How can it be proved that the Green-Lagrange strain components include the rigid body motions?

Can anyone suggest a book or some other source from where i can get the history of the conception of infinitesimal. Sometimes I wonder  how and when (also, why) the great  mathematicians and thinkers conceived this breakthrough.

During the study of FEM, I found the term "weak form" of differential equation, which is the integral form of a differential equation along with the boundary conditions for a boundary value problem. Now my question is that why the term "weak" has been used.

In beam FEM problem generally the nodal displacement (v) and slope (θ) are taken as primary unknowns to determine the element shape function. What will be the obstackle/s if, instead of considering the slope, the nutral axis curvature (κ) is considered as primary unknown along with the displacement?