Tensor Analysis

Needed before any serious Continuum Mechanics.

After got basics of Linear Algebra Done Right (LADR), Linear and Geometric Algebra (LAGA) and Vector and Geometric Calculus (VAGC) consider:

(Mathematical Engineering) Mikhail Itskov (auth.)-Tensor Algebra and Tensor Analysis for Engineers_ With Applications to Continuum Mechanics-Springer International Publishing (2015).pdf

Others above all omit tensors which are likely to be essential.

title = {Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics},
author = {Mikhail Itskov (auth.)},
publisher = {Springer International Publishing},
isbn = {978-3-319-16341-3, 978-3-319-16342-0},
year = {2015},
series = {Mathematical Engineering},
edition = {4},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=f5e3208d229ef9092de8c30566708464}

Perhaps in parallel with Walter Noll “Finite Dimensional Vector Spaces” (FDS) Volumes 1 and 2.


But first do Conceptual Mathematics (CM) and Primer on Infinitesimals (PI)

Also checkout (perhaps in parallel):

title = {An Introduction to Continuum Mechanics — after Truesdell and Noll},
author = {Donald R. Smith (auth.)},
publisher = {Springer Netherlands},
isbn = {978-90-481-4314-6,978-94-017-0713-8},
year = {1993},
series = {Solid Mechanics and Its Applications 22},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=dd309eb52676608e79a02b66a276010c}

An introduction confined to 3D that explains relation to earlier approaches to Geometry including tensors especially for coordinate transformation is:

Kenichi Kanatani-Understanding Geometric Algebra_ Hamilton, Grassmann, and Clifford for Computer Vision and Graphics-A K Peters_CRC Press (2015).pdf

title = {Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics},
author = {Kenichi Kanatani},
publisher = {A K Peters/CRC Press},
isbn = {1482259508,9781482259506},
year = {2015},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=83fdb8d1fb389be5d75de7c61e27188f}

Try to get adequate tensor analysis before getting distracted with too much Universal Geometric Calculus.

See above link which should be combined with this.

Especially before more advanced stuff like Hongbo Li:


Hongbo Li-Invariant Algebras And Geometric Reasoning-World Scientific Publishing Company (2008).pdf

title = {Invariant Algebras And Geometric Reasoning},
author = {Hongbo Li},
publisher = {World Scientific Publishing Company},
isbn = {9812708081,9789812708083,9789812770110},
year = {2008},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=BEEA3E89DFD6771C7A0C7F9AF61C89D3}

Sequel mentioned as reference [125] may be “Symbolic Geometric Reasoning with Advanced Invariant Algebras” pp35-49 in:

doi = {10.1007/978-3-319-32859-1},
title = {[Lecture Notes in Computer Science] Mathematical Aspects of Computer and Information Sciences Volume 9582 ||},
author = {Kotsireas, Ilias S.; Rump, Siegfried M.; Yap, Chee K.},
isbn = {978-3-319-32858-4,978-3-319-32859-1},
year = {2016},
volume = {10.1007/978-3-319-32859-1},
page = {–},
url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1007/978-3-319-32859-1},

But title is mixture of [123] with [125]. Not confirmed. Both IAAGR and this are much too advanced.

Also advanced but much more accessible is Hestenes tutorial:

Advances in Applied Clifford Algebras Volume 24 issue 2 2014 [doi 10.1007%2Fs00006-013-0418-0] Hestenes, David — Tutorial on Geometric Calculus.pdf

doi = {10.1007/s00006-013-0418-0},
title = {Tutorial on Geometric Calculus},
author = {Hestenes, David},
publisher = {Springer},
journal = {Advances in Applied Clifford Algebras},
issnp = {0188-7009},
issne = {1661-4909},
year = {2014},
month = {06},
day = {27},
volume = {24},
issue = {2},
page = {257–273},
url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1007/s00006-013-0418-0},

Perhaps indicates linear maps can be understand as when differential idempotent? Perhaps start here rather than tensors?

First couple of chapters of this may be more helpful as an advanced tutorial:

David Hestenes, Hongbo Li, Alyn Rockwood (auth.), Gerald Sommer (eds.)-Geometric Computing with Clifford Algebras_ Theoretical Foundations and Applications in Computer Vision and Robotics-Springer-Ver.pdf

Perhaps enough to understand Mobius transformations in complex plane?

title = {Geometric Computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics},
author = {David Hestenes, Hongbo Li, Alyn Rockwood (auth.), Gerald Sommer (eds.)},
publisher = {Springer-Verlag Berlin Heidelberg},
isbn = {978-3-642-07442-4,978-3-662-04621-0},
year = {2001},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=ec76652acb2554a063709658e91f980d}

Naeve and Svensson “Geo-MAP Unification” is Ch 5, pp105-126. Appendix has construction cited below.

Sobczyk 2013 NFM probably best after LAGA and VAGC. Starts with spectral on integers so does Jordan normal form etc properly.

Includes Unitary GA (ie complex) and mobius. But I could not see combination with Grassmann-Cayley. No index entries for meet product or double algebra.

Good introduction to that in:

An Algebra of Pieces of Space — Hermann Grassmann to Gian Carlo Rota

See also:
Ester Gasperoni Rota (auth.), Ernesto Damiani, Ottavio D’Antona, Vincenzo Marra, Fabrizio Palombi (eds.)-From Combinatorics to Philosophy-Springer US (2009).pdf
title = {From Combinatorics to Philosophy},
author = {Ester Gasperoni Rota (auth.), Ernesto Damiani, Ottavio D’Antona, Vincenzo Marra, Fabrizio Palombi (eds.)},
publisher = {Springer US},
isbn = {0387887520,9780387887524},
year = {2009},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=54D31C563E951E3E57AD078805B93786}

Anyway, tensors before combinatorics letterplace superalgebra/Whitney Matroids stuff like:

Whitney algebras and Grassmann’s regressive products


(But above may be more philosophically interesting for finitist combinatorial foundations like Grassmann brothers). Perhaps already incorporated in the Grassmann-Cayley/Geometric Algebra dimension 2^{n+1}?

Following are for abstract construction of Clifford Algebras for Universal Geometric Calculus including the usual tensor methods over arbitrary rings and modules instead of just vector spaces over the real field. There is also a “Clifford Analysis” over the complex numbers. Hopefully avoidable. The references included without URLs are added with URLs below.


Ref [1] is only 3pp. Ref [7] includes citation of Naeve and Svensson below.

Abstract. This is a simple way rigorously to construct Grassmann, Clifford
and Geometric Algebras, allowing degenerate bilinear forms, infinite dimen-
sion, using fields or certain modules (characteristic 2 with limitation), and
characterize the algebras in a coordinate free form. The construction is done
in an orthogonal basis, and the algebras characterized by universality. The
basic properties with short proofs provides a clear foundation for further de-
velopment of the algebras.
1. Introduction 1
2. Preliminaries 1
3. Construction 2
4. Characterization of universal Clifford algebras 4
5. Characterization of Geometric algebras 5
6. Construction of Clifford algebras from tensor algebras 6
7. Conclusion 7
8. Appendix 7
References 8

[1] R.D. Arthan, A Minimalist Construction of the Geometric Algebra
[2] Claude Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Springer-Verlag,
Berlin, 1997. Collected Works Vol. 2, Pierre Cartier, Catherine Chevalley Eds.1.
[3] Bertfried Fauser and Rafa l Ab lamowicz, On the decomposition of Clifford algebras of arbi-
trary bilinear form, 1999. http://arxiv.org/abs/math/9911180.
[4] Jacques Helmstetter, Artibano Micali, Quadratic Mappings and Clifford Algebras, Birkhuser,
Basel 2008.
[5] E. Hitzer, Axioms of Geometric Algebra, 2003.
[6] Douglas Lundholm, Lars Svensson, Clifford algebra, geometric algebra, and application, 2009;
[7] Alan Macdonald, An Elementary Construction of the Geometric Algebra, 2002. Adv. Appl.
Cliff. Alg. 12, 1-6 (2002).http://clifford-algebras.org/v12/v121/macdo121.pdf.
[8] Pertti Lounesto, Counterexamples in Clifford algebras,
[9] I. R. Porteous, Clifford algebras and the classical groups, Cambridge University Press, Cam-
bridge, 1995.

David Hestenes, Hongbo Li, Alyn Rockwood (auth.), Gerald Sommer (eds.)-Geometric Computing with Clifford Algebras_ Theoretical Foundations and Applications in Computer Vision and Robotics-Springer-Ver.pdf

Jacques Helmstetter, Artibano Micali-Quadratic mappings and Clifford algebras-Birkhäuser (2008).pdf

title = {Quadratic mappings and Clifford algebras},
author = {Jacques Helmstetter, Artibano Micali},
publisher = {Birkhäuser},
isbn = {3764386053,9783764386054,9783764386061},
year = {2008},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=4729253D175CCE7F0627B760236E1AFB}

(Collected Works of Claude Chevalley) Claude Chevalley, Pierre Cartier, Catherine Chevalley-Collected works. volume 2-Springer (1996).djvu

title = {Collected works},
author = {Claude Chevalley, Pierre Cartier, Catherine Chevalley},
publisher = {Springer},
isbn = {9783540570639,3540570632,0387570632,9780387570631},
year = {1996},
series = {Collected Works of Claude Chevalley},
edition = {1},
volume = {volume 2},
url = {http://gen.lib.rus.ec/book/index.php?md5=B113B1FA38E95AB51B586BD28B58939A}

(Cambridge Studies in Advanced Mathematics 50) I. Porteous-Clifford Algebras and the Classical Groups-Cambridge University Press (1995).djvu

title = {Clifford Algebras and the Classical Groups},
author = {I. Porteous},
publisher = {Cambridge University Press},
isbn = {0521551773,9780521551779},
year = {1995},
series = {Cambridge Studies in Advanced Mathematics 50},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=DF181C41B27E34E1C33BD3876DE3277C}

Lounesto has “Counterexamples in Clifford Algebras”:


Link to main paper currently broken:



Also note claims that Quadratic Form central to Clifford Algebra simplifies elementary trigonometry by using “Quadrance” i.e. GA square instead of taking square root for usual norm. Used with “Spread” for square of sine of angle.



Got Wildberger “Divine Proportions”

N J Wildberger-Divine Proportions_ Rational Trigonometry to Universal Geometry-Wild Egg Books (2005).pdf

Note: For angles use turns with \tau = 2 \cdot \pi


Spread [0, 1/12, 1/8, 1/6, 1/4] turn = Spread [0, 30, 45, 60, 90] degree

= [0, 2, 3, 4, 6]  \cdot \tau/24 = [n/4 for n = 0...4] or just remember Spread(0) = 0 and Spread(1/4) turn = 1 with:

= Spread [2, 3, 4]  \cdot \tau/24 =  Spread((n+1)\tau/24) = n/4 for n = 1…3

Can do trigonometric functions for all multiples of 3 degrees = \tau/120 by adding 36 degrees from pentagon:  Spread latex \tau/10 = (5-\sqrt(5))/8$


For Advanced Linear and Multilinear Algebra:

Volume 2 is multilinear – goes with volume 1 4th edn first. Lots of Clifford Algebra.

(Universitext) Werner Greub-Multilinear Algebra-Springer (1978).pdf

title = {Multilinear Algebra},
author = {Werner Greub},
publisher = {Springer},
isbn = {0387902848,9780387902845, 9781461394259},
year = {1978},
series = {Universitext},
edition = {2nd},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=963A39D57A67BDDE7A8E092DE1588E6B}}

Two below referenced from PlanetMath explanation of free vector space over a set. First has definition (finite support basis).

(Graduate Texts in Mathematics 23) Werner Greub (auth.)-Linear Algebra-Springer-Verlag New York (1975).pdf

title = {Linear Algebra},
author = {Werner Greub (auth.)},
publisher = {Springer-Verlag New York},
isbn = {1468494481, 9781468494488, 9781468494464},
year = {1975},
series = {Graduate Texts in Mathematics 23},
edition = {4},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=04016D07479D8A6709F47A072A457775}}

Second below has simple proof of universality of free vector space over unital R module and Tensor product quotient of bilinearity defined sub-module. Lots of Clifford Algebra.

Ib H. Madsen, Jxrgen Tornehave-From calculus to cohomology_ De Rham cohomology and characteristic classes-Cambridge University Press (1997).djvu

title = {From calculus to cohomology: De Rham cohomology and characteristic classes},
author = {Ib H. Madsen, Jxrgen Tornehave},
publisher = {Cambridge University Press},
isbn = {0521589568,9780521589567},
year = {1997},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=1BABBA2791451933EB4F04449E2B8DC7}


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