Question

Find the Jacobian of the transformation if u=x-y, v=y-z, w=z-x.​

Answer 1

SOLUTION

TO DETERMINE

The Jacobian of the transformation

u = x - y, v = y - z , w = z - x.

EVALUATION

Here the given transformation is

u = x - y, v = y - z , w = z - x

Hence the required Jacobian

$\displaystyle \sf{ = \frac{ \partial (u,v,w)}{ \partial (x,y,z)} }$

$= \displaystyle \begin{vmatrix} \frac{ \partial u}{ \partial x} & \frac{ \partial u}{ \partial y} & \frac{ \partial u}{ \partial z}\\ \\ \frac{ \partial v}{ \partial x} & \frac{ \partial v}{ \partial y} & \frac{ \partial v}{ \partial z} \\ \\ \frac{ \partial w}{ \partial x} & \frac{ \partial w}{ \partial y} & \frac{ \partial w}{ \partial z} \end{vmatrix}$

$= \displaystyle\begin{vmatrix} 1 & - 1 & 0\\ \\ 0 & 1 & - 1 \\ \\ - 1& 0 & 1 \end{vmatrix}$

= 1 + 1 + 0

= 2

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Answer 2

Solution ⤵️

To Find ⤵️

The Jacobian of the transformation,

u = x - y, v = y - z , w = z - x.

Calculation ⤵️

Here the given transformation is,

u = x - y, v = y - z , w = z - x

Hence the required Jacobian,

$\displaystyle \sf{ = \frac{ \partial (u,v,w)}{ \partial (x,y,z)} }$

$= \displaystyle \begin{vmatrix} \frac{ \partial u}{ \partial x} & \frac{ \partial u}{ \partial y} & \frac{ \partial u}{ \partial z}\\ \\ \frac{ \partial v}{ \partial x} & \frac{ \partial v}{ \partial y} & \frac{ \partial v}{ \partial z} \\ \\ \frac{ \partial w}{ \partial x} & \frac{ \partial w}{ \partial y} & \frac{ \partial w}{ \partial z} \end{vmatrix}$

$= \displaystyle\begin{vmatrix} 1 & - 1 & 0\\ \\ 0 & 1 & - 1 \\ \\ - 1& 0 & 1 \end{vmatrix}$

= 1 + 1 + 0

= 2

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