偏导数概念与几何意义

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1、1n!ê©111nnn!!!êêê©©©3.1êVgAÛ¿Â.z=f(x,y)3:M0(x0,y0)?ê½Â½Â3.1z=f(x,y)3:M0(x0,y0),N(M0)þk½Â,y½3y0x3x0?kOþ∆x,A/¼êkOþf(x0+∆x,y0)−f(x0,y0).f(x0+∆x,y0)−f(x0,y0)e4lim3,∆x→0∆xK¡d4¼êz=f(x,y)3:M0(x0,y0)?∂z∂féxê,P,fx(x0,y0)½.∂x∂xM0M0111nnn!!!êêê©©©aq/,z=f(x,y

2、)3:M0(x0,y0)?éyê½Âf(x0,y0+∆y)−f(x0,y0)lim,∆y→0∆y∂z∂fP,fy(x0,y0)½.∂y∂yM0M0111nnn!!!êêê©©©.¼êz=f(x,y)3«DS¼êXJ¼êz=f(x,y)3«DSz:(x,y)?éxêÑ3,@oùê´x,y¼ê,¡¼êz=f(x,y)égCþx∂z∂f¼ê,P,fx(x,y)½.∂x∂xf(x+∆x,y)−f(x,y)∴fx(x,y)=lim,∆x→0∆x(x,y)∈D.111nnn!!!êêê©©©aq/,±½Â

3、¼êz=f(x,y)égCþy∂z∂f¼ê,P,fy(x,y)½.∂y∂y111nnn!!!êêê©©©f(x,y)3:(x0,y0)?éxêfx(x0,y0)Ò´¼êfx(x,y)3:(x0,y0)?¼ê;f(x,y)3:(x0,y0)?éyêfy(x0,y0)Ò´¼êfx(x,y)3:(x0,y0)?¼ê.¼ê{¡ê.111nnn!!!êêê©©©êVg±í2±þõ¼ê.~Xµn¼êu=f(x,y,z)3:(x,y,z)?éxê½Â:f(x+∆x,y,z)−f(x,y

4、,z)fx(x,y,z)=lim.∆x→0∆x111nnn!!!êêê©©©dê½Â,¦f(x,y)3:(x0,y0)?éxê,¢SþÒ´-y=y0,¦¼êf(x,y0)3x0?éxê.Ón,¦fy(x0,y0)Ò´¦¼êf(x0,y)3y0?éyê.Ïd,±^¼ê¦ê{5¦¼êê,ØL3¦êIò,CþÀ~þ.111nnn!!!êêê©©©~~~1.f(x,y)=e−xsin(x+2y),¦f(0,π),x4πfx(0,).4))):f(x,y)=−e−xsin(x+2y)+e

5、−xcos(x+2y),xf(x,y)=2e−xcos(x+2y)yπππ∴fx(0,)=−sin+cos=−1,422ππfy(0,)=2cos=0.42111nnn!!!êêê©©©~~~2.¦e¼êêy(1)z=xy(2)z=arctanx∂z∂zy−1y))):(1)=yx,=xlnx.∂x∂y∂z1yy(2)=y(−)=−,∂x1+()2x2x2+y2x∂z11x=y()=.∂y1+()21x2+y2x111nnn!!!êêê©©©~~~3.PV=RT(R´~ê),¦y:∂P∂V∂T··=−1.∂

6、V∂T∂P∂Pdy5¿:N5é,ØUw∂Vdxdydxû.∂y∂x´vk¿Â.RT∂PRTyyy:P=,⇒=−,V∂VV2RT∂VRPV∂TVV=,⇒=;T=,⇒=;P∂TPR∂PR∂P∂V∂TRTRVRT∴··=−··=−=−1.∂V∂T∂PV2PRPV111nnn!!!êêê©©©xy22,x+y6=0~~~4.f(x,y)=x2+y2,220,x+y=0¦fx(0,0),fy(0,0).f(0+∆x,0)−f(0,0))))µµµfx(0,0)=lim∆x→0∆x0−0=lim=0,∆x→0∆xdx,

7、yÓé¡5,fy(x,y)=0.111nnn!!!êêê©©©3ù~f¥,limf(x,y)Ø3,(x,y)→(0,0)∴f(x,y)3:(0,0)?ØëY.5¿:3¼ê¥,e¼ê3,:,K§3T:7ëY,ù(Øé¼ê5`%ؽ¤á.d~L²:¼êf(x,y)3:(x0,y0)?ê3,¿ØUy¼ê3T:ëY.111nnn!!!êêê©©©n.êAÛ¿Âz=f(x,y),∂z=limf(x0+∆x,y0)−f(x0,y0)∂xM∆x∆x→0½y=y0,z=f(x,y):L:y=y0d¼

8、êêAÛ¿Â:∂z=tanα∂xM111nnn!!!êêê©©©z=f(x,y),∂zf(x0,y0+∆y)−f(x0,y0)=lim∂y∆yM∆y→0½x=x0,z=f