1.设z=z(x,y)是由方程式e的z次方=xyz所含的隐函数,求dz 2.计算出曲面z=2-x^-y^2与xoy坐标面所围成的体积 设函数z=z(x,y)是由方程z+e的z次方=xy所确定的隐...
\u8bbez=z(x,y)\u662f\u7531\u65b9\u7a0b\u5f0fe\u7684z\u6b21\u65b9=xyz\u6240\u542b\u7684\u9690\u51fd\u6570,\u6c42dze^z=xyz
\u4e24\u8fb9\u5bf9x\u6c42\u504f\u5bfc
e^z*z'(x)=y(z+x*z'(x))
z'(x)=yz/(e^z-xy)
∂z/∂x=yz/(e^z-xy)
\u539f\u5f0f\u5bf9y\u6c42\u504f\u5bfc
e^z*z'(y)=x(z+y*z'(y))
∂z/∂y=xz/(e^z-xy)
dz=∂z/∂x*dx+∂z/∂y*dy
=yz/(e^z-xy)*dx+xz/(e^z-xy)*dy
\u4ee4F(x\uff0cy\uff0cz\uff09= z+z^e-xy=0
\u2234Fx=y Fz=-1+e^z\uff0c\u6709\u9690\u51fd\u6570\u8ba2\u7acbZ\u5148\u5bf9x\u504f\u5bfc=y/1+e^z
\u2234Fy=x \u6709\u9690\u51fd\u6570\u8ba2\u7acbZ\u5148\u5bf9y\u504f\u5bfc=x/1+e^z
\u6240\u4ee5Z\u5148\u5bf9x\u518d\u5bf9y\u6c42\u504f\u5bfc(y/1+e^z)dx+(x/1+e^z)dy
\u610f\u4e49\uff1a
\u5fae\u79ef\u5206\u5b66\u7684\u521b\u7acb\uff0c\u6781\u5927\u5730\u63a8\u52a8\u4e86\u6570\u5b66\u7684\u53d1\u5c55\uff0c\u8fc7\u53bb\u5f88\u591a\u7528\u521d\u7b49\u6570\u5b66\u65e0\u6cd5\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u8fd0\u7528\u5fae\u79ef\u5206\uff0c\u8fd9\u4e9b\u95ee\u9898\u5f80\u5f80\u8fce\u5203\u800c\u89e3\uff0c\u663e\u793a\u51fa\u5fae\u79ef\u5206\u5b66\u7684\u975e\u51e1\u5a01\u529b\u3002
\u524d\u9762\u5df2\u7ecf\u63d0\u5230\uff0c\u4e00\u95e8\u5b66\u79d1\u7684\u521b\u7acb\u5e76\u4e0d\u662f\u67d0\u4e00\u4e2a\u4eba\u7684\u4e1a\u7ee9\uff0c\u800c\u662f\u7ecf\u8fc7\u591a\u5c11\u4eba\u7684\u52aa\u529b\u540e\uff0c\u5728\u79ef\u7d2f\u4e86\u5927\u91cf\u6210\u679c\u7684\u57fa\u7840\u4e0a\uff0c\u6700\u540e\u7531\u67d0\u4e2a\u4eba\u6216\u51e0\u4e2a\u4eba\u603b\u7ed3\u5b8c\u6210\u7684\uff0c\u5fae\u79ef\u5206\u4e5f\u662f\u8fd9\u6837\u3002
(e^z -xy)dz=(yz)dx+(xz)dy；
dz=[yz/(e^z-xy)]dx+[xz/(e^z-xy)]dy=[z/(xz-x)]dx+[z/(yz-y)]dy；
（2）曲面 z=2-x²-y² 为一伞形曲面，当 z=2 时，x=y=0；当 z=0（xoy 平面） 时，曲面与 xoy 平面的交线为圆 x²+y²=2，z=0~2 之间曲面围成的封闭空间是一圆锥体；
V=πr²*h/3=π(√2)²*2/3=4π/3；
1 e^z=xyz
e^zz'x=yz+xyz'x z'x=yz/(xy-e^z)=yz/(xy-xyz)=z/(x-xz)
类似 z'y=z/(y-yz)
dz=[z/(x-xz)]dx+[z/(y-yz)]dy
2.立体在xoy坐标面的投影D：x^2+y^2《2
V=∫∫(2-x^2-y^2)dxdy，用极坐标
=∫(0,2π)dθ∫(0,√2)r(2-r^2)dr
=2π(2-1)
=2π
绛旓細棣栧厛灏鏂圭▼ 3xy+2x-4y-z=e^2 绉婚」寰楀埌 z=3xy+2x-4y-e^2銆傚洜姝わ紝鎴戜滑鍙互鐢ㄩ殣鍑芥暟姹傚娉曟潵姹傝Вz鍏充簬x鍜寉鐨勫亸瀵兼暟锛∂z/∂x = 3y + 2 ∂z/∂y = 3x - 4 鍥犳锛寊鍏充簬x鍜寉鐨勫亸瀵兼暟鍒嗗埆涓 3y+2 鍜 3x-4銆傜敱浜 z(1,1)=0锛屾墍浠ユ垜浠彲浠ヨ绠楀嚭锛z(x,...
绛旓細锛1锛夊鏂圭▼x2+y2-z=蠁锛坸+y+z锛変袱杈规眰寰垎寰楋細2xdx+2ydy-dz=蠁'锛坸+y+z锛?锛坉x+dy+dz锛夛紝鍗筹紙蠁'+1锛塪z=锛-蠁'+2x锛塪x+锛-蠁'+2y锛dy鈭磀z锛(?蠁鈥+2x)dx+(?蠁鈥+2y)dy蠁鈥+1锛堚埖蠁'鈮-1锛锛2锛鐢憋紙1锛夌煡?z?x锛?蠁鈥+2x蠁鈥+1锛?z?y锛?蠁鈥+2y蠁鈥...
绛旓細鐢辨柟绋e鈭鈥揷os(xy)=1寰垎寰 e^zdz+sin(xy)*(ydx+xdy)=0,鎵浠z=-e^(-z)*sin(xy)(ydx+xdy).
绛旓細鍥犳锛岀敱璇ラ殣鍑芥暟纭畾鐨勫嚱鏁z=z(x,y)鐨勫亸瀵兼暟涓猴細az/ax=-(af/x)/(af/az)=-[(-yf1-zf2)/x^2]/(f2/x)=[(yf1+zf2)/x^2]/(f2/x)=(yf1+zf2) / xf2 az/ay=-(af/y)/(af/az)=-(f1/x)/(f2/x)=-f1/f2 浜庢槸锛宒z =(az/ax)dx+(az/ay)dy ={[(yf1+zf2)/x...
绛旓細瀵鏂圭▼涓よ竟姹傚井鍒嗭紝寰 F1*(dx-dz)+F2*(dy-dz) = 0锛屾暣鐞嗘垚 dz = ---dx + ---dy锛屽嵆鍙緱鍒 Dz/Dx = ...銆
绛旓細鈭x锛漧nzy鈭磝=lnz-lny鈭翠袱杈瑰x姹傚亸瀵煎緱1锛1z??z?x?1y鍙堝綋x=0锛寉=1锛寊=1鈭?z?x|(0锛1)=1+1=2
绛旓細璁綟(x,y,z)=z^2-2xyz-1 鍒橣x=-2yz,Fy=-2xz,Fz=2z-2xy 伪z/伪x=-Fx/Fz=-(-2yz)/(2z-2xy)=yz/(z-xy)伪z/伪y=-Fy/Fz=xz/(z-xy)鎵浠 dz=伪z/伪x dx+伪z/伪y dy =yzdx/(z-xy)+xzdy/(z-xy)
绛旓細璁緕=z(x,y)鏄敱鏂圭▼z=y+ln锛坸/z锛夌‘瀹氱殑闅愬嚱鏁帮紝姹傚亸z鍋弜锛屽姹傚嚭鏉ョ殑鍋弞鍋弜鍐嶅亸y 濡傛灉鍋氫簡鐨勮瘽锛屼細鏈夐澶栫殑濂栧姳鐨 瑙ｏ細F(x,y,z)=z-y+lnx-lnz鈮0...(1)ӘF/Әx+(ӘF/Әz)(Әz/Әx)=1/x+(1-1/z)(Әz/Әx)=0,...
绛旓細鐢z=未(x-y,y-z),璁疚(u,v)瀵箄銆乿鐨涓闃惰繛缁亸瀵兼暟鍒嗗埆涓何粹1鍜屛粹2锛屽垯 z鈥榵=未鈥1*(x-y)'x+未鈥2*(y-z)'x =未鈥1-未鈥2*z'x 鈭磟鈥榵=未鈥1/(1+未鈥2), (1)瑕佹眰z瀵箈鐨勪簩闃跺亸瀵兼暟锛堢畝鍐欎负z鈥樷欙級锛屛(u,v)椤绘湁浜岄樁鍋忓鏁拔粹樷11銆佄粹欌12=未鈥...
绛旓細z(x)+z(y)=-(f(x)+f(y))/f(z)f(x)=f1(1-z(x)-f2z(x))f(y)=-f1z(y)+f2(1-z(y))f(z)=-f1-f2 鎵浠(x)+z(y)=1+z(x)+z(y)寰梲(x)+z(y)=0.5 娉細鍔犳嫭鍙风殑鍧囦负鍏跺亸瀵兼暟锛宖1f2涔熸槸瀵兼暟銆
