This seems like a trivial question, and I'm just not sure if I'm doing it right.
I have vector in cartesian coordinate system: $\vec{N} =y\vec{a_x} −2x\vec{a_y} + y\vec{a_z}$. And I need to represent it in cylindrical coord.
Relevant equations:
$$A_\rho=A_xcos\phi+A_ysin\phi$$
$$A_\phi=−A_xsin\phi+A_ycos\phi$$
$$A_z=A_z$$
What is cofusing me is this: The formula for $\phi$ is $\phi=arctan(\frac{y}{x})$ .
Are those $x$ and $y$ in fact $a_x$ and $a_y$? If so, then for my problem, wouldn't it be $\phi=arctan(\frac{-2x}{y})$? And do I need to change the unit vectors too?
1 Answer
$\begingroup$A vector field is defined over a region in space $\mathbb{R}^3 :$ $(x,y,z)$ or $(r,\phi, z)$, whichever coordinate system you may choose to represent this space. Your vector $\vec{N}$ should be defined in this space at a position vector $\vec{r} = (x,y,z)$ or $(r,\phi, z)$. So you need to find
$\phi = \arctan \left( \frac{y}{x} \right)$
Simply put: A vector field without reference to its position makes no sense in a transformation of coordinates.
$\endgroup$
