266 Chapter 9Functions of several variables
(9.28)
the equation can also be written as
(9.29)
This form is sometimes called the ‘−1 rule’.
The presence of the – sign in equations (9.27) and (9.29) sometimes causes unease,
but it can be explained geometrically. Thus, for the particular z-surface shown in
Figures 9.3 and 9.6 the slopes along the xand ydirections are both negative at P (the
gradient lines slope ‘downward’), so that (∂z 2 ∂x)
yand (∂z 2 ∂y)
xare negative and their
quotient in equation (9.27) is positive. However, Figure 9.6 shows that for motion
along the contour from P to Q, ∆yis necessarily positive but ∆xnegative (or vice versa
for motion Q to P) so that (∂y 2 ∂x)
zis negative. Hence the – sign. Similar considerations
apply to the other three possible pairs of signs of the slopes.
We note that the distinction between dependent and independent variables has
disappeared from equations (9.26) to (9.29); any one variable can be regarded as a
function of the other two. We note also that whereas (9.26) is readily generalized for
sets of more than three variables, equations (9.27) and (9.29) are true for only three
variables at a time (all others being kept constant).
EXAMPLES 9.15
(i) Givenz 1 = 1 x
2y
3, find
By equation (9.27),
(ii) For , find
∂
∂
=,
∂
∂
=,
∂
∂
=−
r
x
x
r
r
y
y
r
y
x
x
y
r
and
r
y
x
∂
∂
.
rxy=+( )
2212/zyx
y
x
z
x
z
y
xy
x
∂
∂
=−
∂
∂
∂
∂
=−
2
3
32222
3
y
y
x
=−
yx
z
x
xy
z
y
xy
∂
∂
=,
∂
∂
23 =
322z
y
x
∂
∂
.
−=
∂
∂
∂
∂
∂
∂
1
zxy
x
y
y
z
z
x
yy
z
x
x
z
∂
∂
=
∂
∂
1