relationship of polar unit vectors to rectangular I'm looking at page 16 of Fleisch's Student's Guide to Vectors and Tensors.  The author is talking about the relationship between the unit vector in 2D rectangular vs polar coordinate systems.  They give these equations:
\begin{align}\hat{r} &= \cos(\theta)\hat{i} + \sin(\theta)\hat{j}\\
\hat{\theta} &= -\sin(\theta)\hat{i} + \cos(\theta)\hat{j}\end{align}
I'm just not getting it.  I understand how, in rectangular coordinates, $x = r \cos(\theta)$, but the unit vectors are just not computing.

The symbols on the left side of those equations don't make any sense.  If you wanted to change to a new pair of coordinates $(\hat{u}, \hat{v})$ by rotating through an angle $\theta$, then you would have
$$
\left\{\begin{align}
\hat{u} &= (\cos \theta) \hat{\imath} + (\sin \theta)\hat{\jmath} \\
\hat{v} &= (-\sin \theta) \hat{\imath} + (\cos \theta)\hat{\jmath}.
\end{align}\right.
$$
