1 Smooth Functions on a Euclidean Space of C functions will be our primary tool for studyinghigher-dimension manifolds. For this reason,we begin w iew of C func ons on 1.1 Co Versus Analytic Functions Write the coordinates on R asx .x"and let p=(pl )be a point in an open setinIn keeping with the conventions of differential geome ry,the indices on coordinates are superscripts,not subscripts.An explanation of the rules for superscripts and subscripts is given in Section 4.7. ,≤k exist and are continuous at p.The functionf:isatp if it isC for alln other words,its partial derivatives of all orders ∂rf ari.ari exist and are continuous at p.We say that f is Ck on U if it is Ck at every point in U.A similar definition holds for a co function on an open set U.A synonym for Cois“smooth. Example 1.2. (i)A Co function on U is a continuous function on U. (ii)Let f:R-R be f(x)=x1/3.Then f'(x)= 3r-2 3 forx≠0， undefined forx =0. Thus the function f is co but not Cl atx=0
1 Smooth Functions on a Euclidean Space The calculus ofC∞ functions will be our primary tool for studying higher-dimensional manifolds. For this reason, we begin with a review of C∞ functions on Rn. 1.1 C∞ Versus Analytic Functions Write the coordinates on Rn as x1,.,xn and let p = (p1,.,pn) be a point in an open set U in Rn. In keeping with the conventions of differential geometry, the indices on coordinates are superscripts, not subscripts. An explanation of the rules for superscripts and subscripts is given in Section 4.7. Definition 1.1. Let k be a nonnegative integer. A function f : U −→ R is said to be Ck at p if its partial derivatives ∂jf /∂xi1 ··· ∂xij of all orders j ≤ k exist and are continuous at p. The function f : U −→ R is C∞ at p if it is Ck for all k ≥ 0; in other words, its partial derivatives of all orders ∂kf ∂xi1 ··· ∂xik exist and are continuous at p. We say that f is Ck on U if it is Ck at every point in U. A similar definition holds for a C∞ function on an open set U. A synonym for C∞ is “smooth.’’ Example 1.2. (i) A C0 function on U is a continuous function on U. (ii) Let f : R −→ R be f (x) = x1/3. Then f (x) = 1 3 x−2/3 for x = 0, undefined for x = 0. Thus the function f is C0 but not C1 at x = 0
6 1 Smooth Functions on a euclidean space (iii)Let g:R-R be defined by Then g'(x)=f(x)=x1/3,so g(x)is Cl but not C2 atx =0.In the same way one can construct a function that is Ck but not C+!at a given point. The polynomial,sine,cosine.and exponential functions on the real line are all The function f is real-analytic at p if in some neighborhood of p it is equal to its Taylor series at p fx)=f(p)+> p!) 、df 2 2a7pr-pc1-p+ A real-analytic function is necessarily C,because as one learns in real anal- ysis,a convergent power series can be differentiated term by term in its region of convergence.For example,if f=x=-+款- 1 then term-by-term differentiation gives f6)=cosx=1-22+4-. The following example shows that a c function need not be real-analytic.The very flat" near0in the sense that all of its deri atives nish 0 一一一 一一一一一一 Fig.1.1.AC function all of whose derivatives vanish at 0
6 1 Smooth Functions on a Euclidean Space (iii) Let g : R −→ R be defined by g(x) = x 0 f (t) dt = x 0 t 1/3 dt = 3 4 x4/3. Then g (x) = f (x) = x1/3, so g(x) is C1 but not C2 at x = 0. In the same way one can construct a function that is Ck but not Ck+1 at a given point. (iv) The polynomial, sine, cosine, and exponential functions on the real line are all C∞. The function f is real-analytic at p if in some neighborhood of p it is equal to its Taylor series at p: f (x) = f (p) + i ∂f ∂xi (p)(xi − pi ) + 1 2! i,j ∂2f ∂xi∂xj (p)(xi − pi )(xj − pj ) +··· . A real-analytic function is necessarily C∞, because as one learns in real analysis, a convergent power series can be differentiated term by term in its region of convergence. For example, if f (x) = sin x = x − 1 3! x3 + 1 5! x5 −··· , then term-by-term differentiation gives f (x) = cos x = 1 − 1 2! x2 + 1 4! x4 −··· . The following example shows that a C∞ function need not be real-analytic. The idea is to construct a C∞ function f (x) on R whose graph, though not horizontal, is “very flat’’ near 0 in the sense that all of its derivatives vanish at 0. x y 1 Fig. 1.1. A C∞ function all of whose derivatives vanish at 0.
1.2 Taylor's Theorem with Remainder 7 Example 1.3 (A Cfunction very flat at )Define f(x)on R by f(x)= [e-1/x forx>0: l0forx≤0. (See Figure 1.1.)By induction,one can show that f is c on R and that the derivativesf(=0for allk(Problem 1.2). The Taylor series of this function at the origin is identically zero in any neigh borhood of the origin,since all derivatives f()=0.Therefore,f(x)cannot be equal to its Taylor series and f(x)is not real-analytic at 0. 1.2 Taylor's Theorem with Remainder Although a C function need not be equal to its Taylor series,there is a Taylor's the- orem with remainder for Co functions which is often good enough for our r only the constant term f(p) We say tha t a subset S of R"is star-shaped wit respect to a point p in S if for every x in S,the line segment from p to x lies in S (Figure 1.2). Fig.1.2.Star-shaped with respect to p.but not with respect to. Lemma 1.4 (Taylor's theorem with remainder).Let f be a Co function on an open subset Uf star-shaped with respect toa point p=(p")in U. Then there are C functions g).gn(x)on U such that f)=f(p)+-p)g(x).g(p)= i=l ari(p). Proof.Since U is star-shaped with respect to p,for any x in U the line segment p+tx-p),0≤t≤1 lies in U(Figure1.3).Sof(p+tr-p》is defined for 0≤t≤1
1.2 Taylor’s Theorem with Remainder 7 Example 1.3 (A C∞ function very flat at 0). Define f (x) on R by f (x) = e−1/x for x > 0; 0 for x ≤ 0. (See Figure 1.1.) By induction, one can show that f is C∞ on R and that the derivatives f (k)(0) = 0 for all k ≥ 0 (Problem 1.2). The Taylor series of this function at the origin is identically zero in any neighborhood of the origin, since all derivatives f (k)(0) = 0. Therefore, f (x) cannot be equal to its Taylor series and f (x) is not real-analytic at 0. 1.2 Taylor’s Theorem with Remainder Although a C∞ function need not be equal to its Taylor series, there is a Taylor’s theorem with remainder for C∞ functions which is often good enough for our purposes. We prove in the lemma below the very first case when the Taylor series consists of only the constant term f (p). We say that a subset S of Rn is star-shaped with respect to a point p in S if for every x in S, the line segment from p to x lies in S (Figure 1.2). p q Fig. 1.2. Star-shaped with respect to p, but not with respect to q. Lemma 1.4 (Taylor’s theorem with remainder). Let f be a C∞ function on an open subset U of Rn star-shaped with respect to a point p = (p1,.,pn) in U. Then there are C∞ functions g1(x), . . . , gn(x) on U such that f (x) = f (p) +n i=1 (xi − pi )gi(x), gi(p) = ∂f ∂xi (p). Proof. Since U is star-shaped with respect to p, for any x in U the line segment p + t (x − p), 0 ≤ t ≤ 1 lies in U (Figure 1.3). So f (p + t (x − p)) is defined for 0 ≤ t ≤ 1.
1 Smooth Functions on a Euclidean Space Fig.1.3.The line segment from p to x. By the chain rule. d af p+-p》=∑x-pp+x-p If we integrate both sides with respect tot fromto 1.we get o+e-pg-=∑d-p影e+e-p血 (1.1) Let g)='p+e-p》d Jo ax Then gi(x)isCo and (1.1)becomes f(x)-f(p)=(xi-p )gi(x) Moreover, o=ow-影on ◇ In case n=I and p=0,this lemma says that f(x)=f(0)+xfi(x) for some Co function f(x).Applying the lemma repeatedly gives f(x)=f(O)+xf+1(x), wherefi.fi+lare C functions.Hence. f(x)=f(0)+x(fi(0)+xf2(x)) =f0)+xf10)+x2f20)+xf6x) =f0)+fi(0)r+f0)x2+.+f0)x2+fi+1x)x+l. (1.2)
8 1 Smooth Functions on a Euclidean Space p x U Fig. 1.3. The line segment from p to x. By the chain rule, d dt f (p + t (x − p)) = (xi − pi ) ∂f ∂xi (p + t (x − p)). If we integrate both sides with respect to t from 0 to 1, we get f (p + t (x − p))1 0 = (xi − pi ) 1 0 ∂f ∂xi (p + t (x − p)) dt. (1.1) Let gi(x) = 1 0 ∂f ∂xi (p + t (x − p)) dt. Then gi(x) is C∞ and (1.1) becomes f (x) − f (p) = (xi − pi )gi(x). Moreover, gi(p) = 1 0 ∂f ∂xi (p)dt = ∂f ∂xi (p). In case n = 1 and p = 0, this lemma says that f (x) = f (0) + xf1(x) for some C∞ function f1(x). Applying the lemma repeatedly gives fi(x) = fi(0) + xfi+1(x), where fi, fi+1 are C∞ functions. Hence, f (x) = f (0) + x(f1(0) + xf2(x)) = f (0) + xf1(0) + x2(f2(0) + xf3(x)) . . . = f (0) + f1(0)x + f2(0)x2 +···+ fi(0)xi + fi+1(x)xi+1. (1.2)
1.2 Taylor's Theorem with Remainder 9 Differentiating(1.2)repeatedly and evaluating at 0,we get i0=f0k=1.2i So(1.2)is a polynomial expansion of f(x)whose terms up to the last term agree with the Taylor series of f(x)at 0. Remark 1.5.Being star-shaped is not such arestrictive condition.since any open ball B(p,e)=∈R"IIx-pll<∈ is star-shaped with respect to p.If f is a C function defined on an open set U containing p,then there is an e>0 such that p E B(P.E)C U. When its domain is restricted to B(p,e),the function f is defined on a star-shaped neighborhood of p and Taylor's theorem with remainder applies. NorATON.It is customary to write the standard coordinates on R2 as x,y,and the standard coordinates on R3 asx.y. Problems 1.1.A function that is C2 but not C3 Find a function h:R-R that is C2 but not C3 at x =0. 1.2.*AC function very flat at Let f(x)be the function onR defined in Example 1.3. (a)Show by induction that forx>0 and k >0.the kth derivative f()(x)is of the form pak mial pa()of degree iny. c on and that()=0 forak 1.3.A diffe morp sm of an open interval with R Let U C Rn and V C Rn be open subsets.A Coo map F:U-V is called a diffeomorphism if it is bijective and has a co inverse F-:V-U. (a)Show that the function f:(-π/2，π/2)→R,f)=tanx,is a diffeomor pmism. (b)Find a linear function h:(a,b)-(-1,1),thus proving that any two finite open intervals are diffeomorphic. The compositef:(a,b)Ris then a diffeomorphism of an open interval to R
1.2 Taylor’s Theorem with Remainder 9 Differentiating (1.2) repeatedly and evaluating at 0, we get fk(0) = 1 k! f (k)(0), k = 1, 2, . . . , i. So (1.2) is a polynomial expansion of f (x) whose terms up to the last term agree with the Taylor series of f (x) at 0. Remark 1.5. Being star-shaped is not such a restrictive condition, since any open ball B(p, ) = {x ∈ Rn | ||x − p|| < } is star-shaped with respect to p. If f is a C∞ function defined on an open set U containing p, then there is an > 0 such that p ∈ B(p, ) ⊂ U. When its domain is restricted to B(p, ), the function f is defined on a star-shaped neighborhood of p and Taylor’s theorem with remainder applies. Notation. It is customary to write the standard coordinates on R2 as x, y, and the standard coordinates on R3 as x, y, z. Problems 1.1. A function that is C2 but not C3 Find a function h: R −→ R that is C2 but not C3 at x = 0. 1.2.* A C∞ function very flat at 0 Let f (x) be the function on R defined in Example 1.3. (a) Show by induction that for x > 0 and k ≥ 0, the kth derivative f (k)(x) is of the form p2k(1/x) e−1/x for some polynomial p2k(y) of degree 2k in y. (b) Prove that f is C∞ on R and that f (k)(0) = 0 for all k ≥ 0. 1.3. A diffeomorphism of an open interval with R Let U ⊂ Rn and V ⊂ Rn be open subsets. A C∞ map F : U −→ V is called a diffeomorphism if it is bijective and has a C∞ inverse F −1 : V −→ U. (a) Show that the function f : (−π/2,π/2) −→ R, f (x) = tan x, is a diffeomorphism. (b) Find a linear function h: (a, b) −→ (−1, 1), thus proving that any two finite open intervals are diffeomorphic. The composite f ◦ h: (a, b) −→ R is then a diffeomorphism of an open interval to R.