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!

Homework Help: Minimization of the square of the gradient in a volume

  1. Feb 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Find an expression involving the function [tex]\phi(x_1, x_2, x_3)[/tex] that has a minimum average value of the square of its gradient within a certain volume V of space.


    2. Relevant equations

    We are studying functionals, though so far it has only been of one variable. We're considering [tex]J[y]=\int_{x_1}^{x_2}f(y, y'; x)dx[/tex] where [tex]y[/tex] is a function of x. This functional is minimized when f satisfies [tex]\frac{\delta f}{\delta y}-\frac{d}{dx}(\frac{\delta f}{\delta y'})=0[/tex]


    3. The attempt at a solution
    I have no idea how to apply this minimization principle with multiple variables.
     
  2. jcsd
  3. Feb 22, 2008 #2

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    A multivariate function is minimized when its first-order partial derivatives with respect to all of its arguments are simultaneously zero.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Minimization square gradient Date
Minimization of a paper cup Mar 30, 2018
Minimization and least squares/ridge regression Jul 18, 2012
Minimizing infinity norm squared Jun 6, 2011
Minimizing square of deviation / curve fitting May 22, 2009
Minimize the sum of the squares Jul 10, 2006