- #1
bugatti79
- 794
- 1
Homework Statement
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##
Homework Equations
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