Question

If J is Jacobian prove that JJ' = 1.
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Answer 1

SOLUTION

TO PROVE

If J is Jacobian prove that JJ' = 1

EVALUATION

Let u and v are two functions of x and y

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

Now

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

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

Interchanging row and column

∴ JJ'

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

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

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

$= 1$

Hence proved

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