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J. Math. Study Vol. 50, No. 3, pp. 268-276 doi: 10.4208/jms.v50n3.17.04 September2017 OntheChangeofVariablesFormulafor MultipleIntegrals 1,∗ 2 ShiboLiu andYashanZhang 1 Department of Mathematics, Xiamen University, Xiamen 361005,P.R. China; 2 Department of Mathematics, University of Macau, Macau, P.R. China. ReceivedJanuary7,2017;Accepted(revised)May17,2017 Abstract. Wedevelopanelementaryproofofthechangeofvariablesformulainmulti- ple integrals. Our proof is based on an induction argument. Assuming the formula for (m−1)-integrals, we define the integral over hypersurface in Rm, establish the diver- genttheoremandthenusethedivergenttheoremtoprovetheformulaform-integrals. In addition to its simplicity, an advantage of our approachis that it yields the Brouwer Fixed Point Theorem as a corollary. AMSsubjectclassifications: 26B15,26B20 Keywords: Changeofvariables,surfaceintegral,divergenttheorem, Cauchy-Binetformula. 1 Introduction Thechangeofvariables formula for multiple integrals is a fundamental theorem in mul- tivariable calculus. It can be stated as follows. Theorem 1.1. Let D and Ω be bounded open domains in Rm with piece-wise C1-boundaries, 1 ¯ m 1 ¯ ϕ∈C (Ω,R )suchthat ϕ:Ω→DisaC -diffeomorphism. If f ∈C(D),then Z Z ′ Df(y)dy= Ωf(ϕ(x))Jϕ(x)dx, (1.1) where Jϕ(x)=detϕ (x) is the Jacobian determinant of ϕ at x∈Ω. The usual proofs of this theorem that one finds in advanced calculus textbooks in- volves careful estimates of volumes of images of small cubes under the map ϕ and nu- merous annoying details. Therefore several alternative proofs have appeared in recent years. For example, in [5] P. Lax proved the following version of the formula. ∗Correspondingauthor. Emailaddresses: liusb@xmu.edu.cn (S. Liu), colourful2009@163.com (Y.Zhang) c http://www.global-sci.org/jms 268 2017Global-SciencePress S. Liu and Y. Zhang / J. Math. Study, 50 (2017), pp. 268-276 269 Theorem1.2. Let ϕ:Rm→Rm be a C1-map such that ϕ(x)=x for |x|≥R, and f ∈C (Rm). 0 Then ZRm f(y)dy=ZRm f(ϕ(x))Jϕ(x)dx. The requirment that ϕ is an identity map outside a big ball is somewhat restricted. Thisrestriction was also removed by Lax in [6]. Then, Tayor[7] and Ivanov [4] presented different proofs of the above result of Lax [5] using differential forms. See also Ba´ez- Duarte [1] for a proof of a variant of Theorem 1.1 which does not require that ϕ:Ω→D is a diffeomorphism. As pointed out by Taylor [7, Page 380], because the proof relies on integration of differential forms over manifolds and Stokes’ theorem, it requires that one knowsthechangeofvariables formulaasformulatedinourTheorem1.1. In this paper, we will present a simple elementary proof of Theorem 1.1. Our ap- proach does not involve the language of differential forms. The idea is motivated by Excerise 15 of §1-7 in the famous textbook on classical differential geometry [3] by do Carmo. The excerise deals with the two dimensional case m=2. We will perform an induction argument to generize the result to the higher dimensional case m≥2. In our argument,wewillapplytheCauchy-Binetformulaaboutthedeterminantoftheproduct of two matrics. As a byproduct of our approach, we will also obtain the Non-Retraction Lemma(seeCorollary3.2),whichimpliestheBrouwerFixedPointTheorem. 2 Integraloverhypersurface Wewill prove Theorem 1.1 by an induction argument. The case m=1 is easily proved in single variable calculus. Suppose we have proven Theorem1.1 for (m−1)-dimension, wherem≥2.Wewilldefinetheintegraloverahypersurface(ofcodimensionone)inRm andestablishthedivergenttheoreminRm. Then,inthenexttwosectionswewillusethe divergent theoremto prove Theorem1.1 for m-dimension. Let U be a Jordan measurable boundedclosed domain in Rm−1, x:U→Rm, (u1,...,um−1)7→(x1,...,xm) be a C1-map such that the restriction of x in the interior U◦ is injective, and rank∂xi=m−1, (2.1) ∂uj thenwesaythatx:U→RmisaC1-parametrizedsurface. Bydefinition,twoC1-parametr- m ˜ ˜ m 1 ized surfaces x:U→R and x:U→R are equivalent if there is a C -diffeomorphism ˜ ˜ m φ:U→Usuchthatx=x◦φ. Theequivalentclass[x]iscalledahypersurface, and x:U→R ˜ ˜ is called a parametrization of the hypersurface. Since it is easy to see that x(U)=x(U) if ˜ x and x are equivalent, [x] can be identified as the subset S=x(U). 270 S. Liu and Y. Zhang / J. Math. Study, 50 (2017), pp. 268-276 Let S be a hypersurface with parametrization x:U→Rm. By (2.1), for u∈U, N(u)= ∂(x2,...,xm) ,...,(−1)m+1∂(x1,...,xm−1)6=0, (2.2) ∂(u1,...,um−1) ∂(u1,...,um−1) where ∂u1x1 ··· ∂um−1x1 . . . . . . 1 i m i−1 i−1 ˆ ∂(x ,...,x ,...,x ) ∂u1x ··· ∂um−1x =det i+1 i+1 . ∂(u1,...,um−1) ∂u1x ··· ∂um−1x . . . . . m . m ∂u1x ··· ∂um−1x It is well known that N(u) is a normal vector of S at x(u). Now,wecandefinethesurfaceintegral ofacontinuousfunction f :S→R by ZS f dσ=ZU f(x(u))|N(u)|du. (2.3) By the change of variables formular for (m−1)-integrals, it is not difficult to see that if ˜ ˜ m x:U→R isanotherparametrizationofS,then Z Z ˜ ˜ f (x(u))|N(u)|du= f (x(v)) N(v) dv, ˜ U U ˜ where N is definedsimilar to (2.2). Therefore, our surface integral is well defined. If Σ=Sℓ Si,whereSi=xi(Ui)arehypersurfacessuchthatxi(U◦)∩xj(U◦)=∅fori6=j, i=1 i j thenwecall Σ apiece-wise C1-hypersurfaceand definetheintegralof f ∈C(Σ) by Z fdσ= ℓ Z fdσ. ∑ Σ i=1 Si Theorem 2.1 (Divergent Theorem). Let D be bounded open domain in Rm with piece-wise 1 ¯ m 1 C -boundary ∂D, F:D→R bea C -vector field, n is the unit outer normal vector field on ∂D, then ZDdivFdx=Z∂DF·ndσ. Proof. Having defined the surface integral, the proof of the theorem is a standard appli- cation of the Fubini Theorem. We include the details here for completeness. Wesay that F=(F1,...,Fm) is of i-type if Fj =0 for j6=i. We also say that D is of i- type, if there are a bounded closed domain U in Rm−1 with piece-wise C1-boundary and ϕ±∈C1(U)suchthat D=nx|ϕ−(x′)