556 Partial derivatives
which serve a dual purpose: they provide abbreviations for the expressions
on the left sides, and they provide meanings for the abbreviations on the
right sides. It could be supposed that we should insert a comma between
the subscripts x and y to write ur,v in place of u,z,,, but it is customary to
consider the commas to be superfluous.
The result of setting x = y = 0 in (11.131) is
(11.146) au = 0 or uz(0,0) = 0,
and this shows that the "curly dee" notation for partial derivatives is,
after all, a miserably poor purveyor of information. When we see the
first formula in (11.146), there is nothing to tell us that u depends upon
exactly two variables x and y and that 0 is the result of setting x = y = 0
in au/ax. When we keep the curly dees, it is often necessary to put
(11.146) in the form(11.147) au I = 0, uz(0,0) = 0
ax(o,o)so the first formula, like the second, can really mean something. As this
one example may suggest, the curly dees really should be banished from
the universe because they have the habit of giving incomplete and some-
times misleading information. There can be no doubt, however, that
they are so pretty and give information so quickly that they will con-
tinue to survive and be used.
Relatively simple fundamental calculations yield formulas in which
the first two or the second two of the quantities(11.15) aau aau
ay ax' ax ay' u-,-,,(x,Y), U,,-(x,Y)both appear and will cancel out if we can be sure that they are equal.
Taking a partial derivative with respect to x is, like putting on our shoes,
a procedure that is called an operation. Taking a partial derivative with
respect to y is, like putting on our socks, another operation. The ques-
tion whether the mixed derivatives in (11.15) are equal is therefore the
question whether two operations commute, that is, whether the result of
performing the two operations in tandem (that is,one after the other)
is independent of the order in which the operations are performed.
Correct ideas about the problem involving shoes and socks can be
obtained by experimentation. The remainder of the text of this section
is devoted to the problem involving partial derivatives. We begin with
some definitions in which n can be I or 2 or 3 or 416 and the variables
are usually denoted by x, y or x, y, z instead of xl,X2, , xwhen there
are only two or three of them.