# Manipulating Formulas with Derivatives

by Bernie Hunt
Tags: derivatives, formulas, manipulating
 Emeritus Sci Advisor PF Gold P: 16,091 One way to make sense of what they're doing is via differential forms. If the sequence xi are generalized coordinates on (a local patch of) the state space, so that any function f on the state space can be represented as a function of the xi's, then one property of differential forms is that $$df = \sum_i \frac{\partial f}{\partial x_i} dx_i,$$ which, formally, looks just like the chain rule. Because of the formal similarity, differentials share many of the same properties as derivatives, such asd(fg) = f dg + g df.(Some formulations of differential forms take this property as part of the definition) (In differential geometry, it is customary to write i as a superscript, not a subscript. But I wrote it this way becuase I figured it was probably more familiar to you. In particular, so that it doesn't look like exponentiation)