11
Partial derivatives
11.1 Elementary partial derivatives
complicated situations to later sections, we confine attention in this sec-
tion to examples and problems in which the fundamental ideas can be
stated quite simply and it is relatively easy to be completely sure of the
meanings of all of the symbols that are used. We begin with an example.
Suppose a copper rod occupies the interval x, < x S x2 of an x axis and
that we are interested in the temperature u (measured in degrees centi-
grade) at points of the rod at various times t. To be precise about the
matter, we may suppose that the "space coordinate" x is measured in
centimeters with x = 0 at some "space origin" and that the "time
coordinate" t is measured in seconds with t = 0 at some "time origin"
which could be the time at which some particular stop watch was started.
In some problems, it is not presumed that x and i are positive. For
present purposes, we suppose that to each pair of numbers x and t for
which xi S x _< X2 and t > 0 there corresponds exactly one number it
which we may denote byf(x,t) or by u(x,t) and which is the temperature
ssa