nLab
total derivative

Idea

Given a function ff of nn variables, and given nn functions {g i}\{g_i\} of one variable, then the total derivative of the composite function tf(g 1(t),,g n(t))t \mapsto f(g_1(t),\cdots, g_n(t)) is (if it exists) simply its derivative with respect to tt, but understood as a linear combinationof the partial derivatives of ff, via the chain rule:

dfdt= ifg idg idt. \frac{d f}{d t} = \sum_i \frac{\partial f}{\partial g_i} \, \frac{d g_i}{d t} \,.

This has various evident generalizations. One is the horizontal derivative in variational calculus, see at variational bicomplex.

Last revised on September 13, 2017 at 14:28:34. See the history of this page for a list of all contributions to it.