文档介绍：The Calculus of Functions
Section
of
Introduction to Rn
Several Variables
Calculus is the study of functional relationships and how related quantities change with
each other. In your first exposure to calculus, the primary focus of your attention was
on functions involving a single independent variable and a single dependent variable. For
such a function f, a single real number input x determines a unique single output value
f(x). However, many of the functions of importance both within mathematics itself as
well as in the application of mathematics to the rest of the world involve many variables
simultaneously. For example, frequently in physics the function which describes the force
acting on an object moving in space depends on three variables, the three coordinates
which describe the location of the object. If the force function also varies with time,
then the force depends on four variables. Moreover, the output of the force function will
itself involve three variables, the three ponents of the force. Hence the
force function is such that it takes three, or four, variables for input and outputs three
variables. Far plicated functions are easy to imagine: the gross national product
of a country is a function of thousands of variables with a single variable as output, an
airline schedule is a function with thousands of inputs (cities, planes, and people to be
scheduled, as well as other variables like fuel costs and the schedules peting airlines)
and perhaps hundreds of outputs (the particular routes flown, along with their times).
Although such functions may at first appear to be far more difficult to work with than
the functions of single variable calculus, we shall see that we will often be able to reduce
problems involving functions of several variables to related problems involving only single
variable functions, problems which we may then handle using already familiar techniques.
By definition, a function takes a single inpu