''Geometrical Vectors''
by Gabriel Weinreich
University of Chicago Press, Chicago Lectures in Physics series.
1998. Paperback, 115pp. $16.00. ISBN 0-226-89048-1.

http://www.maa.org/reviews/images/vectors.gif

Reviewed in the Mathematical Assocation of America online book review column, see
http://www.maa.org/publications/maa-reviews/geometrical-vectors

This is a very interesting and fun book that sorts out some confusing issues in vector analysis (as the above review says, in some books vectors are displacements between points, in others they seem interchangeable with points, in LinearAlgebra they become abstract, etc.) by taking a subset of differential forms and presenting them geometrically and pictorially.

Due to that approach, he ends up with 4 kinds of vectors:
* arrow (contravariant/line-type vector)
* stack (covariant/plane-type vector)
* sheaf (contravariant vector density)
* thumbtack (covariant vector capacity)

...each of which can be either polar (indicative of direction) or axial (indicative of chirality in a plane).

...And 3 kinds of scalars:
* scalar
* swarm (scalar density)
* scalar capacity

These entities interact in interesting ways, e.g., the cross product of a polar and an axial arrow vector results in a thumbtack vector.

My description in no way does this book justice; the MAA review (http://www.maa.org/reviews/vectors.html) does much better. E.g., it shows an image from the book that gives a pictorial illustration of the construction of a gradient:

http://www.maa.org/reviews/images/gradient.gif

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