Calculus: Analytic Geometry and Calculus, with Vectors

11.3 Formulas involving partial derivatives 585

whenever jOxj < S. Since G(x + Ax, y) is an increasing function ofy and
G(x + Lx, f(x + Ax)) = 0, it follows that

f(x) - E < f(x + Ax) < f(x) +E

whenever jdxj < S. This establishes (20) andour theorem is proved. Since f

is continuous over the interval xo - h < x < xo + h, the graph of y = f (X)
over this interval is a curve C. Moreover, our proof shows that thiscurve C is
the only part of the graph of the equation F(x,y) = 0 which lies inside therec-
tangular region R shown in Figure 11.391.
8 This is another long problem. State and prove a theorem, similar to that
of the preceding problem, in which it is assumed that G=(x,y) 0 0 and thecon-
clusion involves a function 0 for which G(¢ (y), y) = 0.

(^9) Prove that if G is a function of x and y such that G, G., and Gare every-
where continuous, and if (xo,yo) is a point on the graph r of the equation G(x,y)=
0 for which G.(xo,yo) and Gy(xo,yo) are not both 0, then the point (xo,yo) isa
"simple point" on the graph. Remark: This is a theorem in geometry. The
conclusion means that if R is a rectangular (or circular) region which has its
center at (xo,yo) and which has a sufficiently small diameter, then the points of
F that lie in R can be ordered in such a way that they constitute a simple curve
or a Jordan arc; see the last two of Problems 7.19. The theorem implies that
multiple points and isolated points of r can occur only at places where G. and
G are both zero.
10 For each of the equations
(a) xy = 0 (b) x2 + y2 = 0
(c) Y2 -- x3 = 0 (d) y2 - x2(x2 - 1) = 0
(e) y2 - x2(x2 + 1) = 0
the preceding problem allows the possibility that the origin may be an isolated
point or a multiple point. What are the facts:
11 Let
u = [(x - 1)2 + (y - 1)2]x2 + y2 + 3]
and observe that the graph in the xy plane of the equation u = 0 contains only
one point P1. Observe that this result and Problem 9 imply that the two first-
order partial derivatives of u at P1 must be zero. Calculate these derivatives and
show that it is so.
12 Let f be a vector-to-scalar function for which f(r) is a number (or scalar)
whenever r is a vector in the domain of f. We do not need a coordinate system to
define a number D(f, r, u) by the formula

f (r + h) - f(r)

(1) D(f, r, u) = lo

h

whenever r and u are vectors such that the limit exists. In case u is a unit

vector, (1) provides an intrinsic definition of the derivative off at r in the direction

of u. To start acquaintance with this matter, introduce a coordinate system

and notation such that

(2) r = xi + A + zk, f (r) = f (x, y, z)