PX275 - A3e - chain rule additional

comments

more generally:
- $x_{1}, x_{2}, x_{3}, \dots x_{i}$ are used rather than $x, y, t$
- $u_{1}, u_{2}, u_{3}, \dots u_{j}$ are used rather than $ρ, θ ϕ$
- for functions, $f, g$ , just $f$ is used
a more compact notation:

\frac{\partial f}{\partial u_{j}} = \sum_{i} \frac{\partial f}{\partial x_{i}} \frac{\partial x_{i}}{\partial u_{j}}

in operator form:

\frac{\partial}{\partial u_{j}} = \sum_{i} \frac{\partial x_{i}}{\partial u_{j}} \frac{\partial}{\partial x_{i}}

warning

\begin{matrix} x = ρ \cos θ \\ ⟹ \frac{\partial x}{\partial ρ} = \cos θ \\ ρ^{2} = x^{2} + y^{2} \\ or, ρ = (x^{2} + y^{2})^{\frac{1}{2}} \\ ⟹ \frac{\partial ρ}{\partial x} = \frac{x}{(x^{2} + y^{2})^{\frac{1}{2}}} = \frac{x}{ρ} \\ \frac{\partial ρ}{\partial x} = \cos θ \end{matrix}

expected:

\frac{\partial x}{\partial ρ} = {(\frac{\partial ρ}{\partial x})}^{- 1}

the reason:

{(\frac{\partial x}{\partial ρ})}_{θ} and {(\frac{\partial ρ}{\partial x})}_{y}

- different variables are being held constant

what does 'holding variables constant' mean?

consider $(\frac{\partial x}{\partial ρ})_{θ} and (\frac{\partial ρ}{\partial x})_{y} = \cos θ$
holding $θ$ constant: $x \to x + d x$ ; $y \to y + d y$
holding $y$ constant $x \to x + d x$ ; $θ \to θ - d θ$
in general: 'reciprocity relationship' will only hold if the variable being held constant is the same in both sides, ie: ${(\frac{d y}{d x})}_{z} = {(\frac{d x}{d y})}_{z}^{- 1}$