Measure, Integration & Real Analysis, 2021a

142 Chapter 5 Product Measures
This table gives the first five values of
λ n (B n ), using 5.44. The last column of
this table gives a decimal approximation
to λ n (B n ), accurate to two digits after the
decimal point. From this table, you might
guess that λ n (B n ) is an increasing function
of n, especially because the smallest
cube containing the ball B n has n-
dimensional Lebesgue measure 2 n .However,
Exercise 12 in this section shows
that λ n (B n ) behaves much differently.
n λ n (B n ) ≈ λ n (B n )
1 2 2.00
2 π 3.14
3 4π/3 4.19
4 π 2 /2 4.93
5 8π 2 /15 5.26
Equality of Mixed Partial Derivatives Via Fubini’s Theorem
5.46 Definition partial derivatives; D 1 f and D 2 f
Suppose G is an open subset of R 2 and f : G → R is a function. For (x, y) ∈ G,
the partial derivatives (D 1 f )(x, y) and (D 2 f )(x, y) are defined by
(D 1 f )(x, y) =lim
t→0
f (x + t, y) − f (x, y)
t
and
f (x, y + t) − f (x, y)
(D 2 f )(x, y) =lim
t→0 t
if these limits exist.
Using the notation for the cross section of a function (see 5.7), we could write the
definitions of D 1 and D 2 in the following form:
(D 1 f )(x, y) =([f ] y ) ′ (x) and (D 2 f )(x, y) =([f ] x ) ′ (y).
5.47 Example partial derivatives of x y
Let G = {(x, y) ∈ R 2 : x > 0} and define f : G → R by f (x, y) =x y . Then
(D 1 f )(x, y) =yx y−1 and (D 2 f )(x, y) =x y ln x,
as you should verify. Taking partial derivatives of those partial derivatives, we have
(
D2 (D 1 f ) ) (x, y) =x y−1 + yx y−1 ln x
and (
D1 (D 2 f ) ) (x, y) =x y−1 + yx y−1 ln x,
as you should also verify. The last two equations show that D 1 (D 2 f )=D 2 (D 1 f )
as functions on G.
Measure, Integration & Real Analysis, by Sheldon Axler