Consider the scalar field φ=x2+y2-z2-1. Let H be the scalar field defined by
H = -0.5∇.(∇φ/ abs(∇φ)), where abs(∇φ) is the magnitude of ∇φ. Which makes that some sort of unit quantity. When H is evaluated for φ=0 it is the mean curvature of the level surface φ=0.
Calculate H. Write your answer as a function of z only.
Hint: Work in Cartesian coordinates x, y, z throughout the whole of part (c) and make sure that that you work out all of the derivatives before imposing the constraint Φ = 0. You will ﬁnd it useful to use the abbreviation r = √(x2+y2+z2. I didn't use any of this, so I'm doing it wrong!
The Attempt at a Solution
I got (∇φ/ abs(∇φ)) to be 1/√3 x i + 1/√3 y j +1/√3 z k where i, j and k are unit vectors. From this, ∇.(∇φ/ abs(∇φ))=1/√3. This is clearly wrong, wolfram alpha disagrees, I didn't use r, I don't know how to impose the constraint and I think it's something pretty fundamental that I'm missing! I have looked at examples online but they really haven't helped at all!