"Let D be $R^2$ \ {$(x,0) | x \leq 0$}. The polar angle gives for each $(x,y)$ a value $\theta(x,y)$ in the interval $(-\pi, \pi)$. The function $\theta$ is continuous and differentiable on D.
a) Show that $\frac{\partial \theta}{\partial y} = \frac {x}{x^2+y^2} $
b) Find the integral curves (filed lines) for the gradient vector field $\nabla \theta$
Okay, I'm pretty lost here.
a) I would assume that you're going to use implicit differention on $x=rcos\theta, y=rsin\theta, r^2=x^2+y^2$. Don't know where to take it from here though.
b) If I'm correct the integral curves are just $\nabla \theta = (\theta_x, \theta_y)$ so can't really do anything here without a.