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1. Line, surface and volume integrals

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.
2. Simple integral, example or general solution correct?

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)?
3. Integration by parts, can you do this?

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...
4. Simple integral, example or general solution correct?

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...
5. Integration by parts, can you do this?

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.
6. Integration by parts, can you do this?

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) +...
7. Simple integral, example or general solution correct?

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})...