Mathematical Foundations of Network Analysis

1. Front Matter

   Pages I-XI

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2. Introduction

   • Paul Slepian

   Pages 1-2

3. Connected Networks

   • Paul Slepian

   Pages 3-22

4. Loops, Trees, and Cut Sets

   • Paul Slepian

   Pages 23-45

5. Incidence Functions and Incidence Matrices

   • Paul Slepian

   Pages 46-60

6. Linear Algebra Review

   • Paul Slepian

   Pages 61-75

7. Boundary Operator and Coboundary Operator

   • Paul Slepian

   Pages 76-94

8. Axioms of Network Analysis

   • Paul Slepian

   Pages 95-105

9. Existence and Uniqueness of Solutions

   • Paul Slepian

   Pages 106-137

10. Kirchhoff’s Third and Fourth Laws

    • Paul Slepian

    Pages 138-186

11. Back Matter

    Pages 187-196

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In this book we attempt to develop the fundamental results of resistive network analysis, based upon a sound mathematical structure. The axioms upon which our development is based are Ohm's Law, Kirchhoff's Voltage Law, and Kirchhoff's Current Law. In order to state these axioms precisely, and use them in the development of our network analysis, an elaborate mathematical structure is introduced, involving concepts of graph theory, linear algebra, and one dimensional algebraic topology. The graph theory and one dimensional algebraic topology used are developed from first principles; the reader needs no background in these subjects. However, we do assume that the reader has some familiarity with elementary linear algebra. It is now stylish to teach elementary linear algebra at the sophomore college level, and we feel that the require­ ment that the reader should be familiar with elementary linear algebra is no more demanding than the usual requirement in most electrical engineering texts that the reader should be familiar with calculus. In this book, however, no calculus is needed. Although no formal training in circuit theory is needed for an understanding of the book, such experience would certainly help the reader by presenting him with familiar examples relevant to the mathematical abstractions introduced. It is our intention in this book to exhibit the effect of the topological properties of the network upon the branch voltages and branch currents, the objects of interest in network analysis.

• Algebra
• Analysis
• Elektrisches Netzwerk
• calculus
• circuit
• electrical engineering
• graph theory
• linear algebra
• linearity
• network
• network analysis
• operator

• Department of Mathematics, Rensselaer Polytechnic Institute, Troy, USA

  Paul Slepian

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