Answer to Question #268880 in Calculus for Bhuvana

Question #268880

Find the Jacobian ∂(x, y, z)/∂(u, v, w).

(a) u = xy, v = y, w = x + z (b) u = x + y + z, v = x + y − z, w = x − y + z


1
Expert's answer
2021-11-24T17:17:24-0500

Solution;

(a)

Given;

u=xy

v=y

w=x+z

Now;

x=uy=uvx=\frac uy=\frac uv

y=vy=v

z=wx=wuvz=w-x=w-\frac uv

(x,y,z)(u,v,w)=xuxvxwyuyvywzuzvzw\frac{\partial(x,y,z)}{\partial(u,v,w)}=\begin{vmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} &\frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}&\frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u}&\frac{\partial z}{\partial v}&\frac{\partial z}{\partial w} \end{vmatrix} =1vuv200101vuv21\begin{vmatrix} \frac1v & \frac u{v^2}&0 \\ 0 & 1&0\\ -\frac1v&-\frac{u}{v^2}&1 \end{vmatrix} =1v(10)uv2(00)+0(0+1v)\frac1v(1-0)-\frac{u}{v^2}(0-0)+0(0+\frac1v)

Hence;

(x,y,z)(u,v,w)=1v\frac{\partial(x,y,z)}{\partial(u,v,w)}=\frac1v

(b)

u=x+y+z

v=x+y-z

w=x-y+z

Now;

v+w=x+yz+xy+z=2xv+w=x+y-z+x-y+z=2x

Hence;

x=v+w2x=\frac{v+w}{2}

Also;

uv=x+y+zxy+z=2zu-v=x+y+z-x-y+z=2z

Hence;

z=uv2z=\frac{u-v}{2}

Also;

y=x+zwy=x+z-w =v+w2+uv2w\frac{v+w}{2}+\frac{u-v}{2}-w

Hence;

y=uw2y=\frac{u-w}{2}

Therefore;

(x,y,z)(u,v,w)=xuxvxwyuyvywzuzvzw=012121201212120\frac{\partial(x,y,z)}{\partial(u,v,w)}=\begin{vmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} &\frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}&\frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u}&\frac{\partial z}{\partial v}&\frac{\partial z}{\partial w} \end{vmatrix}=\begin{vmatrix} 0 &\frac 12&\frac12\\ \frac12 & 0&-\frac{1}{2}\\ \frac12&-\frac12&0 \end{vmatrix}=0(0+14)12(0+14)+12(140)0(0+\frac14)-\frac12(0+\frac14)+\frac12(-\frac14-0)

Therefore;

(x,y,z)(u,v,w)=14\frac{\partial(x,y,z)}{\partial(u,v,w)}=-\frac14






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