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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 @@ -44,6 +44,24 @@ \section{基本概念} +\subsection{二元函数} + +函数以$f(u,v)$的形式来出现,需要分别对其求偏导。 + +\textbf{例题:}设$z=e^{xy}+f(x+y,xy)$,$f(u,v)$有二阶连续偏导数,求$\dfrac{\partial^2z}{\partial x\partial y}$。 + +解:令$x+y$为$u$,$xy$为$v$,$f(u,v)$对$u$求导就是$f_1'$,对$v$求导就是$f_2'$,求$uv$依次求导就是$f_{12}''$,以此类推。 + +首先求一次偏导:$\dfrac{\partial z}{\partial x}=ye^{xy}+\dfrac{\partial f(u,v)}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial f(u,v)}{\partial v}\dfrac{\partial v}{\partial x}=ye^{xy}+f_1'+f_2'y$。 + +接着对$y$求偏导:$\dfrac{\partial^2z}{\partial x\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'y}{\partial y}$。 + +$=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'}{\partial y}y+f_2'\dfrac{\partial y}{\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial f_1'}{\partial v}\dfrac{\partial v}{\partial y}+\dfrac{\partial f_2'}{\partial u}\dfrac{\partial u}{\partial y}y+\dfrac{\partial f_2'}{\partial v}\dfrac{\partial v}{\partial y}y+f_2'=e^{xy}+xye^{xy}+f_{11}''+f_{12}''x+f_{21}''y+f_{22}''xy+f_2'$。\medskip + +又$f(u,v)$具有两阶连续偏导数,所以$f_{12}''=f_{21}''$。 + +即$=e^{xy}+xye^{xy}+f_{11}''+(x+y)f_{12}''+xyf_{22}''+f_2'$。 + \subsection{复合函数} 函数以复合函数形式$f(g(x,y))$出现,函数的变量是一个整体。 @@ -80,25 +98,27 @@ $\dfrac{\partial^2u}{\partial y^2}=\dfrac{\partial}{\partial y}\left(\dfrac{\par $\therefore\dfrac{\partial z}{\partial x}=e^x+3(x+y)^2-e^{x+y}$。 -\section{二元函数} +\subsection{积分与微分} -函数以$f(u,v)$的形式来出现,需要分别对其求偏导。 +\subsubsection{积分到微分} -\subsection{链式法则} +可能一个函数是积分的形式,又包含多个变量,要求其多元微分值。 -\textbf{例题:}设$z=e^{xy}+f(x+y,xy)$,$f(u,v)$有二阶连续偏导数,求$\dfrac{\partial^2z}{\partial x\partial y}$。 +$\dfrac{\textrm{d}}{\textrm{d}x}\int_{a(x)}^{b(x)}f(t)\,\textrm{d}t=b'(x)f[b(x)]-a'(x)f[a(x)]$。 -解:令$x+y$为$u$,$xy$为$v$,$f(u,v)$对$u$求导就是$f_1'$,对$v$求导就是$f_2'$,求$uv$依次求导就是$f_{12}''$,以此类推。 +\textbf{例题:}设$z=\int_0^1\vert xy-t\vert f(t)\,\textrm{d}t$,$0\leqslant x\leqslant1$,$0\leqslant y\leqslant1$,其中$f(x)$为连续函数,求$z_{xx}''+z_{yy}''$。 -首先求一次偏导:$\dfrac{\partial z}{\partial x}=ye^{xy}+\dfrac{\partial f(u,v)}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial f(u,v)}{\partial v}\dfrac{\partial v}{\partial x}=ye^{xy}+f_1'+f_2'y$。 +解:首先因为$z$是一个绝对值的形式,所以根据积分的性质可以拆开积分区间去掉绝对值:$z=\int_0^{xy}(xy-t)f(t)\,\textrm{d}t+\int_{xy}^1(t-xy)f(t)\,\textrm{d}t=xy\int_0^{xy}f(t)\,\textrm{d}t-\int_0^{xy}tf(t)\,\textrm{d}t+\int_{xy}^1tf(t)\,\textrm{d}t-xy\int_{xy}^1f(t)\,\textrm{d}t$。 -接着对$y$求偏导:$\dfrac{\partial^2z}{\partial x\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'y}{\partial y}$。 +$z_x'=y\int_0^{xy}f(t)\,\textrm{d}t+xy^2f(xy)-xy^2f(xy)-xy^2f(xy)-y\int_{xy}^1f(t)\,\textrm{d}t+xy^2f(xy)=y\int_0^{xy}f(t)\,\textrm{d}t-y\int_{xy}^1f(t)\,\textrm{d}t$。 -$=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'}{\partial y}y+f_2'\dfrac{\partial y}{\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial f_1'}{\partial v}\dfrac{\partial v}{\partial y}+\dfrac{\partial f_2'}{\partial u}\dfrac{\partial u}{\partial y}y+\dfrac{\partial f_2'}{\partial v}\dfrac{\partial v}{\partial y}y+f_2'=e^{xy}+xye^{xy}+f_{11}''+f_{12}''x+f_{21}''y+f_{22}''xy+f_2'$。\medskip +$z_{xx}''=y^2f(xy)+y^2f(xy)=2y^2f(xy)$,同理根据变量对称性$z_{yy}''=2x^2f(xy)$,$z_{xx}''+z_{yy}''=2(x^2+y^2)f(xy)$。 -又$f(u,v)$具有两阶连续偏导数,所以$f_{12}''=f_{21}''$。 +\subsubsection{微分到积分} -即$=e^{xy}+xye^{xy}+f_{11}''+(x+y)f_{12}''+xyf_{22}''+f_2'$。 +注意多元函数进行积分的适合多出来的常数$C$不再是常数,而是与积分变量相关的$C(x)$,$C(y)$,因为对其中一个变量积分时,另一个变量是看作常数的。 + +\textbf{例题:}设$z=f(x,y)$满足$\dfrac{\partial^2z}{\partial x\partial y}=x+y$,且$f(x,0)=x$,$f(0,y)=y^2$,求$f(x,y)$。 \section{多元函数微分应用}