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- www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/...
...Tensors_TM2002211716.pdf.
This is a NASA white paper by Joseph C. Kolecki on the use of tensors in
physics (including electrodynamics) and is quite lovely. It presents
the modern view of tensors as entities linked both traditional bases and
manifolds much as I hope to do here.
- Mathematical Physics by Donald H. Menzel, Dover Press, ISBN
0-486-60056-4. This book was written in 1947 and hence presents both
the ``old way'' and the ``new way'' of understanding tensors. It is
cheap (as are all Dover Press books) and actually is a really excellent
desk reference for both undergraduate and graduate level classical
physics in general! Section 27 in this book covers simple cartesian
tensors, section 31 tensors defined in terms of transformations.
- Schaum's Outline series has a volume on vectors and
tensors. Again an excellent desk reference, it has very complete
sections on vector calculus (e.g. divergence theorem, stokes theorem),
multidimensional integration (including definitions of the Jacobian and
coordinate transformations between curvilinear systems) and tensors (the
old way).
- http://www.mathpages.com/rr/s5-02/5-02.htm
This presents tensors in terms of the manifold coordinate description
and is actually quite lovely. It is also just a part of
http://www.mathpages.com/, a rather
huge collection of short articles on all sorts of really cool problems
with absolutely no organization as far as I can tell. Fun to look over
and sometimes very useful.
- Wikipedia: http://www.wikipedia.org/wiki/Manifold
Tensors tend to be described in terms of
coordinates on a manifold. An -dimensional manifold is
basically a mathematical space which can be covered with locally
Euclidean ``patches'' of coordinates. The patches must overlap so that
one can move about from patch to patch without ever losing the ability
to describe position in local ``patch coordinates'' that are Euclidean
(in mathematese, this sort of neighborhood is said to be ``homeomorphic
to an open Euclidean n-ball''). The manifolds of interest to us in our
discussion of tensors are differentiable manifolds, manifolds on
which one can do calculus, as the transformational definition of tensors
requires the ability to take derivatives on the underlying manifold.
- Wikipedia: http://www.wikipedia.org/wiki/Tensor
This reference is (for Wikipedia) somewhat
lacking. The better material is linked to this page, see e.g.
Wikipedia: http://www.wikipedia.org/wiki/Covariant vector
and
Wikipedia: http://www.wikipedia.org/wiki/Contravariant vector
and much more.
- http://www.mth.uct.ac.za/omei/gr/chap3/frame3.html
This is a part of a ``complete online course in tensors and relativity''
by Peter Dunsby. It's actually pretty good, and is definitely modern in
its approach.
- http://grus.berkeley.edu/jrg/ay202/node183.html
This is a section of an online astrophysics text or set of lecture
notes. The tensor review is rather brief and not horribly complete, but
it is adequate and is in the middle of other useful stuff.
Anyway, you get the idea - there are plentiful resources in the form of
books both paper and online, white papers, web pages, and wikipedia
articles that you can use to really get to where you understand
tensor algebra, tensor calculus (differential geometry), and group
theory. As you do so you'll find that many of the things you've learned
in mathematics and physics classes in the past become simplified
notationally (even as their core content of course does not change).
As footnoted above, this simplification becomes even greater when some
of the ideas are further extended into a general geometric division
algebra, and I strongly urge interested readers to obtain and peruse
Lasenby's book on Geometric Algebra. One day I may attempt to add
a section on it here as well and try to properly unify the geometric
algebraic concepts embedded in the particular tensor forms of
relativistic electrodynamics.
Next: Non-Relativistic Electrodynamics
Up: Mathematical Physics
Previous: Groups of Transformation
Contents
Robert G. Brown
2007-12-28