Jacobian | Sahithyan's Notes

For a transformation $T$ from the $uv$ -plane to the $xy$ -plane defined by $x = x(u,v)$ , $y = y(u,v)$ , the Jacobian of $T$ is

J(u,v) = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\[6pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}

For a transformation $T$ from $uvw$ -space to $xyz$ -space defined by $x = x(u,v,w)$ , $y = y(u,v,w)$ , $z = z(u,v,w)$ , the Jacobian of $T$ is

J(u,v,w) = \frac{\partial(x,y,z)}{\partial(u,v,w)} = \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} & \dfrac{\partial x}{\partial w} \\[6pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} & \dfrac{\partial y}{\partial w} \\[6pt] \dfrac{\partial z}{\partial u} & \dfrac{\partial z}{\partial v} & \dfrac{\partial z}{\partial w} \end{vmatrix}

Examples:

$\dfrac{\partial(x,y)}{\partial(r,\theta)} = r$ , for polar coordinates $x = r\cos\theta$ , $y = r\sin\theta$
$\dfrac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} = \rho^2\sin\phi$ , for spherical coordinates

Change of Variables

Let $T$ be a transformation from the $uv$ -plane to the $xy$ -plane, mapping a region $S$ onto a region $R$ . If:

$T$ is one-to-one on $S$
$T$ has continuous first partial derivatives
$\dfrac{\partial(x,y)}{\partial(u,v)}$ is non-zero and does not change sign on $S$

then

\iint_R f(x,y)\,\text{d}A = \iint_S f(x(u,v), y(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| \text{d}A

Analogously, for a transformation $T$ from $uvw$ -space to $xyz$ -space mapping $S$ onto $R$ , satisfying the same 3 conditions with $\dfrac{\partial(x,y,z)}{\partial(u,v,w)}$ in place of $\dfrac{\partial(x,y)}{\partial(u,v)}$ ,

\iiint_R f(x,y,z)\,\text{d}V = \iiint_S f(x(u,v,w), y(u,v,w), z(u,v,w)) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| \text{d}V