Vector transformation, cylindrical to Cartesian

  1. 1. The problem statement, all variables and given/known data
    I have a result which is in the form (cylindrical coordinates):

    $$ A\boldsymbol{e_{\theta }}=kr\boldsymbol{e_{\theta }} $$

    And I have to provide the answer in cartesian coordinates.


    2. Relevant equations

    I know that the unit vectors:

    $$ \boldsymbol{\hat{\theta} }=\begin{bmatrix}-sin\ \theta
    \\
    cos\ \theta
    \end{bmatrix} $$

    and that
    $$ r=\sqrt{x^{2}+y^{2}} $$


    3. The attempt at a solution

    $$ kr\boldsymbol{e_{\theta }} =k\left (\sqrt{x^{2}+y^{2}}\right ) $$

    $$ \left (-sin\left(tan^{-1}\left(\frac{y}{x}\right ) $$
    $$ \right )\boldsymbol{e_{x}} $$
    $$ +cos\left (tan^{-1}\left (\frac{y}{x}\right )\boldsymbol{e_{y}} \right )\\ $$

    I can't seem to get further than this. I don't know if I've made a mistake, or whether there is some trig identity that can help me simplify further, but I know the final answer and it is much simpler.
    Any help much appreciated.
    Thanks in advance
    P.S. Why does the latex break down when the equation is too long?
     
    Last edited: Mar 11, 2012
  2. jcsd
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