I'm reading about Calculus of Variations and about the general variation of a functional. I bumped into few obstacles in my book I can't get over with.
I have scanned the pages where I have my confusion. I have highlighted my problem areas in different colors :
1. (GREEN) I don't understand why
$$\int_{x_1}^{x_1+\delta x_1}F(x,y+h,y'+h')\;dx-\int_{x_0}^{x_0+\delta x_0}F(x,y+h,y'+h')\;dx$$
equals $$F(x,y,y')|_{x=x_1}\delta x_1 -F(x,y,y')|_{x=x_0}\delta x_0$$
details?
2. (RED) What does this ~ notation actually mean? I'm confused with this...example?
3. (BLUE) To me this is not clear...details?
4. What is meant by the sentence (in page 55): "by a quantity of order higher than 1 relative to $\rho(y,y+h)$"?
In case you don't see the image clearly enough just right-click --> show in another tab, etc. Sorry for posting an image but this would have been quite a task to write in LaTex.
Please let me know if my questions are unclear. Thank you!