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Unformatted text preview: ives of f and g in R at
all x ! U and they are continuous and bounded. Then for every i = 1, . . . , N ,
A
A
A
#g
#f
f (x)
(x) dx = '
g (x)
(x) dx +
f (x) g (x) 1i (x) dHN #1 (x) .
# xi
# xi
U
U
"U
Proof. Fix i ! {1, . . . , N }. We apply the divergence theorem to the function
f : U * RN deﬁned by
&
f (x) g (x) if j = i,
fj (x) :=
0
if j &= i.
Then
div f = N
+ # fj
# (f g )
#g
#f
=
=f
+g
,
# xj
# xi
# xi
# xi
j =1 and so
A,
A
A
A
#g
#f
f
+g
dx =
div f dx =
f · $ dHN #1 =
f g 1i dHN #1 .
# xi
# xi
U
U
"U
"U 106...
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 Spring '14

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