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Another question from a midterm:

Let $f:\mathbb{R}^3 \to \mathbb{R} $ be differentiable. It is also given that $f$ is constant on the following two spheres: $ S_1 = \{(x,y,z)|x^2 + y^2 +z^2 =1\} $ and $ S_2 = \{(x,y,z)| (x-1)^2 + (y-1)^2 + (z-1)^2 =1\} $ .

A. prove that on every point $(x,y,z)\in S_1 $ we have that $\nabla f(x,y,z)$ is a scalar multiple of $(x,y,z)$ .

B. Prove that there must exist a point on $S_1 \cup S_2 $ on which $\nabla f =0$ .

Will someone please help me ?

Thanks !

homogenity
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1 Answers1

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Some hints:

ad A: When $f({\bf p})=C$ and $\nabla f({\bf p})\ne{\bf 0}$ then $\nabla f({\bf p})$ is orthogonal to the level surface $f({\bf x})=C$ at ${\bf p}$.

ad B: Given A some points of $S_1\cup S_2$ are extremely stressed.

Christian Blatter
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