Recall the definition of ordinary differentiation. In light of the treatment above, we now recognize that the ``ordinary'' differentiation we learned in the first year of calculus was ordinary because it was scalar differentiation - differentiation of functions that represent scalar quantities. Given a (continuous, differentiable - we will assume this unless stated otherwise for all functions discussed) function :
Note my explicit and deliberate use of as the independent variable upon which depends. This invites us to think of this as a rate of change in physics where is some physical quantity as a function of the time.
From this one can easily derive all sorts of associated rules, the most important of which are:
We will often express these rules in terms of differentials, not derivatives with respect to specific coordinates. For example:
Most of these simple scalar rules have counterparts when we consider different kinds of vector differentiation.