Derivative I:

Abstract

Let Λ γ denote the Dirichlet-to-Neumann map for an electrical network with conductivity γ. By way of the Kirchhoff matrix K = (κ ij), consider the space of conductivities to be a subset of R N ×N , where N is the number of vertices, We donote the map from γ to Λ γ by L. In this note we compute the directional derivative D L of L. A direction in this context is represented by a matrix , with arbitrary real entries, which is symmetric and has row sum 0. where is symmetric, has row sum 0, and K is a Kirchhoff matrix. Let K(t) = A + tt A B + tt B B T + tt T B C + tt C , and Λ(t) = A + tt A − (B + tt B)(C + tt C) −1 (B T + tt T B). Then D L = Λ (0) = A − B C −1 B T − BC −1 T B + BC −1 C C −1 B T. Proof. Let C(t) = C + tt C. Notice C(t)(C(t)) −1 = I. By the product rule C (t)C(t) + C(t)(C(t) −1) = 0, hence (C −1) (0) = −C −1 C C −1. Using the product rule again