- www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/...

...Tensors_TM2002211716.pdfwww.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.htmhttp://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/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.
- 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. - Tensor This reference is (for Wikipedia) somewhat
lacking. The better material is linked to this page, see e.g.

Covariant vector and

Contravariant vector

and much more. - http://www.mth.uct.ac.za/omei/gr/chap3/frame3.htmlhttp://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.htmlhttp://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.