# Inconsistency going from Polar to Cartesian coordinates

If I have a scalar potential $\phi = \frac{\cos(\theta)}{r}$ and $\vec{V}=\nabla\phi$ and I am going to find the vector field $\vec{V} = \left(\frac{\partial \phi}{\partial r}, \frac{1}{r}\frac{\partial \phi}{\partial \theta}\right)$. I get $\left(-\frac{\cos(\theta)}{r^2},-\frac{\sin(\theta)}{r^2}\right)$. If I change this expression into cartesian coordinates I get $\vec{V} = \left(\frac{-x}{(x^2+y^2)^{3/2}}, \frac{-y}{x^2+y^2)^{3/2}}\right)$

However, if I change $\phi$ to Cartesian coordinates first and use $\vec{V} = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}\right)$, I get $\vec{V} = \left(\frac{y^2 -x^2}{(x^2 + y^2)^2}, \frac{-2xy}{x^2+y^2)^{2}}\right)$

This looks like inconsistency to me...what is going on?

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This is not an inconsistency since the vector is expressed in two different basis. The first one is in the $(\hat{e}_r, \hat{e}_{\theta})$ basis while the second one is in $(\hat{e}_x, \hat{e}_{y})$ basis.
$\vec{V}$ in polar is $V_r \hat{e}_r + V_{\theta} \hat{e}_{\theta}$. Recall that $$\hat{e}_r = \cos(\theta) \hat{e}_x + \sin(\theta) \hat{e}_y = \dfrac{x \hat{e}_x + y \hat{e}_y}{\sqrt{x^2 + y^2}}$$ $$\hat{e}_{\theta} = -\sin(\theta) \hat{e}_x + \cos(\theta) \hat{e}_y = \dfrac{-y \hat{e}_x + x \hat{e}_y}{\sqrt{x^2 + y^2}}$$
$$\vec{V} = V_r \hat{e}_r + V_{\theta} \hat{e}_{\theta} = V_r \left(\dfrac{x \hat{e}_x + y \hat{e}_y}{\sqrt{x^2 + y^2}}\right) + V_{\theta} \left(\dfrac{-y \hat{e}_x + x \hat{e}_y}{\sqrt{x^2 + y^2}} \right) = \dfrac{(xV_r - y V_{\theta}) \hat{e}_x + (yV_r + x V_{\theta}) \hat{e}_y}{\sqrt{x^2 + y^2}}$$