Given the Functional $$V(x_1(t),x_2(t))=\int_0^1\left(e^{x_1(t)-x_2(t)}-e^{x_1(t)+x_2(t)}\right)dt$$ I'm asked to find the first variation of the functional..
I tried it like this , considering $$F(x_1(t),x_2(t))=e^{x_1(t)-x_2(t)}-e^{x_1(t)+x_2(t)}$$ then $$\begin{align}\delta V(x_1(t),x_2(t))=\int_0^1\frac{\partial F}{\partial x_1(t)}\delta x_1(t)+\frac{\partial F}{\partial x_2(t)}\delta x_2(t)&\\=\int_0^1\left\{\left(e^{x_1(t)-x_2(t)}-e^{x_1(t)+x_2(t)}\right)\cdot \delta x_1(t)+\left(-e^{x_1(t)-x_2(t)}-e^{x_1(t)+x_2(t)}\right)\cdot \delta x_2(t)\right\}dt \end{align}$$ is it correct ? or i'm doing something wrong ?