Geometric Methods for Anisotopic Inverse Boundary Value Problems
Electromagnetic fields have a natural representation as differential forms. Typically the measurement of a field involves an integral over a submanifold of the domain. Differential forms arise as the natural objects to integrate over submanifolds of each
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1
Introduetion
Eleetromagnetie fields have a natural representation as differential forms. Typieally the measurement of a field involves an integral over a submanifold of the domain. Differential forms arise as the natural objects to integrate over submanifolds of eaeh dimension. We will see that the (possibly anisotropie) material response to a field ean be naturally assoeiated with a Hodge star operator. This geometrie point of view is now weIl established in eomputational eleetromagnetism, partieularly by Kotoiga [13], and by Bossavit and and others (see for example [26],[22]). The essential point is that MaxweIl's equations ean be formulated in a eontext independent of the ambient Euelidean metrie. This approach has theoretieal eleganee and leads to simplicity of eomputation. In this paper we will review the geometrie formulation of the (scalar) anisotropie inverse eonductivity problem, amplifying some of the geometrie points made in Uhlmann's paper in this volume [28]. We will go on to eonsider generalizations of this anisotropie inverse boundary value problem to systems of Partial Differential Equation, including the result of Joshi and the author on the inverse boundary value problem for harmonie k-forms [8].
2
Review of Geometrie Coneepts and Notation
The eontext for this paper will be a smooth eompaet orientable n dimensional manifold with boundary. We will review briefly some eoneepts and notation from differential geometry essential to the geometrie study of inverse problems. A differential k-form is a seetion of the bundle of skew symmetrie k-linear maps on the tangent spaee to M. The spaee of smooth k forms is denoted by [lk(M). [lO(M) eonsists simply of smooth funetions, and [ll(M) eo-vector fields. The wedge produet a /\ ß of a k form a and an f-form ß is a k + {'-form equal to the skew symmetrie part of a Q9 ß The derivative d : [lO(M) ~ [ll(M) has a natural extension, the exterior derivative d: [lk(M) ~ [lk+l(M) as a derivation on the eomplete algebra of differential forms (1) d(a /\ ß) = da /\ ß + a /\ dß· K. Bingham et al. (eds.), New Analytic and Geometric Methods in Inverse Problems © Springer-Verlag Berlin Heidelberg 2004
W.R.B. Lionheart
:~3tl
The exterior derivative satisfies d 2 = o. Given a local coordinate chart x, a k-form w E flk(NI) can be cxpressed in coordinates as w
L
= l:'Oil
Wi 1
..
ikd:Ci l 1\ ... 1\ dXi~
:'O···:'Oik:'On
The raison d'etre for studying differential forms is that a k-form is the natural object to integrate over k-dimensional submanifolds without the need for any metric or measure. We have the generalized Stoke's (or perhaps NewtonLeibnitz-Gauss-Green-Ostrogradski-Stokes-Poillcare) formula
r
JN
dw=
r
JoN
w
(2)
for a k-form wand a k + I-dimensional submanifold N (or more generally a chain). The space of smooth vector fields on M will be denoted by .I(M) and .Io(NI) will denote vector fields vanishing on [JA1. Thc covariant derivative of a tensor field T will be denoted by 'VT, with componcnts TJ: .....·j;;j. The space of smooth symmetric tenso
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