Some Non-Differentiable Surfaces
Version .75 for MapleVR7
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
Load the plotting package first.
Warning, the name changecoords has been redefined
Singularity along a line.
Singularity at a cone point.
A More Complicated Singularity.
Maple considerrs complex cube roots of -1 as well as real ones.
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;
> 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).