PX275 - B3 - jacobian

the change of variables have been made geometrically, but this can be done in a more generalized way via a method using a construct called a jacobian
to transform from $(x, y, z) \to (u, v, w)$ , the jacobian for this transformation is written as:

J = \frac{\partial (x, y, z)}{\partial (u, v, w)}

d V = d x d y d z = | J | d u d v d w

where, $| J |$ is the absolute value of the jacobian

plane polar

J = \frac{\partial (x, y)}{\partial (p, ϕ)} = | \begin{matrix} \frac{\partial x}{\partial ρ} & \frac{\partial x}{\partial ϕ} \\ \frac{\partial y}{\partial ρ} & \frac{\partial y}{\partial ϕ} \end{matrix} |

remember:
$x = ρ \cos ϕ$
$y = ρ \sin ϕ$

J = ρ \cos^{2 ϕ} - (- ρ \sin)^{2} ϕ = ρ

area:

d A = d x d y = | J | d ρ d ϕ = ρ d ρ d ϕ

spherical polar

J = | \begin{matrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial θ} & \frac{\partial x}{\partial ϕ} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial θ} & \frac{\partial y}{\partial ϕ} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial θ} & \frac{\partial z}{\partial ϕ} \end{matrix} |

remember:
$x = r \sin θ \cos ϕ$
$y = r \sin θ \sin ϕ$
$z = r \cos θ$

\begin{matrix} J = - r^{2} \sin θ \\ d V = r^{2} \sin θ d r d θ d ϕ \end{matrix}