•
Simplifying differentials is dangerous.
In one variable calculus, simplifying differentials (that is, considering
df/dt
as a quotient (which is not: it is the limit of a quotient) is usually harmless: for instance we can ”show”
the chain rule for a function
f
(
x
(
t
)) as follows:
df
dx
dx
dt
=
df
dt
(simplify
dx
in numerator and denominator). This sort of thing produces spectacularly wrong results in mul-
tivariable calculus. If we try to “show” the chain rule for a function
f
(
x
(
u, v
)
, y
(
u, v
)) simplifying differentials,
∂f
∂x
∂x
∂u
+
∂f
∂y
∂y
∂u
=
∂f
∂u
+
∂f
∂u
= 2
∂f
∂u
so we obtain the chain rule with an erroneous factor of 2. A more striking example is the next problem.
♣
PROBLEM 8 (p&p). The equation
F
(
x, y, z
) = 0
can be used to put any of the variables as function of the others:
x
=
x
(
y, z
)
,
y
=
y
(
x, z
)
,
z
=
z
(
x, y
)
.
Assuming that (
x, y, z
) is such that
∂F
(
x, y, z
)
∂x
= 0
,
∂F
(
x, y, z
)
∂y
= 0
,
∂F
(
x, y, z
)
∂z
= 0
,
show
∂x
∂z
∂z
∂y
∂y
∂x
=
−
1
(not 1 as if we simplify differentials).
IMPLICIT FUNCTIONS
Problems 1.5, 1.7 (both p&p), 1.6, 1.8 (both
M
)
.
TAYLOR EXPANSIONS IN TWO VARIABLES,
Problems 1, 2, 3, 4, 5 (all p&p). You can check
your computations with
MATHEMATICA
as in
TAYLOR EXPANSIONS.