Exercises

1. 1. Generate a surface plot and contour plot for the following functions on the given domains:
      1. $$f(x,y) = \frac{x}{1+x^2+y^2}$$ for $-3 \leq x \leq 3$ and $-3 \leq y\leq 3$.
      2. $g(x,y)= \cos(x) \cos(y)$ for $0 \leq x \leq 2 \pi$ and $0 \leq y \leq 2 \pi$.

   2. What does the contour plot look like in the regions where the surface plot has a steep incline? What does it look like where the surface plot is almost flat?
   3. What can you say about the surface plot in a region where the contour plot looks like a series of nested circles?

2. Generate a surface plot for the following functions on the domains given.
   1. $r=z^2(1+\cos(2\theta))$ for $0 \leq \theta \leq 2\pi$ and $-1 \leq z \leq 1$. Use cylindrical coordinates.
   2. $$\rho = \exp(\frac{\theta}{3}) \sin(\phi)$$ for $0 \leq \theta \leq 2\pi$ and $0 \leq \phi \leq \pi$. Use spherical coordinates.

3. Consider the following function $$f(x,y) = \exp(-x^2) \cos(y)$$ for $-2 \leq x \leq 2$ and $0 \leq y \leq 2 \pi$ which looks like a deep valley with a mountain opposite it. Is it possible to find a path from the point $$(1,\frac{\pi}{2},0)$$ to $$(-1,\frac{3\pi}{2},0)$$ such that the value of $z$ is always between $-0.25$ and $0.25$ ? You do not have to find a formula for your path, but you must present convincing evidence that it exists. For example, you might want to sketch your path in by hand on an appropriate countour plot.