Proof that Variation of Integral is Equal to Integral of the Variation

[Arman777]

Homework Statement

I need to proof that $$\delta(\int_a^b F(x)dx)=\int_a^b\delta F(x)dx$$

Relevant Equations

The variation comes from the calculus of variation. For a given path the extremum occcurs on $$\delta I=\delta \int_{x_1}^{x_2}f (y,y';x) dx=0$$

I actually don't know how to proceed.

I tried something like this

The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$
where $f'(x)=F(x)$

However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$

where $f'(x)=F(x)$. For the right side of the equation can I say that
$$\int_a^b\delta F(x)dx = \int_a^b F'(x)\delta x dx = \delta (F(b)-F(a))$$

Is this true ? Thanks

 

[stevendaryl]

I think you're confused. The $\delta$ doesn't mean a derivative, so you can't write $\delta f = f' \delta x dx$.

The $\delta$ just means a change. Not a change as a function of $x$, but a change in the entire function. Imagine that you have two functions, an original function, $F(x)$, and a slightly different function $\overline{F}(x)$. Then $\delta F(x) = \overline{F}(x) - F(x)$. $\delta ( \int F(x) dx) = \int \overline{F}(x) dx - \int F(x) dx$. So you're just asking to prove that:

$\int \overline{F}(x) dx - \int F(x) dx = \int (\overline{F}(x) - F(x)) dx$

 

[Arman777]

oh I see. In examples such as for $F(x)=x^2$ we are writing $\delta F(x) = 2x\delta x$ So I thought I can apply it here.

From your last equation it seems that prove is finished since its just the sum rule of the integration

 

[Orodruin]

Typically, the integration variable is not what is being varied. What is being varied is usually the function that the functional depends on.

 

[stevendaryl]

oh I see. In examples such as for $F(x)=x^2$ we are writing $\delta F(x) = 2x\delta x$ So I thought I can apply it here.

Well $\delta$ means a difference in two values. Sometimes it means the difference between the same function at two different points, and sometimes it means the difference between two different functions at the same point. The latter is what is meant here.