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)