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**How, in general, do you "expand" an expression in powers of some variable?**

I'm studying mechanics right now out of the Landau-Lifschitz book, and he often talks about expanding expressions. But I honestly don't know how to do this, in general. I know about forming Taylor expansions of functions of one variable, but when things get to more than one variable, I start to get confused. Let me give you an example: Apparently, if we expand a function

[tex]

L(\vec{v}^2 + 2\vec{v} \cdot \vec{\epsilon} + \vec{\epsilon}^2)

[/tex]

"in powers of [tex]\vec{\epsilon}[/tex]," to first order, where [tex]\vec{\epsilon}[/tex] is some infinitesimal vector, we get

[tex]

L(\vec{v}^2) + \frac{\partial L}{\partial \vec{v}^2} 2 \vec{v} \cdot \vec{\epsilon}.

[/tex]

Could someone

*please*explain why that is? You know, how to

*get*that? And what's so special about expanding "in powers of [tex]\vec{\epsilon}[/tex]?" As best I can tell, we are letting [tex]\vec{\epsilon} = 0[/tex] when we do the expansion. So are we expanding [tex]L[/tex] about [tex]\vec{\epsilon} = 0[/tex]? If so, why are we taking a partial of [tex]L[/tex] with respect to [tex]\vec{v}^2[/tex]?

I have tried to use the information provided at http://en.wikipedia.org/wiki/Taylor_Series to figure this out - particularly what's at the bottom of the page - but without much success.