# Gradient of an inverse vector function?

1. Oct 10, 2012

### CAF123

1. The problem statement, all variables and given/known data
Consider $$f(\vec{x}) = |\vec{x}|^r,$$ where $\vec{x} \in ℝ^n$ and $r \in ℝ$.
Find $\vec{∇}f$
3. The attempt at a solution
I know a vector function maps real numbers to a set of vectors, but here I believe we have the opposite. (inverse of a vector function, assuming inverse exists?)
I am unsure of where to go next.

2. Oct 10, 2012

### Dr. T

Assuming the usual Euclidean norm on ℝ^n, the output is just a scalar. Computing the gradient, then, is a matter of taking partial derivatives. For example, the first component of the gradient is the partial of f with respect to x_1. If f= (sqrt(x_1^2 + ... + x_n^2))^r = (x_1^2 + ... + x_n^2) ^ (r/2) then the partial with respect to x_1 is r/2(x_1^2 + ... + x_n^2)^(r/2-1)*(2x_1). Continuing in this fashion gives the gradient.

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