Surfaces in Rⁿ

Title page

Divergence type equations

Index

Partial integration in Rⁿ
[chap:PartialIntegration]

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The goal of this chapter is, to prove an n -dimensional version of the "Fundamental Theorem of Calculus"

∫

f′(x) dx = f(b) - f(a) ,

which is known from Analysis I.

Motivation

As we have seen, the generalization of the "derivative" in one space dimension to the n -dimensional case is the "divergence operator" (which already has been introduced in Analysis II). The corresponding generalization of the "Fundamental Theorem of Calculus" will be "Divergence Theorem" (or "Theorem of GAUSS"). This is theorem satz:6-7 below.

Definition (Divergence operator)

For a continuously differentiable vector field q: Rⁿ → Rⁿ

div q(x) :=

∑

i=1

∂_i q_i(x) (Other notation: ∇•q(x))

is called the divergence of q at the point x , and div is called divergence operator. It follows, that for every orthonormal basis {ẽ₁, ..., ẽ_n} of Rⁿ

div q(x) =

∑

i=1

e_i•D q(x)(e_i) =

∑

i=1

ẽ_i •Dq(x)(ẽ_i) .

Here {e₁, ..., e_n} denotes the canonical basis of Rⁿ .

Proof. The vectors ẽ_i have a representation ẽ_i = ∑ _k=1ⁿ a_ik e_k . Thus

∑

i=1

ẽ_i•Dq(x)(ẽ_j) =

∑

k,l = 1

(

∑

a_ik a_il)e_k•Dq(x)(e_l) =

∑

k,l = 1

δ_kl e_k•Dq(x)(e_l),

since the matrix (a_ik)_ik is orthogonal.

The proof of the "Divergence theorem" satz:6-7 will be reduced to the following two essential special cases. The first one is the local case in the interior of the domain Ω (see theorem satz:6-1). This applies to situations, where the vector field has compact support in Ω . The second one is the local case at the "smooth" boundary of Ω (see theorem satz:6-3). Here the vector field has support in a neighbourhood of a point at this boundary (see definition definition:6-2).

Interior case

Definition ( C¹ -boundary)

Local case at boundary

In the proof of the general divergence theorem we shall reduce the situation to the local ones in satz:6-1 and satz:6-3. In order to be able to proceed like this, one has to decompose C¹ -functions in C¹ -functions with local support. This is the content of the following statement.

Partition of unity

We now would be able to prove the divergence theorem for domains Ω with C¹ -boundary. However, in many applications this is not the case, often the boundary contains corners and edges. These corners and edges are lower dimensional objects, a general definition for such singular sets is the following:

m -dimensional zero sets

The central theorem of this chapter is the following "Divergence theorem" (also called "GAUSS theorem" or "Theorem of GREEN-OSTROGRADSKI-GAUSS"):

Divergence theorem (Theorem of GAUSS) [satz:6-7]

Assume Ω⊂Rⁿ and f: clos(Ω) → Rⁿ satisfy:

[satz:6-7-(1)] Ω is a bounded GAUSS domain (see definition below).

[satz:6-7-(2)] f ∈C⁰(clos(Ω);Rⁿ) ∩C¹(Ω;Rⁿ) with div (f) ∈L¹(Ω) .

Then

∫

div (f) dLⁿ =

∫

∂Ω

f •ν_Ω dH^n-1 .

Remark 1: In the above formula the integral over ∂Ω is defined as the surface integral over ∂Ω∖A , where A is the (n-1) -dimensional zero set from the definition. (This definition of the integral is consistent with definition:6-5-(2).)

Remark 2: Property satz:6-7-(1) is satisfied, e.g. for boxes in Rⁿ . Property satz:6-7-(2) is satisfied, for example, if f∈C¹(clos(Ω);Rⁿ) (note, that Ω is assumed to be bounded).

Definition (GAUSS domain) [definition:6-7]

Let Ω⊂Rⁿ . We call it a GAUSS domain, if Ω is an open set and if the boundary ∂Ω consists of an (n-1) -dimensional zero set A and a C¹ -boundary with the property, that locally ∂Ω∖A has a finite surface area, that is H^n-1((∂Ω∖A)∩B_R(0))<∞ for all R>0 .

Ausblick: The divergence theorem can also be understood as the identity

div ( Χ_Ω f) = Χ_Ω div f + f ∇ Χ_Ω

in distributional sense over the entire space Rⁿ , which is a product formula (for distributions see [Analysis IV:Distribution] ).

Moreover, let us mention, that within the HILBERT space theory for partial differential equations (see the script PDE I, WS 2002/03) one works with LIPSCHITZ domains, a different class of domains. Also for LIPSCHITZ domains there is a version of the divergence theorem.

Proof of the divergence theorem

The divergence theorem leads to formulas concerning "Partial Integration" in Rⁿ . For n=1 and Ω=]a,b[⊂R the formula for integration by parts (see Analysis I) is