In theory one can also integrate a scalar or vector field within volume with some directional vector. This would come up if you extended the divergence theorem to a four-dimensional space.
Yes of course. v is the indefinate integral of v' and not over the boundary of the initial region. I should have realize this before. Thanks!
Then, is there anything to be done without knowing the exact form of f(k,x)?
Yes, it's a book. Usually for graduate level, but the section on calculus of variations is just an ellaboration of what you've probably already learned.
This method uses integrating along a parametized line, which is something you learn early on in vector calculus. I think what throws many...
Well, I'm taking u as k_x and v' as f in the standard notation
\int u v' = u v| - \int u' v
therefore I will not have a derivative of f.
Not sure what you mean by surface term. I do realize this is an integral over three variables and I should use the divergence theorem in general. But...
So, to generalize it, \int \frac{dA}{dt}B\ dt = - \int \frac{dB}{dt}A\ dt is true if one of the functions A or B vanishes at both endpoints, which is what LCKurtz showed. You just have to remember in the variational principle A and B will be derivatives.
You derivation is not quite correct. I would advise you to see Goldstein (Classical Mechanics). It has a thorough explanation of Lagrangian mechanics starting with the variational principle.
You will be making variations with respect to \varepsilon , using x(t, \varepsilon ) = x(t, 0) +...
This should be very simple, but I can find a simple example that violates my general conclusion. I clearly must be doing something wrong with my integration by parts. Any suggestions would be great.
Define a distribution such that the density;
\eta(\vec{x})=\int d\vec{k} f(\vec{x},\vec{k})...