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: Continuously differentiable function

  1. Apr 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that if ##f## is a continuously differentiable real valued function on an open interval in ##E^2## and ##\partial^2f/\partial x\partial y=0,## then there are continuously differentiable real-valued functions ##f_1,f_2## on open intervals in ##\mathbb{R}## such that ##f(x,y)=f_1(x)+f_2(y).##

    How can I prove this?

    2. Relevant equations

    None

    3. The attempt at a solution

    Let ##(x_0,y_0)\in E^2## and integrate twice:

    ##0=\int_{y_0}^y\int_{x_0}^x\partial_x(\partial_yf(x',y'))dx'dy'=\int_{y_0}^y(\partial_yf(x,y')-\partial_yf(x_0,y'))dy'=f(x,y)-f(x,y_0)-f(x_0,y)+f(x_0,y_0).##
     
  2. jcsd
  3. Apr 7, 2014 #2
    This looks fine. Was there any problem with this?
     
  4. Apr 7, 2014 #3
    Nope, I was just confirming. Thanks for confirming!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted