Here we will often refer to ∇f for a scalar valued f. It equals the vector
( )T
∂f-,⋅⋅⋅,-∂f-
∂x1 ∂xn
when f is a differentiable function of n variables. It is just the transpose of the matrix of the
derivative of f taken with respect to the usual coordinates.
Recall the definition of the integral and measure on a manifold.
∑N ∫ ( ) ( )
σn (E) ≡ ψi R−i1 (u ) XE R −i1(u) Ji(u)du
i=1 Ri(Ui∖L )
where
( ( ))
Ji(u) ≡ det DR −i1(u)∗DR −i1(u) 1∕2
and
{ψi}
was a partition of unity as described above and L was a set closed in Ω such that
for each i,R_{i}
(Ui ∩L )
has measure zero. Here it will be assumed that L = ∅. In the following
definition, the atlass will have a special form.
Definition 11.1.1A bounded open set, U ⊆ ℝ^{n}is said to have a C^{1}boundary andto lie on one side of its boundary if the following conditions hold. There exist open boxes,Q_{1},
⋅⋅⋅
,Q_{N},
n
Qi = ∏ (ai,bi)
j=1 j j
such that ∂U ≡U∖ U is contained in their union. Also, for each Q_{i}, there exists k and afunction, g_{i}such that U ∩ Q_{i}is of the form
(
{ k−∏ 1( )
x : (x1,⋅⋅⋅,xk− 1,xk+1,⋅⋅⋅,xn ) ∈ aij,bij ×
( j=1
n )}
∏ (ai,bi) and ai< xk < gi(x1,⋅⋅⋅,xk−1,xk+1,⋅⋅⋅,xn) (11.1)
j=k+1 j j k )
or else of the form
(
{ k−∏ 1( i i)
(x : (x1,⋅⋅⋅,xk−1,xk+1,⋅⋅⋅,xn) ∈ a j,bj ×
j=1 )
∏n }
(aij,bij) and gi(x1,⋅⋅⋅,xk−1,xk+1,⋅⋅⋅,xn) < xk < bij . (11.2)
j=k+1 )
The function g_{i}has a continuous derivative on∏_{j=1}^{k−1}
(ai,bi)
j j
×∏_{j=k+1}^{n}
(ai,bi)
j j
≡ B_{i}.Assume g_{i}∈ C^{1}
(--)
Bi
. That is, g_{i}is the restriction to B_{i}of a function in C_{c}^{1}
( n−1)
ℝ
. Inparticular, Dg_{i}will be bounded. We let Q_{0}be an open set such that Q_{0}⊆ U andU⊆∪_{i=0}^{N}Q_{i}.
Then on Q_{i}∩ ∂U = U_{i}, what is the mapping to ℝ^{n−1}? The point on the manifold
is
indicates that the variable x_{k(i)
} is ommitted.
Also σ_{n−1} is inner and outer regular on ∂U because it is a finite Borel measure on this set.
Note that the Borel sets refer to the smallest σ algebra which contains the sets open in
∂U.
Lemma 11.1.2Suppose U is a bounded open set as described above. Then there exists aunique function, n
(y)
for y ∈ ∂U such that
|n (y )|
= 1,n is Borel measurable, and for everyw ∈ ℝ^{n}satisfying
|w|
= 1, and for every f ∈ C_{c}^{1}
(ℝn)
,
∫ ∫
f (x-+-tw-)−-f (x)
lt→im0 U t dx = ∂U f (n⋅w )dσn−1
Proof: The integrand on the left is bounded because of the mean value inequality.
Therefore, letting t → 0 be any sequence, one can apply the dominated convergence theorem
and obtain that the limit on the left equals
∫
U ∇f (x)⋅wdx
Next this will be shown to equal an appropriate integral over ∂U. Let U⊆ V ⊆V⊆∪_{i=0}^{N}Q_{i}
and let
{ψi}
_{i=0}^{N} be a C^{∞} partition of unity on V such that spt
(ψi)
⊆ Q_{i}. Then for all t
small enough and x ∈ U,
N
f (x-+tw-)−-f (x)-= 1∑ ψif (x+ tw) − ψif (x).
t t i=0
Thus using the dominated convergence theorem,
∫ f (x + tw)− f (x )
lim ----------------dx (11.5)
t→0 U ( t )
∫ 1∑N
= ltim→0 U t ψif (x + tw )− ψif (x) dx
i=0
∫ ∑N ∑n
= Dj (ψif)(x)wjdx
U i=0j=1
∫ ∑n N∑ ∫ ∑n
= Dj (ψ0f)(x)wjdx + Dj (ψif)(x)wjdx (11.6)
Uj=1 i=1 Uj=1
(11.6)
Since spt
(ψ0)
⊆ Q_{0}, it follows the first term in the above equals zero. In the second term, fix
i. Without loss of generality, suppose the k
(i)
in the above definition, referring to the variable
left out, equals n and 11.1 holds. This just makes things a little easier to write. Thus g_{i} is a
function of
n∏−1( )
(x1,⋅⋅⋅,xn−1) ∈ aij,bij ≡ Bi
j=1
Then
∫ ∑n
Dj(ψif)(x)wjdx
U j=1
∫ ∫ gi(x1,⋅⋅⋅,xn−1)∑n
= B ai Dj(ψif)(x)wjdxndx1 ⋅⋅⋅dxn−1
i n j=1
∫ ∫
gi(x1,⋅⋅⋅,xn−1)∑n
= Bi −∞ Dj (ψif)(x)wjdxndx1 ⋅⋅⋅dxn− 1
j=1
Letting x_{n} = y + g_{i}
(x1,⋅⋅⋅,xn−1)
and changing the variable, this equals
∫ ∫ n
= 0 ∑ D (ψ f)(x ,⋅⋅⋅,x ,y+ g (x ,⋅⋅⋅,x ))w dydx ⋅⋅⋅dx (11.7)
Bi − ∞ j=1 j i 1 n−1 i 1 n−1 j 1 n−1
(11.7)
Recall D_{j} refers to the partial derivative taken with respect to the entry in the j^{th} slot. In the
n^{th} slot is found not just x_{n} but y + g_{i}
(x1,⋅⋅⋅,xn−1)
so a differentiation with respect to
x_{j},j < n, will not be the same as D_{j}. In fact, it will introduce another term involving g_{i,j}.
Thus
∂
∂x-(ψif (x1,⋅⋅⋅,xn−1,y+ gi(x1,⋅⋅⋅,xn−1)))
j
= Dj (ψif)(x1,⋅⋅⋅,xn−1,y+ gi(x1,⋅⋅⋅,xn−1))