Question

F=xu +0-y, G = u^2 + vy +w, H =zu – v + vw, compute Ə(F, G, H)/(u, v, w).​

Answer 1

SOLUTION

GIVEN

$\sf{F = xu + v - y}$

$\sf{ G= {u}^{2} + vy + w}$

$\sf{H = zu - v + vw}$

TO DETERMINE

$\displaystyle \sf{ \frac{ \partial (F, G, H)}{ \partial (u,v,w)} }$

EVALUATION

Here it is given that

$\sf{F = xu + v - y}$

$\sf{ G= {u}^{2} + vy + w}$

$\sf{H = zu - v + vw}$

Now

$\displaystyle \sf{ \frac{ \partial (F, G, H)}{ \partial (u,v,w)} }$

$= \displaystyle \begin{vmatrix} \frac{ \partial F}{ \partial u} & \frac{ \partial F}{ \partial v} & \frac{ \partial F}{ \partial w}\\ \\ \frac{ \partial G}{ \partial u} & \frac{ \partial G}{ \partial v} & \frac{ \partial G}{ \partial w} \\ \\ \frac{ \partial H}{ \partial u} & \frac{ \partial H}{ \partial v} & \frac{ \partial H}{ \partial w} \end{vmatrix} \:$

$= \displaystyle \begin{vmatrix} \sf{x } & 1 & 0\\ \\ \sf{2u + y} & 0 & 1 \\ \\ \sf{z} & \sf{- 1 + w} & \sf{v} \end{vmatrix} \:$

$\sf{ = x(0 + 1 - w) - 1(2uv + vy - z) + 0}$

$\sf{ = - xw -2uv - vy + z}$

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Answer 2

this is the required solution.