2434.sagews

var('x,y,z')

(x, y, z)

## Sage uses "abs" rather than vertical bars for the absolute-value function
## We'd write this as f(x,y) = | |x-y|-|x+y| |
f(x,y) = (abs(x)-abs(y))^2

## The graph of f has a sharp point at the origin,
## and isn't differentiable there
FPlot = plot3d(f, (x,-1,1), (y,-1,1))
FPlot

## What happens if we zoom in near (0,0,0)?
## The surface does not flatten out, which suggests that f is not differentiable.
## On the other hand, the z-coordinates are getting smaller and smaller.
## This suggests that the function ys that the function
r = 0.1
plot3d(f, (x,-r,r), (y,-r,r))

## No matter how far you zoom in, the surface still has a crease!
r = 0.01
plot3d(f, (x,-r,r), (y,-r,r))

## On the other hand, the partial derivatives f_x(0,0) and f_y(0,0) both exist
## (and both equal zero).  What this means geometrically is the following: if
## you slice the surface with the plane y=0 or x=0, you will get a curve that
## DOES have a tangent line.

r=1
XPlane = implicit_plot3d(x==0, (x,-r,r), (y,-r,r), (z,-r,r), color="red")
FPlot + XPlane

YPlane = implicit_plot3d(y==0, (x,-r,r), (y,-r,r), (z,-r,r), color="orange")
FPlot + XPlane + YPlane

##  But putting those slices together doesn't necessarily give you a tangent plane.  Don't be misled!