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Differential Forms in Electromagnetics

Differential Forms in Electromagnetics

Ismo V. Lindell

ISBN: 978-0-471-72308-0

Oct 2004, Wiley-IEEE Press

253 pages



An introduction to multivectors, dyadics, and differential forms for electrical engineers

While physicists have long applied differential forms to various areas of theoretical analysis, dyadic algebra is also the most natural language for expressing electromagnetic phenomena mathematically. George Deschamps pioneered the application of differential forms to electrical engineering but never completed his work. Now, Ismo V. Lindell, an internationally recognized authority on differential forms, provides a clear and practical introduction to replacing classical Gibbsian vector calculus with the mathematical formalism of differential forms.

In Differential Forms in Electromagnetics, Lindell simplifies the notation and adds memory aids in order to ease the reader's leap from Gibbsian analysis to differential forms, and provides the algebraic tools corresponding to the dyadics of Gibbsian analysis that have long been missing from the formalism. He introduces the reader to basic EM theory and wave equations for the electromagnetic two-forms, discusses the derivation of useful identities, and explains novel ways of treating problems in general linear (bi-anisotropic) media.

Clearly written and devoid of unnecessary mathematical jargon, Differential Forms in Electromagnetics helps engineers master an area of intense interest for anyone involved in research on metamaterials.


1 Multivectors.

1.1 The Grassmann algebra.

1.2 Vectors and dual vectors.

1.3 Bivectors.

1.4 Multivectors.

1.5 Geometric interpretation.

2 Dyadic Algebra.

2.1 Products of dyadics.

2.2 Dyadic identities.

2.3 Eigenproblems.

2.4 Inverse dyadic.

2.5 Metric dyadics.

2.6 Hodge dyadics.

3 Differential Forms.

3.1 Differentiation.

3.2 Differentiation theorems.

3.3 Integration.

3.4 Affine transformations.

4 Electromagnetic Fields and Sources.

4.1 Basic electromagnetic quantities.

4.2 Maxwell equations in three dimensions.

4.3 Maxwell equations in four dimensions.

4.4 Transformations.

4.5 Super forms.

5 Medium, Boundary, and Power Conditions.

5.1 Medium conditions.

5.2 Conditions on boundaries and interfaces.

5.3 Power conditions.

5.4 The Lorentz force law.

5.5 Stress dyadic.

6 Theorems and Transformations.

6.1 Duality transformation.

6.2 Reciprocity.

6.3 Equivalence of sources.

7 Electromagnetic Waves.

7.1 Wave equation for potentials.

7.2 Wave equation for fields.

7.3 Plane waves.

7.4 TE and TM polarized waves.

7.5 Green functions.


Appendix A: Multivector and Dyadic Identities.

Appendix B: Solutions to Selected Problems.


About the Author.

“…a modern, clear and well-organised account…in an easily mastered notation…” (Ultramicroscopy, Vol 104, 2005)