Question

If x=u^2-v^2,y=2uv find the jacobian of x, y with respect to u and v​

Answer 1

SOLUTION

GIVEN

x = u² - v² , y = 2uv

TO DETERMINE

The jacobian of x, y with respect to u and v

EVALUATION

Here it is given that x = u² - v² , y = 2uv

So the jacobian of x, y with respect to u and v

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

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

$= \displaystyle\begin{vmatrix} 2u & - 2v \\ \\ 2v & 2u \end{vmatrix}$

$= 4 {u}^{2} + 4 {v}^{2}$

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