设z=x^3f(xy,y x^2),f具有连续二阶偏导数
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xy+xz=8-x²yx+yz=12-y²zy+zx=-4-z²x(x+y+z)=8y(x+y+z)=12z(x+y+z)=-4(x+y+z)²=8+12-4=
由于x-y=a,z-y=10得x-z=a-10并且由x²+y²+z²-xy-yx-zx=1/2[(x-y)²+(y-z)²+(z-x)²]=
对方程两边求全微分得:(e^z-1)dz+y^3dx+3xy^2dy=0(方法和求导类似)移项,有dz=-(y^3dx+3xy^2dy)/(e^z-1)
e^z-z+xy^3=0偏z/偏x:z'e^z-z'+y^3=0y^3=z'(1-e^z)z'=y^3/(1-e^z)偏z/偏y:z'e^z-z'+3xy^2=0z'=3xy^2/(1-e^z)偏z/
先求一阶导数,由于f有两个分量,要先对f的两个分量求导,再根据复合函数求导,两个分量对x求导,也就是z对x的一阶导数是:f1*y-f2*y/x^2,接下来再让这个式子对x求导,注意,这里利用乘法的导数
x^2+y^2+z^2-3xyz=0两边对x求偏导,2x+2z*dz/dx-3yz-3xydz/dx=0从中解得:dz/dx=(3yz-2x)/(2z-3xy)(1)同理:dz/dy=(3xz-2y)
x=6-3y &nbs
令u=xy,v=x+yz=f(u,v)az/ax=y(fu)+(fv)a^2z/axay=a(az/ax)/ay=a(y(fu)+(fv))/ay=(fu)+y(a(fu)/ay)+a(fv)/ay=
x+4y+3z=0x=-4y-3z3x-2y-5z=03x=2y+5zx=(2y+5z)/3所以-4y-3z=(2y+5z)/3-12y-9z=2y+5z14y=-14zy=-z所以x=-4*(-z)
=xy-3xy+2xy-xy=-xy
设u=xy,v=y/x,则z=f(u,v),所以ðz/ðx=f'1*ðu/ðx+f'2*ðv/ðx=yf'1-yf'2/x^2,注意到f'1
题目有问题,yx/(y+x)=4/3应该是yz/(y+z)=4/3xy/(x+y)=-2(x+y)/(xy)=-1/21/x+1/y=-1/2(1)yz/(y+z)=4/3(y+z)/(yz)=3/4
因为y=3xy+x,所以x-y=-3xy,当x-y=-3xy时,2x+3xy−2yx−2xy−y=2(x−y)+3xy(x−y)−2xy=2(−3xy)+3xy−3xy−2xy=35.
设u=xy,v=y/x,则z=x³f(u,v),au/ax=y,av/ax=-y/x²故az/ax=3x²f(u,v)+x³f'u(u,v)(au/ax)+x&
1.z'x=3x²y²z'y=2x³y2.z'x=4x³z'y=3y³3.z'x=ye^(xy)+2xyz'y=xe^(xy)+x²4.u'
∵x-y=4xy,∴2x+3xy-2yx-2xy-y=2(x-y)+3xyx-y-2xy=8xy+3xy4xy-2xy=112.故答案为:112.
3xy-3xy-xy+2yx=-xy+2xy=xy
XY=XZ+YX?那么也就是XY=X(Z+Y)咯,Y=Z+Y?无法证明的.题抄错啦~`
u=x^2+y∂u/∂x=2x∂u/∂y=1du=(∂u/∂x)dx+(∂u/∂y)dy=2xdx+dy