Total Derivative - The Total Derivative Via Differentials

The Total Derivative Via Differentials

Differentials provide a simple way to understand the total derivative. For instance, suppose is a function of time t and n variables as in the previous section. Then, the differential of M is

This expression is often interpreted heuristically as a relation between infinitesimals. However, if the variables t and pj are interpreted as functions, and is interpreted to mean the composite of M with these functions, then the above expression makes perfect sense as an equality of differential 1-forms, and is immediate from the chain rule for the exterior derivative. The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if then . In particular, if the variables pj are all functions of t, as in the previous section, then

\operatorname d M = \frac{\partial M}{\partial t} \operatorname d t + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\frac{\partial p_i}{\partial t}\,\operatorname d t.

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