PX153 - H6 - gradient operator

from PX153 - H3 - gradients of functions of three variables:

\vec{\nabla} f = ({\hat{\vec{e}}}_{x} \frac{\partial}{\partial x} + {\hat{\vec{e}}}_{y} \frac{\partial}{\partial y}) f

we have,

{\hat{\vec{e}}}_{x} = \cos θ {\hat{\vec{e}}}_{r} - \sin θ {\hat{\vec{e}}}_{θ}, {\hat{\vec{e}}}_{y} = \sin θ {\hat{\vec{e}}}_{r} + \cos θ {\hat{\vec{e}}}_{θ}

to find: ${\hat{\vec{e}}}_{x} \frac{\partial f}{\partial r}$ in terms of $r, θ$ :
- using chain rule:
$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial f}{\partial θ} \frac{\partial θ}{\partial x}$
- from $r = \sqrt{x^{2} + y^{2}}$ :
$\frac{\partial r}{\partial x} = \frac{2 x}{2 \sqrt{x^{2} + y^{2}}} = \frac{r \cos θ}{r} = \cos θ$
- from $θ = \arctan (\frac{y}{x})$ :
$\frac{\partial θ}{\partial x} = \frac{1}{1 + (\frac{y}{x})^{2}} \cdot \frac{- y}{x^{2}} = \frac{- r \sin θ}{r^{2}} = - \frac{\sin θ}{r}$
- we get:
${\hat{\vec{e}}}_{x} \frac{\partial f}{\partial y} = (\cos θ {\hat{\vec{e}}}_{r} - \sin θ {\hat{\vec{e}}}_{θ}) (\cos θ \frac{\partial f}{\partial r} + \frac{\sin θ}{r} \frac{\partial f}{\partial θ})$
similarly, wrt $y$ :

\frac{\partial r}{\partial y} = \sin θ, a n d \frac{\partial θ}{\partial y} = \frac{\cos θ}{r}

- we get:
$$\hat{\vec e}{y} \ \frac{\partial f}{\partial y} = (\sin\theta ; \hat{\vec e} + \cos\theta; \hat{\vec e}_{\theta}) (\sin\theta \frac{\partial f}{\partial r} + \frac{\cos\theta}{r}\ \frac{\partial f}{\partial \theta})$$

combine and simplify to get:

\vec{\nabla} f (r, θ) = ({\hat{\vec{e}}}_{r} \frac{\partial}{\partial r} + {\hat{\vec{e}}}_{θ} \frac{1}{r} \frac{\partial}{\partial θ}) f

eg: for $U (r) = \frac{q_{1} q_{2}}{4 π ϵ_{0} r}$
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