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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany

Surfaces in  Rn 
[chap:Surfaces]


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In this chapter we introduce the notion of a surface in EUKLIDean spaces. We define a measure on such surfaces using local parametrizations. This allows us, to apply the construction of LEBESGUE's integral in chapter chap:LebesgueIntegral. As a consequence, we obtain an integral over surfaces, for which all central theorems about integrals hold (e.g. the dominated convergence theorem). After this we apply the theory by computing some important special surface integrals.

We start this chapter with the definition of surfaces.

Definition ( Ck -surfaces)    [definition:5-1]
Let  0≦m≦n  and  k≧1  be integers. A subset  M⊂Rn  is called  m -dimensional  Ck -surface patch (or  m -dimensional local  Ck -surface ), if there exists an open connected set  D⊂Rm , and a bijective map  γ:D → M , such that the following holds:
  • [definition:5-1-(i)]  γ  is  k -times continuously differentiable,
  • [definition:5-1-(ii)]  γ-1  is continuous,
  • [definition:5-1-(iii)]  Dγ(y):Rm → Rn  is injective for all  y∈D .
Moreover, a subset  M⊂Rn  is called  m -dimensional  Ck -surface, if  M  locally is an  m -dimensional surface patch, that is for all  x0∈M  there is an open set  U⊂Rn  with  x0∈U , such that  U∩M  is an  m -dimensional surface patch.
Special definitions: A  0 -dimensional surface patch is a point, a  1 -dimensional surface is called curve. An  (n-1) -dimensional surface is called hypersurface. A subset in  Rn  is an  n -dimensional surface (resp. an  n -dimensional surface patch), if and only if it is an open set (resp. an open connected set) in  Rn . This follows from definition:5-1-(i) und definition:5-1-(iii) applying the theorem of the inversen mapping (see Analysis II).
Remark: Above the linear map  Dγ(y):Rm → Rn  is the derivative of  γ  in  y . It follows from definition:5-1-(iii), that the image  Dγ(y)(Rm)  is an  m -dimensional subspace of  Rn .
Remark: In the above definition it has been assumed, that the set  D  is connected. Since  D  is an open set, this is equivalent to the fact, that  D  is path connected, that is every to points  D  can be connected within  D  by a continuous curve. (Via approximation by convolution it follows, that it is possible to connect these two points by differentiable curves.)

In nature, that is for  n=3 , surfaces occur in many circumstances. Often  2 -dimensional surfaces occur as interface between two media, for example, the surface of a solid or water, where the second medium is a gas. Of special interest are interfaces between a solid and a fluid or between two different fluids, for example, water and oil. Moreover,  2 -dimensional surfaces may occur as as a medium by its own, for example, a thin paper sheet or a thin membrane, where the surrounding medium is a gas. Examples of  1 -dimensional surfaces are fibres and thin wires.

For all these examples it is assumed, that the physical properties can be formulated on a macroscale, for which the thickness of the interface can be neglected. On a molecular scale these surfaces do not exist.

If one considers the time evolution of such surfaces, they become  3 -dimensional, resp.  2 -dimensional, surfaces in  R4=R×R3 , that is in time and space.

The derivative of a mapping is the linear approximation of this mapping. In analogy, we define the linear approximation of a surface in a given point of this surface. This approximation is called "tangent space". For general sets this linear approximation needs not to be a subspace, it is only a cone. Therefore the more general notion of a tangent space is the "tangent cone". This notion is not only needed in this chapter, it also plays an important role in the subsequent chapters chap:DifferentialEquations and chap:SurfaceIntegration.

Tangent space    [sect:tangentialraum]
Let  M ⊂Rn  and  x0 ∈M . Then
Tx0(M) := {v ∈Rn  ; 
∃  (xk)k ∈N ∃  (rk)k ∈N  :  xk∈M, rk>0 ,
xk → x0, 
xk - x0
 
rk
 
 → v as k → ∞}
is called the tangent cone of  M  at  x0 . The set  Tx0(M)  is a cone with vertex  0 , that is
v ∈Tx0(M), r ≧0      ==>     rv ∈Tx0(M) .
If the tangent cone  Tx0(M)  is a subspace of  Rn , it is called the tangent space of  M  at  x0 .

In definition:5-1 surfaces have been defined via local parametrizations. The following theorem provides three alternative definitions, a definition using graphs, another definition using zero sets, and a third definition using diffeomorphisms.

Theorem (Representation of surfaces)    [satz:5-4]
Let  M ⊂Rn  and  x0 ∈M . Then the following statements are equivalent:
  • [satz:5-4-(i)] Representation by parametrization.  There exists an open set  U ⊂Rn  with  x0 ∈U , such that  U ∩M  is an  m -dimensional  Ck -surface patch.
  • [satz:5-4-(ii)] Representation as graph.  There exists an open set  U ⊂Rn  with  x0 ∈U , such that  U ∩M  is an  m -dimensional  Ck -graph, that is:

    There is an open and connected set  D ⊂Rm , an orthogonal transformation  Q  of  Rn  and a  k -times continuously differentiable map  g: D → Rn-m , such that

    {(y,g(y))   ;   y ∈D }= {Qx∈Rn   ;   x ∈U ∩M }.
    Here  (y,g(y))=(y1,...,ym,g1(y),...,gn-m(y)) .
    Remark 1: The orthogonal transformation  Q  can be chosen as permutation of the standard basis in  Rn . This gives, that there exist  i1,...,in∈N  with  {i1,...,in}={1,...,n} , such that
    U ∩M = {
    m
    ∑
    k=1
    		 yk eik +
    n
    ∑
    k=m+1
    		 gk-m(y) eik   ;   y = (y1,...,ym) ∈D },
    where  ej ,  j=1,...,n , are the canonical unit vectors in  Rn .
    Remark 2: The orthogonal transformation  Q  can be chosen, such that  Q(Tx0(M)) = {(y,0) ∈Rn   ;   y ∈Rm } . This implies, that there exists an orthonormal basis  {e1,...,en}  of  Rn , such that
    U ∩M = {
    m
    ∑
    k=1
    		 yk ek +
    n
    ∑
    k=m+1
    		 gk-m(y) ek   ;   y = (y1,...,ym) ∈D }.
  • [satz:5-4-(iii)] Representation as zero set.  There exists an open set  U ⊂Rn  with  x0 ∈U , such that  U ∩M  is an  m -dimensional  Ck -zero set, that is:

    There exists a  k -times continuously differentiable map  ϕ: U → Rn-m , such that  Dϕ(x): Rn → Rn-m  is surjective for all  x ∈U∩M , and such that

    U∩M = {x ∈U ; ϕ(x) = 0} .
  • [satz:5-4-(iv)] Representation by diffeomorphisms.  There exists an open set  U ⊂Rn  with  x0 ∈U , such that  U ∩M  is  Ck -diffeomorph to a relative open set in  Rm ×{0} , that is:

    There exists an open connected set  V⊂Rn  with  0∈V  and a  Ck -diffeomorphism  τ:V → U , such that

    U∩M = {τ(y,0)  ; y∈Rm, (y,0)∈V} .
    Here  (y,0)=(y1,...,ym,0,...,0) .
Now we are ready to define a surface measure for subsets  E  of an  m -dimensional surface  M . First let us consider the affine case.

HAUSDORFF measure on surfaces    [sect:5-7]
Let  M  be an  m -dimensional  C1 -surface. A set  E⊂M  is called a patch set (in analogy to box sets for the LEBESGUE measure, see sect:1-2-(iii)), if  E  as a subset of  Rn  is a BOREL set, and if  clos(E)  is a compact subset of  γ(D)  for some parametrization  γ  of a patch of  M . Then, for  m≧1 , the HAUSDORFF measure of  E  is defined by
Hm(E):=
 
∫
D
 Χγ-1(E) (y) sqrt( det(DγT(y)D γ(y))) dLm(y).
(For  m=0  the set  γ(D)  is a point. Therefore  H0(E):=1 , if  E≠∅ , thus consisting of a single point, and  H0(E):=0 , if  E=∅ .) Then the following holds:
  • [sect:5-7-(i)] The value  Hm(E)  is independent of the choice of the parametrization  γ .
  • [sect:5-7-(ii)] On every fixed surface patch  E |→ Hm(E)  is  ϭ -additive.
  • [sect:5-7-(iii)] For  m=n  the HAUSDORFF measure coincides with the LEBESGUE measure, that is  Hn=Ln .

Now we are able to apply the construction of LEBESGUE' integral at the beginning of this lecture (see the first three chapters chap:LebesgueIntegral, chap:MeasurableSets, chap:MeasurableFunctions). Without any additional work we obtain an integral on surfaces: We start with the measure  Hm  in sect:5-7 on the ring of patch sets of  M  (this in fact is a  ϭ -ring). For the construction of the theory in chap:LebesgueIntegral the necessary property eq:1-essential has been essential, which here follows from sect:5-7-(ii). is satisfied.

Thus the entire LEBESGUE theory (that is, all theorems concerning general measures) is applicable, e.g. FATOU's lemma and the dominated convergence theorem.

Integral on surfaces    [sect:5-8]
Let  M⊂Rn  be an  m -dimensional  C1 -surface. Moreover, let
B:= {
k
∪
j=1
 Ej;  k∈N, Ej are patch sets } .
Then  B  is a ring, and the measure  Hm  is  ϭ -additive on  B  (see sect:5-7-(ii)). It follows (as in sect:1-2): There is exactly one additive map
Hm:B → R,
which on patch sets has the representation as in sect:5-7.

Further, the theory in chapter chap:LebesgueIntegral provides the existence of LEBESGUE' integral with respect to  Hm  on  M , and all general theorems of chapter chap:MeasurableSets and chapter chap:MeasurableFunctions hold.

The corresponding set of  Hm -integrable functions  f:M → Y  we denote by

L(Hm┗ M;Y) ,
where  Hm┗ M  (that is,  Hm  restricted on  M ) is called the  m -dimensional HAUSDORFF measure on  M . The corresponding integral of an  Hm -integrable function  f  on  M  is
 
∫
M
 f dHm =
 
∫
M
 f(x) dHm(x) .

Special cases for  m  are  m=1 , that is curves, and  m=n-1 , that is hypersurfaces. They are treated in the following.

We continue with some applications using surface integrals.

Version 1.7
H.W. Alt - 02.01.2007

Multiple integrals Multiple integrals
Title page Title page
Partial integration in  Rn  Partial integration in  Rn 
Partial integration in  Rn  Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany