REAL ANALYSIS
FUNCTIONS OF SEVERAL VARIABLES
We denote (
x
1
. . . x
2
) by
X
and
f
(
x
1
. . . x
n
) by
f
(
X
). We may think of
X
as a vector or a point.
If
A
= (
a
1
. . . a
n
)
B
= (
b
1
. . . B

n
) then
A

B
= (
a
1

b
1
. . . a
n

b
n
)
A
+
B
= (
a
1
+
b
1
. . . a
n
+
b
n
)
A.B
=
a
1
b
1
+
. . .
+
a
n
b
n
 a scalar

A

=
q
a
2
1
+
. . .
+
a
2
n
norm of
A

A

B

=
q
(1
a

a
b
)
2
+
. . .
+ (
a
n

b
n
)
2
is the distance
AB

A.B
 ≤ 
A
 
B

Cauchy’s inequality.
Suppose we have
m
functions of
n
variables
(1)
f
(
X
)
,
(2)
f
(
X
)
. . .
(
m
)
F
(
X
).
We shall denote by the vector function
F
(
X
) = (
(1)
f
(
x
1
. . . x
n
)
, . . .
(
m
)
f
(
x
1
. . . x
n
))
Theorem 1
If
f
(
X
)
g
(
X
) are continuous at
A
relative to
S
then so are
f
(
X
)
±
g
(
X
)
, f
(
X
)
g
(
X
) and, if
g
(
A
)
6
= 0
,
f
(
X
)
g
(
X
)
.
Theorem 2
Suppose that the components
(1)
f
(
X
)
. . .
(
m
)
f
(
X
) of the vector
function
F
(
X
) are continuous at
A
relative to
S
. Let
B
=
F
(
A
) and let
T
be the set of all points
F
(
X
) with
X
in
S
. Then if
g
(
Y
) =
g
(
y
1
. . . y
m
)
is continuous at
B
relative to
T
, it follows that
g
(
F
(
x
)) is continuous
at
A
relative to
S
.
Differentiability
f
(
X
) is differentiable at
X
+
A
⇔ ∃
a vector
G

f
(
X
)

f
(
A
)

G
(
X

A
)

X

A

→
0 as
X
→
A
.
If
f
is differentiable then
∂f
∂x
1
. . .
∂f
∂x
n
all exist and the vector
G
is
‡
∂f
∂x
1
. . .
∂f
∂x
n
·
.
We call this (
grad f
(
X
))
X
=
A
or (
∇
f
)
X
=
A
.
Thus
f
(
X
) is differentiable
⇔
δf
∇
fδX

δX

→
0 as
δX
→
0.
Theorem 3
If
∂f
∂x
1
, . . .
∂f
∂f
∂x
n
are continuous at
X
=
A
, then
f
(
X
) is differ
entiable at
X
=
A
.
1
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Proof
Suppose
H
6
= 0 and

H

is sufficiently small. Consider
1

H

fl
fl
fl
fl
fl
n
X
r
=1
{
f
(
a
1
+
h
1
. . . , a
r
+
h
r
, a
r
+1
. . . a
n
)

f
(
a
1
+
h
1
. . . a
r

1
+
h
r

1
a
r
. . . a
n
)

h
r
f
r
(
A
)
}
(This is
1

H

{
f
(
A
+
h
)

f
(
A
)

A
∇
f
}
)
≤
1

H

fl
fl
fl
fl
fl
n
X
r
=1
[
{
f
r
(
a
1
+
h
1
. . . a
r

1
+
h
r

1
a
r
+
θ
r
f
r
1
a
r
+1

a
n
)

f
r
(
A
)
}
h
r
]
fl
fl
fl
fl
fl
0
< θ
r
<
1
Let
V
be the vector with components
f
r
(
a
1
+
h
1
, . . . a
r
θ
f
h
r
, a
r
+1
. . . a
n
)

f
r
A
(
r
= 1
,
2
, . . . n
)
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 Spring '98
 pfitz

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