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a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex +++ b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -166,6 +166,20 @@ $\therefore z=\dfrac{1}{2}x^2y+\dfrac{1}{2}xy^2+x+y^2$。 通过微分定义和极限即可证明。 +\subsubsection{隐函数存在定理} + +当给出一个隐函数方程$f(x,y)$或$f(x,y,z)$时,各自对变量求偏导得到偏导函数,然后将判断的点的值代入偏导函数,若得到的值不为0则在该点的某个邻域内能确定关于这个变量的隐函数,否则不能。 + +\textbf{例题:}已知方程$xy-z\ln y+e^{xz}=1$,若存在点$(0,1,1)$的一个邻域,求该邻域内该方程能确定哪些隐函数。 + +解:对于$F=xy-z\ln y+e^{xz}-1$,则$F(0,1,1)=0$。 + +分别求偏导:$F_x'=y+ze^{xz}$,$F_y'=x-\dfrac{z}{y}$,$F_z'=-\ln y+xe^{xz}$。 + +代入点:$F_x'=1+1=2\neq0$,$F_y'=-1\neq0$,$F_z'=0$。 + +所以根据隐函数存在定理$x$、$y$的隐函数存在。 + % \subsection{极限} \subsection{微分}