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′)