1、Section 9.4FermatJacobi,Jakob 2Partial Derivatives and Total Differentials of Multivariable Composite FunctionsTheorem are bothSuppose that(,)uu x y and(,)vv x y(,).u vThen the compositedifferentiable at the corresponding point (,),(,)zf u x y v x y(,)x yis also differentiable at the point function
2、isdifferentiable at the point while the function(,)zf u v(,),x yand its total differential is.zuzvzuzvdzdxdyu xvxu yvy and.v u then the functions u and v have corresponding incrementsProof Let the increments of the variables x and y be and,y x 3Partial Derivatives and Total Differentials of Multivar
3、iable Composite FunctionsProof(continued)zuzvzuzvdzdxdyu xvxu yvy Since u and v are both differentiable at(,),x y2(),vvvxyoxy 1(),uuuxyoxy Since f is differentiable at the22()().xy where point(,),u vz 22()().zzuvouvuv corresponding4Partial Derivatives and Total Differentials of Multivariable Composi
4、te FunctionsProof(continued)Then,we compose the functions u and v into the function f and zuzvzuzvzxyu xvxu yvy where 2212()()()().zzooouvuv Now,we need only verify that the a is a higher-order infinitesimalThat is0lim0.2(),vvvxyoxy 1(),uuuxyoxy w.r.t.5Partial Derivatives and Total Differentials of
5、Multivariable Composite FunctionsProof(continued)Notice thatSince 22222222()()()()()().()()ouvouvuvuv 1,uuxy1|()|ouxuyxy|u Thus then,|u is bounded.|v Similarly,is also bounded.This implies the result.2222()()uvuv is bounded.2212()()()()zzooouvuv 6Partial Derivatives and Total Differentials of Multiv
6、ariable Composite Functionszuzvzuzvdzdxdyu xvxu yvy By the formula we know that,zzuzvxu xvx .zzuzvyu yvy and12(,),1,2,.,iinuu xxximMore generally,if 12(,)myf u uu are both differentiable,then the composite function is also differentiable,1212nnyyydydxdxdxxxxwhere1212,1,2,.mjjjmjuuuyyyyjnxuxuxux 7Par
7、tial Derivatives and Total Differentials of Multivariable Composite FunctionsSolutionFindExample(,)zf u v Letwhere is differentiable.,.zzxy(,),zf xy xy(,)zf xy xy is differentiable becauseThe composite function Then,we havevxy uxy andare both differentiable.zx ffyuvfufvuxvxzy fufvuyvyffxuv12 fyf12 f
8、xf8Partial Derivatives and Total Differentials of Multivariable Composite FunctionsExampleFindLetwhere is differentiable.2(,sin),zf xx Solution 2(,sin)zf xx is differentiable.It is obvious that the function21212()(sin)2cos.dzd xdxffxfxfdxdxdxThen,.dzdxfFinish.9Partial Derivatives and Total Different
9、ials of Multivariable Composite FunctionsExampleProve thatLetwhere is derivable.22(),uxy Proof 22()uxy as a composite functionRegard the functionfunction composed by 22.zxyand()uz Derivation with respect to x and y respectively gives()2,uzyy ()2,uzxx so that0.uuxyyx0.uuxyyx 2()2()xyzxyz Finish.10Par
10、tial Derivatives and Total Differentials of Multivariable Composite FunctionsExample Letwhere the second order partial derivatives(,),zf u x y If of the function f are continuous with respect to each variable.,yuxe find 2.zy x We have Solution zx fufuxx12.yf efwe have12(,),(,),gf u x y hfu x yUsing
11、111312123().yyyyf xefef ef xef2zy x ()ygehy yyghegeyyPartial Derivatives and Total Differentials of Multivariable Composite FunctionsExampleThe second order partial derivatives f,g are continuous with respect to each variable,1)(,),zf u v ux vxy findzx .zy andffyuvfufvuxvxzy fufvuyvy0ffxuv12fyf2xf z
12、x 1122)(,),sin,zg u x y uxy find2zy x 2.zx y and12Invariance of the total differential formRational operation rules for total differentials2();();1(),0.d uvdudvd uvvduudvudvduudv vvvExample Find().d xyyzxz13Rational operation rules for total differentialsExampleis differentiable,find the partial derivatives of(,)f u vIf the function,.x yzfy x Solution By the invariance of the total differential form we havedz121222111,yffdxffdyyxyx then1221,zyffxyx 12211.zffyyx 12xyf df dyx1222ydxxdyxdyydxffyx
