Vector field in cylindrical coordinates

[Les talons]

Homework Statement

Sketch each of the following vector fields.
$E_5 = \hat \phi r$
$E_6 = \hat r \sin(\phi)$

I wish to determine the $\hat x$ and $\hat y$ components for the vector fields so that I can plot them using the quiver function in MATLAB.

Homework Equations

A cylindrical coordinate is given by $(r, \phi, z)$
$r = \sqrt[+]{x^2 + y^2}$
$\phi = \arctan(y/x)$
$z = z$
Transformation from cylindrical coordinates into Cartesian coordinates:
$\hat x = \hat r \cos(\phi) - \hat \phi \sin(\phi)$
$\hat y = \hat r \sin(\phi) + \hat \phi \cos(\phi)$

The Attempt at a Solution

$E_5 = \hat \phi r$
Since the given vector field $E_5$ only has a $\hat \phi$ component,
$\hat x = \hat r \cos(\phi) - \hat \phi \sin(\phi) = 0 \cos(\phi) - r \sin(\phi) = - \sqrt[+]{x^2 + y^2} \sin(\phi)$
Then I transformed $\phi$ in terms of $x$ and $y$ based on the definition of a cylindrical coordinate
$\hat x = - \sqrt{x^2 + y^2} \sin(\arctan(y/x)) = \displaystyle \frac{-y \sqrt{x^2 + y^2} }{x \sqrt{1 +y^2 /x^2} }$
I followed the same approach for the $\hat y$ component
$\hat y = \hat r \sin(\phi) + \hat \phi \cos(\phi) = 0 \sin(\phi) + r \cos(\phi) = \sqrt[+]{x^2 + y^2} \cos(\phi)$
$\hat y = \sqrt{x^2 + y^2} \cos(\arctan(y/x)) = \displaystyle \frac{\sqrt{x^2 + y^2} }{\sqrt{1 +y^2 /x^2} }$
Then
$E_5 = - \hat x \displaystyle \frac{y \sqrt{x^2 + y^2} }{x \sqrt{1 +y^2 /x^2} } + \hat y \displaystyle \frac{\sqrt{x^2 + y^2} }{\sqrt{1 +y^2 /x^2} }$

Is this approach valid? Thanks for helping. I'll post my attempt for $E_6$ in a little while.

I'm having trouble thinking about the vector field since I'm used to working in Cartesian coordinates, so I can't tell if this makes sense.

 

[Les talons]

$E_6 = \hat r \sin(\phi)$
$\hat x = \sin(\phi) \cos(\phi) = \sin(\arctan(y/x)) \cos(\arctan(y/x))$
$\hat y = \sin^2(\phi) = \sin^2(\arctan(y/x))$

 

[Les talons]

If anyone happens upon this thread looking for help in a similar question, the correct conversions are as follows for the $\hat x$ and $\hat y$ components of $\vec E_5$:
$E_{5x} = E_{5r} \cos \phi - E_{5\phi} \sin \phi$
$E_{5y} = E_{5r} \sin \phi + E_{5\phi} \cos \phi$

and similarly for $\vec E_6$ (or another vector in cylindrical coordinates)