differential forms and E&M - book recommendations?
A C
John Baez about the fact that the magnetic field is a pseudovector, and
how this means that one can avoid having to pick a right-hand rule to
play with electromagnetism (if I understood the post correctly).
That was the proverbial camel-breaking straw, and I've now decided I
really want to learn about electromagnetism using differential forms.
My problem is that I'm finding it difficult to find a book to use. I
know John Baez has a book that talks about this (and other things),
called "Knots, Gauge Fields, and Quantum Gravity". On the one hand, I'm
an avid reader of the "This Week's Finds" columns, so I very much enjoy
John Baez's writing style. However, the title sounds rather
intimidating, and I'm curious to know what the prerequisites for reading
the book are. I've tried searching google's archives, but I haven't
been able to find anything about prerequsities for the book. The
closest thing I've found was in a review excerpt on amazon.com, which
says it's suitable for 'advanced math/physics undergraduates and
graduate students'.
I've got a decent background in calculus (including vector calculus),
some differential equations, and some linear alebra, and I've taken
introductory college physics courses on electromagnetism and waves. I'm
currently studying books on complex analysis (Needham's book,
supplemented by Brown and Churchill), a real analysis book (the one by
Strichartz), and I've recently started trying to work my way through the
first volume of Spivak's Introduction to Differential Geometry. I'm
finding I like geometrical approaches to math - the more opportunities
to scribble pictures, the more fun I have.
Is it feasible (or sensible) for me to try to read Knots, Gauge Fields
and Quantum Gravity given my background? Does anyone have any other
recommendations for books to read about an approach to electromagnetism
that uses differential forms?
Thanks in advance for your time and any help,
-A.C.
* - the post I'm referring to is this one:
http://www.lns.cornell.edu/spr/2002-02/msg0039345.html
Stephen Speicher
> I know John Baez has a book that talks about this (and other
> things), called "Knots, Gauge Fields, and Quantum Gravity".
You seem to be conflating two different titles:
"Knots and Quantum Gravity," for which John Baez is editor, and
"Gauge Fields, Knots, and Gravity," which John coauthored with
Javier Muniain. I will just assume it is this latter which you
meant.
> On the one hand, I'm an avid reader of the "This Week's Finds"
> columns, so I very much enjoy John Baez's writing style.
> However, the title sounds rather intimidating, and I'm curious
> to know what the prerequisites for reading the book are.
>
> I've got a decent background in calculus (including vector
> calculus), some differential equations, and some linear alebra,
> and I've taken introductory college physics courses on
> electromagnetism and waves.
You do not mention familiarity with special relativity, but aside
from that you seem to have the minimum background which John
lists as prerequisites for his book. (Although, he does hope the
rest is accompanied by "that undefinable commodity known as
'mathematical sophistication'.") If, as you say, you enjoy
John's writings, then coupled with your expressed interest and
motivation, I suspect you will love this book.
There are other books which cover in much more detail many of the
individual subjects, but I have really come to appreciate the
succinct yet proper introduction this book provides. I find
myself going back to parts of the book for quick reference, even
though I have other books which specialize in each individual
area. "Gauge Fields, Knots, and Gravity" is a wonderful blend of
mathematics and physics, and I think it will suit you well for
some time to come.
If you decide to get the book, be sure to write to John for the
errata in electronic form.
p.s. I happen to have an extra copy I acquired (long story) which
I was going to put up on eBay, so if you are interested let me
know privately by e-mail.
--
Stephen
s...@compbio.caltech.edu
Printed using 100% recycled electrons.
-----------------------------------------------------------
~
A.J. Tolland
On Tue, 2 Jul 2002, A C wrote:
> I've got a decent background in calculus (including vector calculus),
> some differential equations, and some linear alebra, and I've taken
> introductory college physics courses on electromagnetism and waves. I'm
> currently studying books on complex analysis (Needham's book,
> supplemented by Brown and Churchill), a real analysis book (the one by
> Strichartz), and I've recently started trying to work my way through the
> first volume of Spivak's Introduction to Differential Geometry. I'm
> finding I like geometrical approaches to math - the more opportunities
> to scribble pictures, the more fun I have.
You'll be able to read the book and do the exercises (which are
integrated quite nicely). Let me know if you find a solution to the last
problem.
> Is it feasible (or sensible) for me to try to read Knots, Gauge Fields
> and Quantum Gravity given my background? Does anyone have any other
> recommendations for books to read about an approach to electromagnetism
> that uses differential forms?
I actually wouldn't recommend any other book. In fact, I'd
recommend trying it yourself first. There's not really all that much to
it. Electromagnetism itself doesn't change. If you know the basics,
Maxwell's equations, potentials, charge conservation, etc, you won't need
anything new physically. You just screw around with the notation a
little. It's actually something you can work out for yourself once you've
achieved a little proficiency with differential forms.
In fact, I'd encourage you to try. (Read Baez's book anyways;
there's tons of cool stuff in there.) I'll write a little guide below,
where you do all the work, and I just ask dumb questions. Don't page down
if you don't want to see it first here.
Combine the vector potential A and the scalar potential phi into a
1-form
A = phi dt + A_x dx + A_y dy + A_z dz
Now define F = dA. Show that F has components which are given by
the usual formulae for E & B field components. Note also that if we shift
A by df, where f is some function, F' = d(A + df) = F. This simple fact
has a long name.
Since, d^2 = 0, dF = 0. Work out the components see if they look
familiar.
Now hit F with the Hodge *-operator.
J = d(*F) doesn not trivially vanish. It's a 3-form. Find an
interpretation for its components. Since J = d(a 2-form), dJ = 0.
Interpret this in more familiar words.
That's essentially it.
Now work out the analogous formulae in 3d Minkowski space. It's
simpler but still kinda cool. Then try them in spherical coordinates (t,
r, theta, phi) on 4d Minkowski space, then on a curved spacetime, R x S^3,
for instance. Then try to figure out if electromagnetism is any different
on a spacetime that looks like R x T^3 instead of regular Minkowski space.
--A.J.
John Baez
A C <aSpamMeN...@comcast.net> wrote:
>I know John Baez has a book that talks about this (and other things),
>called "Knots, Gauge Fields, and Quantum Gravity".
Buy it! I need the money! Operators are standing by.
By the way, it's called "Gauge Fields, Knots and Gravity"
Don't accept cheap imitations.
>On the one hand, I'm
>an avid reader of the "This Week's Finds" columns, so I very much enjoy
>John Baez's writing style. However, the title sounds rather
>intimidating, and I'm curious to know what the prerequisites for reading
>the book are.
Vector calculus, matrices, special relativity, Maxwell's equations,
and a smidgen of that indefinable commodity called "mathematical
sophistication".
This book has three big sections: the first on Maxwell's equations,
the second on the Yang-Mills equations, and the third on Einstein's
equations. It sounds like you want to read the first one: it teaches
differential forms assuming that you know Maxwell's equations and want
to see how they get nicer when you use differential forms.
It's not as scary as it sounds.
Gary Pajer
Two others to look at: Bamberg and Sternberg "A Course in Mathematics for
Students of Physics" covers everything from scratch, and in great detail.
It reads like an undergrad text because it is, but it's got incredible
detail, and covers E&M in an interesting way, explaining what E and D really
are. Also, Schultz, "Geometrical Methods of Physics" (I think). This
one's the opposite. Broad brush strokes, and little detail, but deep, it's
a book you can sit back in an easy chair and read.
See also a review paper in an IEEE journal about 20 years ago by Deschamps
called "Electromagnetic Theory and Differential Forms" or something close to
that. I wish I could give you the exact reference. If you have trouble,
let me know. I like the appendix of this paper as a summary of the whole
business.
"A C" <ache...@comcast.net> wrote in message
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