(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Folks, how is the following expansion obtained for the following function

##F(x,u,u')## where x is the independent variable.

The change ##\epsilon v## in ##u## where ##\epsilon## is a constant and ##v## is a function is called the variation of ##u## and denoted by ##\delta u \equiv \epsilon v##

2. Relevant equations

3. The attempt at a solution

##\Delta F=F(x,u+\epsilon v, u'+\epsilon v')-F(x,u,u')##.

Expanding in powers of ##\epsilon## (treating ##u+\epsilon v## and ##u'+\epsilon v'## as dependent functions)

##\displaystyle \Delta F= F(x,u,u')+ \epsilon v \frac {\partial F}{\partial u}+ \epsilon v' \frac{\partial F}{\partial u'}+ \frac{(\epsilon v)^2}{2!} \frac{\partial^2 F}{\partial u^2}+\frac{(\epsilon v)(\epsilon v')}{2!} \frac{\partial^2 F}{\partial u \partial u'}+\frac{(\epsilon v')^2}{2!} \frac{\partial^2 F}{\partial u'^2}+....-F(x,u,u')##

How is the above line obtained...I looked at Taylor expansion but could not recognize its application to this function. Any links or tips will be appreciated.

Regards

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# Homework Help: Variational Operator/First Variation - Taylor Expansion

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