Editing Exterior algebra (section)

== References ==

=== Mathematical references ===
{{refbegin|30em}}
* {{citation |last1=Bishop |first1=R. |author1-link=Richard L. Bishop |last2=Goldberg |first2=S.I. |year=1980 |title=Tensor analysis on manifolds |publisher=Dover |isbn=0-486-64039-6 |url-access=registration |url=https://archive.org/details/tensoranalysison00bish }}
*: Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.
* {{citation |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |year=1989 |title=Elements of mathematics, Algebra I |publisher=Springer-Verlag |isbn=3-540-64243-9}}
*: This is the ''main mathematical reference'' for the article.  It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure.  See §III.7 and §III.11.
* {{citation |last1=Bryant |first1=R.L. |author1-link=Robert Bryant (mathematician) |last2=Chern |first2=S.S. |author-link2=Shiing-Shen Chern |last3=Gardner |first3=R.B. |last4=Goldschmidt |first4=H.L. |last5=Griffiths |first5=P.A. |author-link5=Philip A. Griffiths |year=1991 |title=Exterior differential systems |publisher=Springer-Verlag}}
*: This book contains applications of exterior algebras to problems in [[partial differential equations]].  Rank and related concepts are developed in the early chapters.
* {{citation |last1=Mac Lane |first1=S. |author-link1=Saunders Mac Lane |last2=Birkhoff |first2=G. |author-link2=Garrett Birkhoff |year=1999 |title=Algebra |publisher=AMS Chelsea |isbn=0-8218-1646-2}}
*: Chapter XVI sections 6–10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.
* {{citation |last=Sternberg |first=Shlomo |author-link=Shlomo Sternberg |year=1964 |title=Lectures on Differential Geometry |publisher=Prentice Hall}}
*: Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.
{{refend}}

=== Historical references ===
{{refbegin|30em}}
* {{wikicite |ref={{harvid|Bourbaki|1989b}} |reference={{harvtxt|Bourbaki|1989|loc=Historical note on chapters II and III}}}}
* {{citation |last=Clifford |first=W. |author-link=William Kingdon Clifford |year=1878 |title=Applications of Grassmann's Extensive Algebra |journal=American Journal of Mathematics |publisher=The Johns Hopkins University Press |doi=10.2307/2369379 |jstor=2369379 |volume=1 |issue=4 |pages=350–358}}
* {{citation |last=Forder |first=H.G. |authorlink=Henry Forder|year=1941 |title=The Calculus of Extension |publisher=[[Internet Archive]]|url=https://archive.org/details/in.ernet.dli.2015.77536/page/n5/mode/2up?view=theater}}
* {{citation |last=Grassmann |first=Hermann |author-link=Hermann Grassmann |year=1844 |title=Die Lineale Ausdehnungslehre – Ein neuer Zweig der Mathematik |language=de |url=https://books.google.com/books?id=bKgAAAAAMAAJ&q=Die+Lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik&pg=PA1}} (The Linear Extension Theory – A new Branch of Mathematics) [http://resolver.sub.uni-goettingen.de/purl?PPN534901565 alternative reference]
* {{citation |last=Kannenberg |first=Lloyd |year=2000 |title=Extension Theory (translation of Grassmann's ''Ausdehnungslehre'') |publisher=American Mathematical Society |isbn=0-8218-2031-1}}
* {{citation |last=Peano |first=Giuseppe |author-link=Giuseppe Peano |year=1888 |title=Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva}}; {{citation |last=Kannenberg |first=Lloyd |year=1999 |title=Geometric calculus: According to the Ausdehnungslehre of H. Grassmann |publisher=Birkhäuser |isbn=978-0-8176-4126-9 |url-access=registration |url=https://archive.org/details/geometriccalculu0000pean }}.
* {{citation |last=Whitehead |first=Alfred North |author-link=Alfred North Whitehead |year=1898 |title=A Treatise on Universal Algebra, with Applications |journal=Nature |volume=58 |issue=1504 |page=385 |publisher=Cambridge |doi=10.1038/058385a0 |bibcode=1898Natur..58..385G |s2cid=3985954 |url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01950001&seq=5}}
{{refend}}

=== Other references and further reading ===
<!--For works inessential to the article, though these may also have been referenced in passing.-->
{{refbegin|30em}}
* {{citation |last=Browne |first=J.M. |year=2007 |title=Grassmann algebra – Exploring applications of Extended Vector Algebra with Mathematica |url=http://www.grassmannalgebra.info/grassmannalgebra/book/index.htm |access-date=2007-05-09 |archive-date=2009-02-19 |archive-url=https://web.archive.org/web/20090219180241/http://grassmannalgebra.info/grassmannalgebra/book/index.htm |url-status=usurped }}
*: An introduction to the exterior algebra, and [[geometric algebra]], with a focus on applications.  Also includes a history section and bibliography.
* {{citation |last=Spivak |first=Michael |author-link=Michael Spivak |year=1965 |title=Calculus on manifolds |publisher=Addison-Wesley |isbn=978-0-8053-9021-6}}
*: Includes applications of the exterior algebra to differential forms, specifically focused on [[integral|integration]] and [[Stokes's theorem]].  The notation <math display=inline>{\textstyle\bigwedge}^{\!k} V </math> in this text is used to mean the space of alternating ''k''-forms on ''V''; i.e., for Spivak <math display=inline>{\textstyle\bigwedge}^{\!k} V </math> is what this article would call <math display=inline>{\textstyle\bigwedge}^{\!k} V^*. </math>  Spivak discusses this in Addendum 4.
* {{citation |last=Strang |first=G. |author-link=Gilbert Strang |year=1993 |title=Introduction to linear algebra |publisher=Wellesley-Cambridge Press |isbn=978-0-9614088-5-5}}
*: Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.
* {{springer |id=E/e037080 |title=Exterior algebra |author=Onishchik, A.L. }}
* {{citation |first=Fleming |last=Wendell |chapter=7. Exterior algebra and differential calculus |chapter-url={{GBurl|v-QlBQAAQBAJ|p=275}} |title=Functions of Several Variables |publisher=Springer |edition=2nd |orig-year=1977 |date=2012 |isbn=978-1-4684-9461-7 |pages=275–320 |url= }}
*: This textbook in [[multivariate calculus]] introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges.
* {{cite book | last1 = Shafarevich | first1 = I.R. | author-link1 = Igor Shafarevich | last2 = Remizov | first2 = A.O. | year = 2012 | title = Linear Algebra and Geometry | publisher = [[Springer Science+Business Media|Springer]] | isbn = 978-3-642-30993-9 | url = https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9 }}
*: Chapter 10: The Exterior Product and Exterior Algebras
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf "The Grassmann method in projective geometry"] A compilation of English translations of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"] An English translation of an early book on the geometric applications of exterior algebras
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/grassmann_-_mechanics_and_extensions.pdf "Mechanics, according to the principles of the theory of extension"] An English translation of one Grassmann's papers on the applications of exterior algebra
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{{Linear algebra}}
{{tensors}}

[[Category:Algebras]]
[[Category:Multilinear algebra]]
[[Category:Differential forms]]