Hey guys, neat forum! I've got a quick calculus question that I'd love to get some help on. Suppose that I have some function [itex]f[/itex] of two variables [itex]x[/itex] and [itex]y[/itex]. I can write this function as [itex]f=f(x,y)[/itex]. Now suppose that this function satisfies [itex]f(x,y)|_{x=0}=f(0,y)=0[/itex]. This means that I can write the function as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(x,y)=\int_{0}^{1}d t\frac{d}{d t}f( t x,y).

[/tex]

At least I think this is true because I can evaluate the integral on the r.h.s. as

[tex]

\int_{0}^{1}d t\frac{d}{d t}f( t x,y)=f( t x,y)|_{ t=0}^{ t=1}=f(x,y)-f(0,y)=f(x,y).

[/tex]

My problem is that I have a suspicion that [itex]f(x,y)[/itex] can also be written in the form

[tex]

f(x,y)=x\int_{0}^{1}d t\frac{\partial}{\partial x}f( t x,y),

[/tex]

but I can't quite see how one does this. For instance, an obvious approach is to try

[tex]

f(x,y) = \int_{0}^{1}d t\frac{d}{d t}f( t x,y)[/tex]

[tex]= \int_{0}^{1}d t\left[\frac{\partial( t x)}{\partial t}\frac{d}{d( t x)}f( t x,y)+\frac{\partial y}{\partial t}\frac{\partial}{\partial y}f( t x,y)\right][/tex]

[tex] = x\int_{0}^{1}d t\frac{d}{d( t x)}f( t x,y),[/tex]

but this isn't quite what I want because the derivative of [itex]f[/itex] inside the integral is with respect to [itex]tx[/itex], andnotwith respect to [itex]x[/itex], which is what I really want.

I'm probably missing something blindingly obvious. Any help with this (even a simple hint or two) would be cool.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Pretty simple question (I hope)

**Physics Forums | Science Articles, Homework Help, Discussion**