Calculus: Analytic Geometry and Calculus, with Vectors

11.1 Elementary partial derivatives 557

Definition 11.16 14 function f of n variablesx1, x2, , x is said

to have the limit L as x1, x2j , x approaches al, a2, , an and we

write
lim f (xl,x2, ,xn) = L
xi,x2,

if to each e > 0 there corresponds a S > 0 such that

If(X1,X2,... x,)- LI < e

whenever

0 < (xl - ai)2 + (x2 - a2)2 +... + (xn - an)2 < S.

Definition 11.17 A function f of n variables x1, x2, , x is said
to be continuous at a,, a2, , an if

lim f(xl)x2j ,xn) = f(a1)a2, .. an)-

x,.24, ... x-4ai,a2, ... ,an

We are now prepared to state a fundamental theorem which guarantees
equality of f,, and fxy whenever these and some other derivatives exist
and are continuous; the fact that the theorem gives additional informa-
tion is interesting but less important.
Theorem 11.18 If u(x,y), u,,(x,y), uy(x,y), and uxy(x,y) all exist and
are continuous over some circular disk consisting of points (xl,yl) for which
(xl - x)2 + (yl - y)2 < S, then exists and

(11.181) uyx(x,Y) = uz (x,Y)

Proof of this theorem is quite tricky because it requires two applica-
tions of the mean-value theorem 5.52, and the first of these applications
must be made in a particular special way to produce the required result.
We shun the curly dees and use the subscript notation so we can know
what we are doing. To approach the derivative uyx(x,y) about which
we must learn, we observe that the derivatives in

(11.182) uy(x,y) = lim

u(x, y + Ay) - u(x,y)

Ay- o AY

(11.183) uy(x + Ax, y) = lim

u(x + Ax, Y + Ay) - u(x + Ax, y)

Ay- o AY

exist when IOxl is sufficiently small. We must prove that the limit in

(11.184) uyx(x,y) = lim

uy(x + Ox, y) - uv(x)y)

AX-0 Ax

exists and is uxy(x,y). Substituting for the terms in the numerator of
the right member gives

(11.185)

uyx(x,y) = lim lim u(x+Ox, y±Dy)-u(x+tx, y)-u(x, y-1- y)+u(x,Y)
Ax-+o oy-aO Ox Dy