The following is a question on a problem sheet I received this week for an advanced Vector Calculus unit. It's something of a "warm up" sheet.
Let $U$ be defined by $U = \mathbb{R}^2$ \ $\{(x,0):x \leq 0\}$, and define $\theta : U \rightarrow (-\pi,\pi)$ by requiring that $\theta(p)$ is an argument of $p$ for any $p \in U$.
The question continues after that.
I have absolutely no idea what "argument" means in this context, and my google searches are fruitless (mostly because argument isn't exactly a word in infrequent use). Any help?
The term "argument of $p$" is commonly used for the angle between the positive $x$-axis and the ray from the origin through $p$. I have mostly heard it used in the context of the complex plane, although it certainly works in the regular $\Bbb R^2$.