268 Chapter 9Functions of several variables
This expression is closely related to that in Example 9.13 (walking on a circle) for
dz 2 dθ. Thus, when uis constant (= 1 a
2, say),u 1 = 1 x
21 + 1 y
21 = 1 a
2is the equation of a circle
of radius a, and displacements at constant uare therefore constrained to the circle,
withx 1 = 1 a 1 cos 1 θanddx 2 dθ 1 = 1 −y. Then,
For example, ifz 1 = 1 xythen and
0 Exercises 44–46
Change of independent variables
Letz 1 = 1 f(x, y)be a function of the independent variables xand ywith total differential
of zwith respect to xand y,
(9.17)
Let the variables xand ybe themselves functions of two other independent variables,
uand v:
x 1 = 1 x(u, 1 v), y 1 = 1 y(u, 1 v) (9.31)
Then zcan be treated as a function of uand v, and its total differential with respect to
the new variables is
(9.32)
The relationships between the partial derivatives in (9.32) and those in (9.17) can be
obtained in the following way. Divide the total differential (9.17) by duat constant v
to give equation (9.33a), and by dvat constant uto give (9.33b):
(9.33a)
(9.33b)
u
yux
zz
x
xz
y
∂
∂
=
∂
∂
∂
∂
+
∂
∂
vv
∂
∂
u
y
v
vv
∂
∂
=
∂
∂
∂
∂
+
∂
∂
z
u
z
x
x
u
z
y
yx
∂
∂
v
y
u
dz
z
u
du
z
d
u
=
∂
∂
+
∂
∂
v
v
v
dz
z
x
dx
z
y
dy
yx
=
∂
∂
+
∂
∂
dz
d
xy
θ
=−
22u
z
x
yxy
∂
∂
=−
2dz
d
z
x
dx
d
y
z
x
uu
θθ
=
∂
∂
=−
∂
∂