Differentiation under the integral

Here is a PDF with an informal proof of a very general form of differentiation under the integral.

It’s formulated using geometric algebra and provides a simple demonstration of using some of the basic identities in geometric calculus.

The end result is:

d/(dt) int_(D(t)) L_t(x; d^m x) = int_(D(t)) dot L_t(x; (d^m x ^^ (del x)/(del t)) * dot grad_x) + int_(del D(t)) L_t(x; d^(m-1) x ^^ (del x)/(del t)) + int_(D(t))(del L_t(x; d^m x))/(del t)