Question

By a change of variable $x(u,v)=uv,y(u,v)=v/u$ in double integral, the integrand $f(x,y)$ changes to $f(uv,v/u)\varphi (u,v)$. Then, $\varphi (u,v)$ is

• A. $2v/u$
• B. $2uv$
• C. $v^2$
• D. $1$

Answer 1

The correct option is A $2v/u$

$\int \int f(x,y)dx.dy=\int \int f(uv,v/u)\varphi (u,v)du.dv$
where $\varphi (u,v)=$Jacobian function
$\therefore J=\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right|$
$\therefore x(u,v)=uv,y(u,v)=v/u$
$\Rightarrow J=\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}v& u\\ \frac{-v}{u^2}& \frac{1}{u}\end{array}\right|=\frac{2v}{u}$