Images and other neat stuff

• 01/28/16: I sketched the graph of \(f(x,y)=x^2-y^2\), which I claimed is a saddle shape (or technically, a hyperbolic paraboloid).
  • Here is a computer drawing of the graph.
  • Here is the same graph intersected with the plane \(x=1\), giving the cross-section \(f(1,y)=1-y^2\).
  • Here is the same graph intersected with the plane \(y=-1\), giving the cross-section \(f(x,-1)=x^2-1\).
• 02/09/16: I discussed the level sets of the function \(f(x,y,z)=x^2+y^2-z^2\). Here are the three types of level set in the same picture.
• 02/11/16: We looked at continuity of some functions. In particular
  • \(f(x,y)=\frac{x^3}{x^2+y^2}\) is continuous at \((0,0)\). This can be seen by looking at the graph of the function, which has no holes in it, or by looking at the contours of the function and seeing how they behave near the origin.
  • \(f(x,y)=\frac{x^2-y^2}{x^2+y^2}\) is not continuous at \((0,0)\). Notice that there is a tear in the graph of the function and that the contours all seem to cross through the origin, suggesting that the limit cannot exist there.