Maxwells equations and differential forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:49:24Z http://mathoverflow.net/feeds/question/72160 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms Maxwells equations and differential forms Markus Ulke 2011-08-05T09:09:47Z 2011-08-05T15:35:19Z <p>Hi,</p> <p>is there a textbooks that explains the maxwell equations in differential form?</p> <p>What I understood so far is, that the \$E\$ and \$B\$ fields can be assembled to a differential 2 Form \$F\$, and the Maxwell Equations can be written quite nicely with the Hodge \$*\$ and the exterior deriative \$d\$. Going further the equations can be derived as an Euler Lagrange (or Yang Mills?) equation from a connection of a fibre bundle.</p> <p>I am searching for a book that describes how the geometric entities are mapped to the physical entities with a focus on mathematical exactness.</p> <p>Thanks</p> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72166#72166 Answer by Ben McKay for Maxwells equations and differential forms Ben McKay 2011-08-05T10:30:26Z 2011-08-05T10:30:26Z <p>Bernard F. Schutz, Geometrical methods of mathematical physics, p 175, chapter 5.11 Rewriting Maxwell's equations using differential forms.</p> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72168#72168 Answer by Cristi Stoica for Maxwells equations and differential forms Cristi Stoica 2011-08-05T11:07:57Z 2011-08-05T11:16:57Z <p><a href="http://books.google.com/books?id=YiGoQgAACAAJ&amp;dq=knot+gauge&amp;hl=en&amp;ei=eM47Tp_vIo3B8QP60fHwAg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=3&amp;ved=0CDgQ6AEwAg" rel="nofollow">Baez &amp; Muniain, Gauge Fields, Knots and Gravity</a>, chapter 5, p. 69, Rewriting Maxwell's equations.</p> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72174#72174 Answer by Fedor Petrov for Maxwells equations and differential forms Fedor Petrov 2011-08-05T12:15:22Z 2011-08-05T12:15:22Z <p>Bolibruch's nice explanation is here: www.mccme.ru/free-books/dubna/bol1.pdf I do not know whether it has English translation, sorry.</p> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72175#72175 Answer by Qfwfq for Maxwells equations and differential forms Qfwfq 2011-08-05T12:37:18Z 2011-08-05T12:51:08Z <ul> <li><p>M. Nakahara, <a href="http://books.google.it/books?id=cH-XQB0Ex5wC&amp;dq=inauthor%3A%2522Mikio+Nakahara%2522&amp;hl=it&amp;ei=HOg7TrbtG8G28QPc_pD3Ag&amp;sa=X&amp;oi=book_result&amp;ct=book-thumbnail&amp;resnum=1&amp;ved=0CCwQ6wEwAA" rel="nofollow"><em>Geometry, topology and physics</em></a>. Paragraph 10.5 "Gauge theories", specifically 10.5.1 "\$U(1)\$ gauge theories".</p></li> <li><p>R.S. Palais, <em>The geometrization of physics</em>, lecture notes from a course at National Tsing Hua University Hsinchu, Taiwan June-July 1981 [available on the internet, I think] Specifically, the paragraph "Generalized Maxwell equations"</p></li> <li><p>G.Svetlychny, <em>Preparation to gauge theory</em> [freely available on the ArXiv]. Chapter 7 "electromagnetism" paragraph 7.1 "Maxwell's Equations".</p></li> <li><p>H.Youk, <em>A survey on gauge theory and Yang-Mills equations</em> [available on the internet, I think]. Paragraph 7.1 "The Hodge-star operator and Maxwell's equations".</p></li> <li><p>Eguchi et al., <em>Gravitation, gauge theories and differential geometry</em>.</p></li> </ul> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72178#72178 Answer by Stefan Behrens for Maxwells equations and differential forms Stefan Behrens 2011-08-05T13:27:49Z 2011-08-05T13:27:49Z <p>I remember that when I was studying for an exam in electrodynamics I really liked this book:</p> <ul> <li>Parrott, Stephen: <em>"Relativistic electrodynamics and differential geometry"</em> (Springer, 1987) <a href="http://www.ams.org/mathscinet-getitem?mr=867408" rel="nofollow">(MathSciNet Review)</a></li> </ul> <p>It aims to give a mathematically precise treatment of the fundamentals of classical electrodynamics in the language of Lorentzian geometry. Unfortunately, it seems to be difficult to come by. Neither Google books nor Amazon have a preview for it.</p> <p>But it's really worth a look, not only because of the mathematical language, but because it discusses something that usually gets swept under the rug, namely that a charged "test particle" in an electromagnetic field is not only affected by the field but actually interacts with the field! This has always bugged me in my physics courses.</p> http://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/72187#72187 Answer by Paul Siegel for Maxwells equations and differential forms Paul Siegel 2011-08-05T15:35:19Z 2011-08-05T15:35:19Z <p>I really strongly recommend chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". In this book and its companion volume "Topology, Geometry, and Gauge Fields: Foundations", Naber provides a detailed, self-contained introduction to topology and geometry with a view toward physics. Both books are extremely detailed and strike an excellent balance between mathematical sophistication and physical motivation (with a bias toward the math). I can't recommend them enough.</p>