1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Gradient of an inverse vector function?

  1. Oct 10, 2012 #1

    CAF123

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Consider [tex] f(\vec{x}) = |\vec{x}|^r, [/tex] where [itex] \vec{x} \in ℝ^n [/itex] and [itex] r \in ℝ[/itex].
    Find [itex] \vec{∇}f [/itex]
    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. jcsd
  3. Oct 10, 2012 #2
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook