__ ____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).