264 Chapter 9Functions of several variables
A special case of the total derivative (9.21) is obtained if tis replaced by x. Then
y 1 = 1 y(x)is an explicit function of xandz 1 = 1 f(x, y(x))can be treated as a function of
the single variable x. Then, from (9.21),
(9.23)
We note that ifx 1 = 1 x(t)then this total derivative with respect to xis related to the total
derivative (9.21) with respect to tby the chain rule:
(9.24)
EXAMPLE 9.14
(i) Givenz 1 = 1 x
21 + 1 y
3, wherey 1 = 112 x, finddz 2 dx. Then, (ii) ifx 1 = 1 e
t, finddz 2 dt.
(i) By equation (9.23), since ,
(ii) Ifx 1 = 1 e
tthen and
This is identical to the result of Example 9.12, sincex 1 = 112 y.
0 Exercise 41
Walking on a contour
Consider changes in xand ythat leave the value of the functionz 1 = 1 f(x, y)unchanged.
In Figure 9.6, the plane ABC is parallel to the xy-plane, so that all points on the curve
APB on the representative surface are at constant value of z. The displacement P to Q
therefore lies on a contour of the surface. Then, by (9.16),
(9.25)
∆= ≈
∂
∂
∂
∂
z
z
x
x
z
y
y
yx
0 ∆∆
dz
dt
dz
dx
dx
dt
==− 23 xxy
24dx
dt
ex
t==,
dz
dx
z
x
z
y
dy
dx
= xy y xy
∂
∂
∂
∂
=+ 23 ×− =− 23
22 4()( )
dy
dx
x
=− =−y
1
22dz
dt
dz
dx
dx
dt
z
x
dx
dt
z
y
dy
yx
==
∂
∂
+
∂
∂
ddx
dx
dt
dz
dx
z
x
z
y
dy
dx
yx
=
∂
∂
∂
∂