《高数双语》课件section 9.5.pptx
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1、Section 9.5FermatJacobi,Jakob 2Differentiation of Implicit Functions隐函数隐函数For an equation 1212,(,)0,nnF xxxxxx if there exists a function of n variables 12(x),x(,)R,nnyx xx such that when we substitute it into the equation,the equationbecomes an identity 12(,)0,nF xxxy is called an implicit function
2、 determined by the equation.(x)y then Differentiation of Implicit FunctionsTheorem(Existence of implicit functions 隐函数存在定理隐函数存在定理)(2)Both partial derivatives of the function F are continuous in asatisfies the conditions:Suppose that the function(,)F x y00(,)0.yFxy(3)Then(1)00(,)0;F xy neighborhood o
3、f the point 00(,);xyand ,()0.F x f x in a()yf x (1)There exists one and only one function has a continuous derivative in the()yf x(2)The function 00(,),xy00()yf x such that neighbourhood of the point 00(,)U xy00(,),U xyandneighbourhood.xyFdydxF 34Differentiation of Implicit FunctionsThe proof of thi
4、s theorem is beyond the scope of this book.We justwith a continuous derivative.determines a implicit function()yf x deduce the formula under the assumption that the equation(,)0F x y ),0.(f xF x()yf x we obtain Substituting into(,)0F x y Differentiate both sides of the above identity using the chain
5、 rule,we have0.xydyFFdxthere exists a neighbourhood00(,)0,yFxy Since yFis continuous and 00(,);U xythus 0yF such that 00(,)U xyin.xyFdydxF 5Differentiation of Implicit Functionsis determined by(,)zf x y Similarly,if the function of two variables ,(,)0.F x y f x y in three variables,then we havean eq
6、uation(,)0F x y z Differentiate both sides of the above identity,we have0.yzzFFy 0,xzzFFx so that,0.yxzzzFFzzFxFyF Suppose that the function z=z(x,y)is determined bythe equationFind the total differential dz2222.xyzxyzat the point(1,0,-1).6Differentiation of Implicit FunctionsExampleSolution(I)Using
7、 the formula for derivation2220.xyzxyzx 2220.zxzzxyzxyxxyz (1,0,1)1.zx 222(1,0,1)0.zxzzxyzxyxxyz 7Differentiation of Implicit FunctionsSolution(I)Using the formula for derivation2220.xyzxyzy 2220.zyzzyxzxyyxyz (1,0,1)2.zy 222(1,0,1)0.zyzzyxzxyyxyz Therefore,(1,0,1)2.dzdxdy 8Differentiation of Implic
8、it FunctionsSolution(II)Using the invariance of the total differential form 2222.d xyzxyzd2222220.2xdxydyzdzyzdxxzdyxydzxyz222(1,0,1)0.xdxydyzdzyzdxxzdyxydzxyz(1,0,1)0.2dxdzdy (1,0,1)2.dzdxdy Finish.9Differentiation of Implicit FunctionsSolution Obviously,all the first order,.ucxaz vcybzLet has cont
9、inuous first Example Suppose that the function(,)u v is determined by order partial derivatives and the function(,)zz x y constants.Find.xyazbz where a,b and c are allthe equation(,)0,cxaz cybz(,)cxaz cybz exist and partial derivatives of the composite function we haveare continuous.221212,yyzFcczFa
10、bab 111212,xxzFcczFabab so.xyazbzc(,)(,),F x y zcxaz cybz Let10Differentiation of Implicit Functions Defined By More Than One EquationConsider the system of two equations of functions(,)0,(,)0.F x y u vG x y u vIn general,two of theThese two equations contain four variables.If they can determine two
11、variables can be determined by the others.functions of two variables with continuous partial derivatives(,),(,),uu x yvv x yso thatDifferentiate both sides of above identities,we have ,(,),(,)0.G x y u x y v x y ,(,),(,)0,F x y u x y v x y 0.xuvuvGGGxx0,xuvuvFFFxx11Differentiation of Implicit Functi
12、ons Defined By More Than One EquationThese two equations form a linear algebraic system for the two unknownand.vx quantitiesux If the determinant of coefficients(,)0,(,)uvuvFFF GJGGu v then,by the Cramers rule,we obtain(,)(,).F Gvu yyJ (,)(,),F Guy vyJ By the same way,we have(,)(,).F Gvu xxJ (,)(,),
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