Question

the jacobian of p,q,r with respect to x,y,z where p=x+y+z ,q=y+z,r=z​

Answer 1

SOLUTION

TO DETERMINE

The Jacobian of p,q,r with respect to x,y,z where p=x+y+z ,q=y+z,r=z

EVALUATION

Here it is given that p=x+y+z ,q=y+z,r=z

So the required Jacobian

= J

$= \displaystyle \: \frac{ \partial (p,q,r)}{ \partial(x,y,z)}$

$= \displaystyle\begin{vmatrix} \frac{ \partial p}{ \partial x} & \frac{ \partial p}{ \partial y} & \frac{ \partial p}{ \partial z} \\ \\ \frac{ \partial q}{ \partial x} & \frac{ \partial q}{ \partial y} & \frac{ \partial q}{ \partial z} \\ \\ \frac{ \partial r}{ \partial x} & \frac{ \partial r}{ \partial y} & \frac{ \partial r}{ \partial z} \end{vmatrix}$

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

$= 1$

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Answer 2

$\huge \rm {\underline{\underline{ \red{Solution}}}}$

Given that :

$\boxed{{ \purple{p = x + y + z , \: q = y+z, \: r=z}}}$

The required Jacobian :

$\rm{ \color{navy}{ = J}}$

${ \color{navy}{= \displaystyle \frac{ \partial (p,q,r)}{ \partial(x,y,z)}=∂(x,y,z)∂(p,q,r)}}</p><p>$

$\begin{gathered}{ \color{navy}{= \displaystyle\begin{vmatrix} \frac{ \partial p}{ \partial x} & \frac{ \partial p}{ \partial y} & \frac{ \partial p}{ \partial z} \\ \\ \frac{ \partial q}{ \partial x} & \frac{ \partial q}{ \partial y} & \frac{ \partial q}{ \partial z} \\ \\ \frac{ \partial r}{ \partial x} & \frac{ \partial r}{ \partial y} & \frac{ \partial r}{ \partial z} \end{vmatrix}}}\end{gathered}$

$\begin{gathered}{ \color{navy}{= \displaystyle\begin{vmatrix} 1 & 1 & 1\\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{vmatrix}}}\end{gathered}$