PX275 - B1d - time derivatives

in cartesians

\vec{r} = r_{x} \hat{i} + r_{y} \hat{j} + r_{z} \hat{k}

\begin{aligned} \vec{v} & = (\frac{d r_{x}}{d t} \hat{i} + \frac{d \hat{i}}{d t} r_{x}) + (\frac{d r_{y}}{d t} \hat{j} + \frac{d \hat{j}}{d t} r_{y}) + (\frac{d r_{z}}{d t} \hat{k} + \frac{d \hat{k}}{d t} r_{z}) \\ = \frac{d r_{x}}{d t} \hat{i} + \frac{d r_{y}}{d t} \hat{j} + \frac{d r_{z}}{d t} \hat{k} \end{aligned}

\vec{r} = ρ {\hat{e}}_{ρ} (ϕ) + z {\hat{e}}_{z}

\vec{v} = (\frac{d ρ}{d t} {\hat{e}}_{ρ} + ρ \frac{d {\hat{e}}_{ρ}}{d t}) + (\frac{d z}{d t} {\hat{e}}_{z} + z \frac{d {\hat{e}}_{ρ}}{d t})

\begin{aligned} {\hat{e}}_{ρ} & = \cos ϕ \hat{i} + \sin ϕ \hat{j} \\ \frac{d {\hat{e}}_{ρ}}{d ϕ} & = - \sin θ \hat{i} + \cos ϕ \hat{j} \\ \equiv {\hat{e}}_{ϕ} \\ ⟹ \frac{d {\hat{e}}_{ρ} (ϕ)}{d t} & = \frac{d {\hat{e}}_{ρ}}{d ϕ} \frac{d ϕ}{d t} \\ = \frac{d ϕ}{d t} {\hat{e}}_{ϕ} \end{aligned}

∴ \vec{v} = \frac{d ρ}{d t} {\hat{e}}_{ρ} + ρ \frac{d ϕ}{d t} {\hat{e}}_{ϕ} + \frac{d z}{d t} {\hat{e}}_{z}