PX275 - A3c - change of variables and coordinate system

cylindrical coordinates

P (x, y) \to P (ρ, θ)

$x = ρ \cos θ$
$y = ρ \sin θ$
$ρ^{2} = x^{2} + y^{2}$
$\tan θ = y / x$

$x$ and $y$ are functions of $ρ$ and $θ$
at any point, $P$ , the function has a particular value, independent of the co-ordinate system used: $f (x, y) = g (ρ, θ)$
for a small step: $ρ \to ρ + d ρ$ , $θ \to θ + d θ$
using the chain rule: $f (x, y) = g (ρ, θ) :$

\begin{matrix} \frac{\partial g}{\partial ρ} & = \frac{\partial f}{\partial x} \frac{\partial x}{\partial ρ} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial ρ} & = \cos θ \frac{\partial f}{\partial x} + \sin θ \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial θ} & = \frac{\partial f}{\partial x} \frac{\partial x}{\partial θ} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial θ} & = - ρ \sin θ \frac{\partial f}{\partial x} + ρ \cos θ \frac{\partial f}{\partial y} \end{matrix}

\begin{aligned} \frac{\partial}{\partial ρ} & = \cos θ \frac{\partial}{\partial x} + \sin θ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial x} & = \cos θ \frac{\partial}{\partial ρ} - \frac{1}{ρ} \sin θ \frac{\partial}{\partial θ} \\ \frac{\partial}{\partial y} & = \sin θ \frac{\partial}{\partial ρ} + \frac{1}{ρ} \cos θ \frac{\partial}{\partial θ} \end{aligned}