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# Scalar Differentiation

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:

• The Chain rule. Suppose we have a function where is itself a function of (and there is no separate'' time dependence in ). Then:

• The Sum rule. Suppose we have two functions, and . Then:

• The Product rule. Suppose we have two functions, and . Then:

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.

Next: Vector Differentiation Up: Scalar and Vector Calculus Previous: Scalar and Vector Calculus   Contents
Robert G. Brown 2017-07-11