554 Partial derivatives
at the "space-time place" having "space-time coordinates" x and t.
Thus the temperature u is a function of the two "variables" x and t.
We are now in a realm where ideas of major importance can appear all
over the place. Some students who are now dubious about the possi-
bility of doing anything useful or interesting with these functons will,
in two or three years, have substantial information about the possibility
of starting with positive numbers a and L and a given function g and then
determining a positive integer n and constants A1, A2,. , A, such
that the particular function u defined by
n j kT
(11.11) u(x,t) _ 41F L= sin
k-1 T x
will be a good approximation to g(x) when t = 0. In any case, (11.11)
exhibits important examples (one for each choice of n and A1j X12, ,
A) of functions u to which our work applies.
We continue study of our example in which temperature u (measured
in degrees) is a function of x (measured in centimeters) and t (measured
in seconds). If we wish to study the temperature of the rod at some
particular time to, we can set t = to and, without bothering to be fussy
about the distinction between afunc-
u
u(x to) tion and values of the function, con-
sider u(x,to) to be a function of x
alone. If the graph of u(x,to) versus
x1 x2 x x happens to be that shown in Figure
Figure 11.12 11.12, we can look at the graph to see
where the temperature is increasing,
but it would not be too easy to determine the rate of change of u with
respect to x. To do this and get a number of degrees per centimeter, we
would want to differentiate u(x,to) with respect to x. Thus we are led to a
very important idea. When u is a function of x and some other variables
(in our case, just one other variable), it may make sense to assign fixed
values to all of the variables except x and differentiate the result with
respect to x. In particular, the idea does make sense when we know what
these other variables are and, in addition, we know that the resulting func-
tion of x is a differentiable function of x. When we know what we are
doing (this is a conservative statement providing for the possibility that
we may sometimes be puzzled by situations in thermodynamics) the result-
ing derivative is called the partial derivative of u with respect to x. While
there are other and more informative symbols for partial derivatives, the
simplest and most ingenious one is 8u/Ox. The "curly dees" in this
symbol are unusual Greek deltas, and the symbol is usually read "par-
tial of u with respect to x."
We are already in a position to understand statements and make cal-
culations. Partial derivatives are important things that abound in