Suppose
M
,
N
are open subsets of
R
n
. A smooth map
F
:
M
→
N
is conformal iff
∀
x
∈
M
∀
u
,
v
∈
R
n
:
⟨
D
F
(
x
)
u
,
D
F
(
x
)
v
⟩
=
h
(
x
)
2
⟨
u
,
v
⟩
Fix
x
; the mapping
B
:
(
u
,
v
)
↦
⟨
D
F
(
x
)
u
,
D
F
(
x
)
v
⟩
is bi-linear and symmetric. By the
polarization formula
we conclude that
B
(
u
,
v
)
=
h
(
x
)
2
⟨
u
,
v
⟩
if and only if
B
(
u
,
u
)
=
h
(
x
)
2
⟨
u
,
u
⟩
.