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In Gelfand and Fomin (Calculus of Variations) at page 14 they derive a formula for a certain variation. My problem is just one part of that derivation.

$$\Delta J= J[y+h]-J[y]=\int_{a}^{b} [F(x,y+h,y'+h')-F(x,y,y')]\ dx$$

They say that it follows by using Taylor's theorem that

$$\Delta J=\int_{a}^{b}[F_{y}(x,y,y')h+F_{y'}(x,y,y')h']\ dx +\ldots ,$$

where the subscripts denote partial derivatives with respect to the corresponding arguments, and the dots denote terms of order higher than 1 relative to h and h'.

They omit the rest of the second equation, since it is actually the variation of $J[y]$, but I am just wondering what they actually omitted?

How the Taylor's theorem is formulated for functionals?

Zzz
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  • I think what's going on is just the usual Taylor's theorem for multivariable functions. Expand $F(x,y,y')$ and the other function to one power in $h$ and you get the result. – amcalde Jun 30 '16 at 17:03
  • https://en.wikipedia.org/wiki/Taylor%27s_theorem#Example_in_two_dimensions – amcalde Jun 30 '16 at 17:04
  • Do you mean that they just omit the remainder? – Zzz Jul 01 '16 at 16:57
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    Yes they are throwing away terms with extra powers of $h$, perhaps without saying so explicitly. Granted, this is what you almost always do. – amcalde Jul 01 '16 at 17:11

1 Answers1

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$$\Delta J= J[y+h]-J[y]=\int_{a}^{b} [F(x,y+h,y'+h')-F(x,y,y')]\ dx \\ = \int_{a}^{b} [(F(x,y,y') + F_y(x,y,y') h + F_{y'}(x,y,y') y' + \mathcal{O}(h^2,h'^2) )-F(x,y,y')]\ dx \\ = \int_{a}^{b} [F_y(x,y,y') h + F_{y'}(x,y,y') y' + \mathcal{O}(h^2,h'^2)]\ dx \\ = \int_{a}^{b} [F_y(x,y,y') h + F_{y'}(x,y,y') y']\ dx + \ldots $$

They just threw away the higher order terms.

That is what is usually meant by the trailing dots in your equation.

amcalde
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  • What does $\mathcal{O}(h^{2},h'^{2})$ mean in your response? – Zzz Oct 03 '16 at 16:50
  • Here it means terms that will have a factor of $h^2$ or $h'^2$. Look up Taylor's expansion theorem for more information. – amcalde Oct 03 '16 at 23:41