558 Partial derivatives
Supposing for the moment that y and y + Ay remain fixed like good
numbers usually do, we define a function q, by the formula(11.186) ¢(t) = u(t, Y + Dy) - u(t,Y).The numerator N of the ponderous quotient in (11.185) is then found to
be the second member of the formula(11.187) N=O(x+Ax)-i(x)=O'()Ox
= [um( , Y +'ay) - u.t(,Y)] Ax,
and an application of the mean-value theorem and (11.186) then gives
the rest of the formula in which is a number between x and x + Ax.
Substituting the last member of (11.187) for the numerator in (11.185)
gives(11.188) ut,Z(x,y) = lim lim u=(, Y + AY) - u=( ,Y)
nz--*o AY-0 AY
Since uz(,t) is a differentiable function of t over the interval from y to
y + Ay, another application of the mean-value theorem gives(11.189) uyx(x,y) = lira lim u2y(,n),
Lx-.o ny--.O
where n lies between y and y + Ay. Matters are complicated by the
fact that both i and 7l can depend upon Ay, and the next step is a delicate
one that demands careful attention in a rigorous course in advanced
calculus. The fact that the limit in the left member of the formula(11.1891) lim us U 'n) = uxy(*,y)
av- o
exists and the fact that u=y is continuous imply that there is a number
for which Ix - i;*j < lixi and the formula holds. Since usy is contin-
uous, this and (11.189) give(11.1892) uyz(x,y) = lim u=y(i*,y) = uxy(x,Y).Thus the limits exist and have the required value. It is not expected
that the above proof of Theorem 11.18 will be "learned" in this course,
but we need not be blissfully unaware of the facts that the theorem is
important and that we could learn very much about partial derivatives
and limits if we would (as is often done in advanced calculus) invest
enough time to make a thorough study of the proof.
Two observations can be made. In the first place, it is easy to insert
an extra variable z in Theorem 11.18 and its proof to obtain the formula
(11.1893) u:y(x,Y,z) = uys(x,Y,z)when the appropriate functions and derivativesare continuous. In the
second place, continuity of appropriate functions and derivatives allows