ∈ K, (i) (associativity) (a × b) × c = a × (b × c), (ii) (distributivity) (a + b) × c = a × c + b × c and a × (b + c) = a × b + a × c, (iii) (homogeneity) r(a × b) = (ra) × b = a × (rb). Equivalently, an algebra over a field K is a ring A which is also a vector space over K such that the ring multiplication satisfies the homogeneity condition (iii). Thus, an algebra has three operations: the addition and multiplication of a ring and the scalar multiplication of a vector space. Usually we omit the multiplication sign and write ab instead of a × b. Addition and multiplication of functions induce corresponding operations on C∞ p , making it into an algebra over R (Problem 2.2)
14 2 Tangent Vectors in R"as Derivations 2.3 Derivations at a Point A mapL:VW between vector spaces over a field K is called a linear map or a linear operator if for any r∈K and u,v∈V, (i)L(u+v)=L(u)+L(v): (ii)L(rv)=rL(v). To emphasize the fact that the scalars are in the field K,such a map is also said to be K-linear. For each tangent vector vat a point p in R",the directional derivative at p gives a map of real vector spaces Du:Cpo-R. By (2.1),D is R-linear and satisfies the Leibniz rule Du(fg)=(Duf)g(p)+f(p)Dvg. (2.2) essentially because the partial derivatives a/ax'lp have these properties. In general,any linear map D:CR satisfying the Leibniz rule(2.2)is called a derivation at p or a point-derivation of C.Denote the set of all derivations at p by D).This set is in fact a real vector space. since the sum of two derivation P and a at p are again derivatic (Problem2.3). Thus far.we know that directional derivatives at p are all derivations at p.so there is a map 中：T(R")→Dp(R"), (2.3) vnD=∑ta, Since D is clearly linear in v.the mapis a linear operator of vector spaces. Lemma 2.2.If D is a point-derivation of Co.then D(c)=0 for any constant function c. Proof.As we do not know if every derivation at p is a directional derivative,we need to prove this lemma using only the defining properties of a derivation at p. By R-linearity,D(c)=cD(1).So it suffices to prove that D(1)=0.By the Leibniz rule D(I)=D(1×1)=D(I)×1+1×D(I)=2D(I). Subtracting D(1)from both sides gives0=D(1). Theorem 23.The linear mapT)defined in (2.3)is an isomor phism of vector spaces
14 2 Tangent Vectors in Rn as Derivations 2.3 Derivations at a Point A map L: V −→ W between vector spaces over a field K is called a linear map or a linear operator if for any r ∈ K and u, v ∈ V , (i) L(u + v) = L(u) + L(v); (ii) L(rv) = rL(v). To emphasize the fact that the scalars are in the field K, such a map is also said to be K-linear. For each tangent vector v at a point p in Rn, the directional derivative at p gives a map of real vector spaces Dv : C∞ p −→ R. By (2.1), Dv is R-linear and satisfies the Leibniz rule Dv(fg) = (Dvf )g(p) + f (p)Dvg, (2.2) essentially because the partial derivatives ∂/∂xi |p have these properties. In general, any linear map D: C∞ p −→ R satisfying the Leibniz rule (2.2) is called a derivation at p or a point-derivation of C∞ p . Denote the set of all derivations at p by Dp(Rn). This set is in fact a real vector space, since the sum of two derivations at p and a scalar multiple of a derivation at p are again derivations at p (Problem 2.3). Thus far, we know that directional derivatives at p are all derivations at p, so there is a map φ : Tp(Rn) −→ Dp(Rn), (2.3) v → Dv = vi ∂ ∂xi p . Since Dv is clearly linear in v, the map φ is a linear operator of vector spaces. Lemma 2.2. If D is a point-derivation of C∞ p , then D(c) = 0 for any constant function c. Proof. As we do not know if every derivation at p is a directional derivative, we need to prove this lemma using only the defining properties of a derivation at p. By R-linearity, D(c) = cD(1). So it suffices to prove that D(1) = 0. By the Leibniz rule D(1) = D(1 × 1) = D(1) × 1 + 1 × D(1) = 2D(1). Subtracting D(1) from both sides gives 0 = D(1). Theorem 2.3. The linear map φ : Tp(Rn) −→ Dp(Rn) defined in (2.3) is an isomorphism of vector spaces.