Some Non-Differentiable Surfaces

Version .75 for MapleVR7

Summary

This worksheet includes:
a surface with a singularity along a line.
a surface with a cone point.
analysis of a non-differentiable surface with a more complicated
singularity.

Load the plotting package first.

> with(plots):

Warning, the name changecoords has been redefined

> setoptions3d(axes=framed);

Singularity along a line.

> plot3d(abs(x+y),x=-1..1,y=-1..1,view=0..1,orientation=[-62,75]);

Singularity at a cone point.

> plot3d(sqrt(x^2+y^2),x=-1..1,y=-1..1,view=0..1,orientation=[36,69],shading=zhue);

A More Complicated Singularity.

Maple considerrs complex cube roots of -1 as well as real ones.

> (-1.0)^(1/3);

So we define a cube root function which gves the real cube root of a negative number.

> real_cube_root := x -> if evalf(x) > 0 then evalf(x^(1/3)) else -(evalf((-x)^(1/3))) fi;

> real_cube_root(-1);

> f := (x,y) -> real_cube_root(x^3-3*x*y^2);

> plot3d(f,-.1.. .1,-.1.. .1,shading=zhue);

An argument for why this function is not differentiable at the origin:
Along the line y = k*x, this function is just linear - namely x*(1-3*k^2)^(1/3).
So directional derivatives exist in every direction.
Looking at the partial derivatives, one sees grad f would have to be (1 0) if it existed.
Hence the directional derivative in the direction (cos(theta),sin(theta)) would be
cos(theta) if f were differentiable.
But for the unit vector v =(cos(theta),sin(theta)), the function at tv is
t* cos(theta) *(1- 3 * tan^2(theta))^(1/3), so the directional derivative in the direction v
is cos(theta) *(1- 3 * tan^2(theta))^(1/3) rather than cos(theta).
Thus f is not differentiable at (0,0).