- #1
Lee33
- 160
- 0
Homework Statement
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?
Homework Equations
None
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).##