EDIT: Sorry, I think I posted this in the wrong section. I have moved it to "
Analysis, Topology and Differential Geometry". Please delete it!
Hi,
I have a past tier exam problem which I would like to check my solution for.
The question: Let p be real. Suppose is continuously differentiable, and satisfies
for all and for all .
Let denote the gradient of f at x and the dot product. Prove that
for all .
I first considered the n=1 case: Fixing , define 0,\infty)\to \mathbb{R}" alt="g0,\infty)\to \mathbb{R}" /> by . We have for all . Differentiating with respect to gives .
On the other hand, . I do not know how to justify this step however: f is defined on R-0, so how does one go about differentiating with respect to a variable that has a restricted domain?
I would then have , and taking gives the desired result.
Onto the general case, fixing and defining again, we have and on the other hand, by the chain rule,
. Taking again gives the result.
Is the above jump to Rn-0 ok?
Thanks so much for your help!