diff --git a/.gitignore b/.gitignore index 62dcff1..4495bc9 100644 --- a/.gitignore +++ b/.gitignore @@ -25,7 +25,6 @@ aom.pdf ## Bibliography auxiliary files (bibtex/biblatex/biber): -*.bbl *.bcf *.blg *-blx.aux diff --git a/7-integration.tex b/7-integration.tex index a9844c9..4f58849 100644 --- a/7-integration.tex +++ b/7-integration.tex @@ -1,2 +1,12 @@ +We finally have all the main ingredients to generalize our line integral detour and discuss integration of $n$-forms over $n$-dimensional manifolds. + \section{Orientation} + +\newthought{We know from calculus one}, or our line integral examples, that the direction in which we traverse the interval, or a curve, can actually make a difference. +Indeed, the sign of the integral of a differential $n$-form is only fixed after choosing an orientation of the manifold. + +If for a curve an orientation is simply a choice of a direction along it, so we can make sense of it in terms of clockwise or counter-clockwise, generalising the concept will require an extra abstraction step. +Not just that, you have seen already that in $\R^n$ there is a standard orientation, but in other vector spaces we may need to make arbitrary choices. +For manifolds, the situation is much more complicated: for example, on a M\"obius strip it is impossible to make any such choice, as it turns out, it is non-orientable. + \section{Stokes' Theorem} diff --git a/aom.bbl b/aom.bbl new file mode 100644 index 0000000..bd70d08 --- /dev/null +++ b/aom.bbl @@ -0,0 +1,451 @@ +% $ biblatex auxiliary file $ +% $ biblatex bbl format version 2.9 $ +% Do not modify the above lines! +% +% This is an auxiliary file used by the 'biblatex' package. +% This file may safely be deleted. It will be recreated by +% biber as required. +% +\begingroup +\makeatletter +\@ifundefined{ver@biblatex.sty} + {\@latex@error + {Missing 'biblatex' package} + {The bibliography requires the 'biblatex' package.} + \aftergroup\endinput} + {} +\endgroup + + +\refsection{0} + \datalist[entry]{anyt/global//global/global} + \entry{book:abrahammarsdenratiu}{book}{} + \name{author}{3}{}{% + {{hash=a662cea2e646a5fd5025f8ebbdaaae10}{% + family={Abraham}, + familyi={A\bibinitperiod}, + given={Ralph}, + giveni={R\bibinitperiod}}}% + {{hash=02d8e9c18a21963d5d0460f3bfef567f}{% + family={Marsden}, + familyi={M\bibinitperiod}, + given={Jerold\bibnamedelima E.}, + giveni={J\bibinitperiod\bibinitdelim E\bibinitperiod}}}% + {{hash=1887ef8423f419033e7a7aa700f5ca03}{% + family={Ratiu}, + familyi={R\bibinitperiod}, + given={Tudor\bibnamedelima S.}, + giveni={T\bibinitperiod\bibinitdelim S\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer, New York}% + } + \strng{namehash}{2a647dd3e00e15bb547f20c5113ce125} + \strng{fullhash}{2a647dd3e00e15bb547f20c5113ce125} + \strng{bibnamehash}{2a647dd3e00e15bb547f20c5113ce125} + \strng{authorbibnamehash}{2a647dd3e00e15bb547f20c5113ce125} + \strng{authornamehash}{2a647dd3e00e15bb547f20c5113ce125} + \strng{authorfullhash}{2a647dd3e00e15bb547f20c5113ce125} + \field{labelalpha}{AMR04} + \field{sortinit}{A} + \field{sortinithash}{d77c7cdd82ff690d4c3ef13216f92f0b} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{edition}{2nd} + \field{series}{Applied mathematical sciences} + \field{title}{Manifolds, tensor analysis, and applications} + \field{volume}{75} + \field{year}{2004} + \verb{doi} + \verb 10.1007/978-1-4612-1029-0 + \endverb + \endentry + \entry{book:arnold}{book}{} + \name{author}{1}{}{% + {{hash=792a2142e186bdd494bf17f95f32ab79}{% + family={Arnold}, + familyi={A\bibinitperiod}, + given={Vladimir\bibnamedelima I.}, + giveni={V\bibinitperiod\bibinitdelim I\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer-Verlag, New York}% + } + \strng{namehash}{792a2142e186bdd494bf17f95f32ab79} + \strng{fullhash}{792a2142e186bdd494bf17f95f32ab79} + \strng{bibnamehash}{792a2142e186bdd494bf17f95f32ab79} + \strng{authorbibnamehash}{792a2142e186bdd494bf17f95f32ab79} + \strng{authornamehash}{792a2142e186bdd494bf17f95f32ab79} + \strng{authorfullhash}{792a2142e186bdd494bf17f95f32ab79} + \field{labelalpha}{Arn89} + \field{sortinit}{A} + \field{sortinithash}{d77c7cdd82ff690d4c3ef13216f92f0b} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{edition}{2nd edition} + \field{series}{Graduate Texts in Mathematics} + \field{title}{Mathematical methods of classical mechanics} + \field{volume}{60} + \field{year}{1989} + \verb{doi} + \verb 10.1007/978-1-4757-2063-1 + \endverb + \endentry + \entry{book:bonfigliolifulci}{book}{} + \name{author}{2}{}{% + {{hash=9bbdd2bd9ed7dc42e3328938055abce0}{% + family={Bonfiglioli}, + familyi={B\bibinitperiod}, + given={Andrea}, + giveni={A\bibinitperiod}}}% + {{hash=bc22ceb3999a239f41c7e962ce48fd0f}{% + family={Fulci}, + familyi={F\bibinitperiod}, + given={Roberta}, + giveni={R\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer Berlin Heidelberg}% + } + \strng{namehash}{5385c1e584ae09e295d9d2ecfd774571} + \strng{fullhash}{5385c1e584ae09e295d9d2ecfd774571} + \strng{bibnamehash}{5385c1e584ae09e295d9d2ecfd774571} + \strng{authorbibnamehash}{5385c1e584ae09e295d9d2ecfd774571} + \strng{authornamehash}{5385c1e584ae09e295d9d2ecfd774571} + \strng{authorfullhash}{5385c1e584ae09e295d9d2ecfd774571} + \field{labelalpha}{BF12} + \field{sortinit}{B} + \field{sortinithash}{276475738cc058478c1677046f857703} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{title}{Topics in Noncommutative Algebra} + \field{year}{2012} + \verb{doi} + \verb 10.1007/978-3-642-22597-0 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007%2F978-3-642-22597-0 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007%2F978-3-642-22597-0 + \endverb + \endentry + \entry{lectures:hitchin}{misc}{} + \name{author}{1}{}{% + {{hash=d0af63589d0e61aaeda653b8860bf668}{% + family={Hitchin}, + familyi={H\bibinitperiod}, + given={Nigel}, + giveni={N\bibinitperiod}}}% + } + \strng{namehash}{d0af63589d0e61aaeda653b8860bf668} + \strng{fullhash}{d0af63589d0e61aaeda653b8860bf668} + \strng{bibnamehash}{d0af63589d0e61aaeda653b8860bf668} + \strng{authorbibnamehash}{d0af63589d0e61aaeda653b8860bf668} + \strng{authornamehash}{d0af63589d0e61aaeda653b8860bf668} + \strng{authorfullhash}{d0af63589d0e61aaeda653b8860bf668} + \field{labelalpha}{Hit14} + \field{sortinit}{H} + \field{sortinithash}{5f15a7bc777ad49ff15aa4d2831b1681} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{note}{Unpublished lecture notes} + \field{title}{Differentiable manifolds} + \field{year}{2014} + \verb{urlraw} + \verb https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Differentiable_manifolds/manifolds2014.pdf + \endverb + \verb{url} + \verb https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Differentiable_manifolds/manifolds2014.pdf + \endverb + \endentry + \entry{book:knauf}{book}{} + \name{author}{1}{}{% + {{hash=9a3dd08d6e1a58606049ffd1bd306bd6}{% + family={Knauf}, + familyi={K\bibinitperiod}, + given={Andreas}, + giveni={A\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer, New York}% + } + \strng{namehash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \strng{fullhash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \strng{bibnamehash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \strng{authorbibnamehash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \strng{authornamehash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \strng{authorfullhash}{9a3dd08d6e1a58606049ffd1bd306bd6} + \field{labelalpha}{Kna18} + \field{sortinit}{K} + \field{sortinithash}{9fd838a31ba64d981e8f44562bd33f89} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{series}{UNITEXT} + \field{title}{Mathematical physics: classical mechanics} + \field{volume}{109} + \field{year}{2018} + \verb{doi} + \verb 10.1007/978-3-662-55774-7 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007%2F978-3-662-55774-7 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007%2F978-3-662-55774-7 + \endverb + \endentry + \entry{book:lee:topology}{book}{} + \name{author}{1}{}{% + {{hash=32fbdee07151a2283306531745546911}{% + family={Lee}, + familyi={L\bibinitperiod}, + given={John\bibnamedelima M.}, + giveni={J\bibinitperiod\bibinitdelim M\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer New York}% + } + \strng{namehash}{32fbdee07151a2283306531745546911} + \strng{fullhash}{32fbdee07151a2283306531745546911} + \strng{bibnamehash}{32fbdee07151a2283306531745546911} + \strng{authorbibnamehash}{32fbdee07151a2283306531745546911} + \strng{authornamehash}{32fbdee07151a2283306531745546911} + \strng{authorfullhash}{32fbdee07151a2283306531745546911} + \field{labelalpha}{Lee11} + \field{sortinit}{L} + \field{sortinithash}{2c7981aaabc885868aba60f0c09ee20f} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{title}{Introduction to Topological Manifolds} + \field{year}{2011} + \verb{doi} + \verb 10.1007/978-1-4419-7940-7 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007/978-1-4419-7940-7 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007/978-1-4419-7940-7 + \endverb + \endentry + \entry{book:lee}{book}{} + \name{author}{1}{}{% + {{hash=32fbdee07151a2283306531745546911}{% + family={Lee}, + familyi={L\bibinitperiod}, + given={John\bibnamedelima M.}, + giveni={J\bibinitperiod\bibinitdelim M\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer, New York}% + } + \strng{namehash}{32fbdee07151a2283306531745546911} + \strng{fullhash}{32fbdee07151a2283306531745546911} + \strng{bibnamehash}{32fbdee07151a2283306531745546911} + \strng{authorbibnamehash}{32fbdee07151a2283306531745546911} + \strng{authornamehash}{32fbdee07151a2283306531745546911} + \strng{authorfullhash}{32fbdee07151a2283306531745546911} + \field{labelalpha}{Lee13} + \field{sortinit}{L} + \field{sortinithash}{2c7981aaabc885868aba60f0c09ee20f} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{series}{Graduate Texts in Mathematics} + \field{title}{Introduction to Smooth Manifolds} + \field{volume}{218} + \field{year}{2013} + \verb{doi} + \verb 10.1007/978-1-4419-9982-5 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007/978-1-4419-9982-5 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007/978-1-4419-9982-5 + \endverb + \endentry + \entry{book:McInerney}{book}{} + \name{author}{1}{}{% + {{hash=bef419e9621a47e7b327d465eb8ec17a}{% + family={McInerney}, + familyi={M\bibinitperiod}, + given={Andrew}, + giveni={A\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer New York}% + } + \strng{namehash}{bef419e9621a47e7b327d465eb8ec17a} + \strng{fullhash}{bef419e9621a47e7b327d465eb8ec17a} + \strng{bibnamehash}{bef419e9621a47e7b327d465eb8ec17a} + \strng{authorbibnamehash}{bef419e9621a47e7b327d465eb8ec17a} + \strng{authornamehash}{bef419e9621a47e7b327d465eb8ec17a} + \strng{authorfullhash}{bef419e9621a47e7b327d465eb8ec17a} + \field{labelalpha}{McI13} + \field{sortinit}{M} + \field{sortinithash}{cfd219b90152c06204fab207bc6c7cab} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{title}{First Steps in Differential Geometry} + \field{year}{2013} + \verb{doi} + \verb 10.1007/978-1-4614-7732-7 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007/978-1-4614-7732-7 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007/978-1-4614-7732-7 + \endverb + \endentry + \entry{lectures:merry}{misc}{} + \name{author}{1}{}{% + {{hash=8d7ebf70d21bdd08061e63dfb55c2a0a}{% + family={Merry}, + familyi={M\bibinitperiod}, + given={Will}, + giveni={W\bibinitperiod}}}% + } + \strng{namehash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \strng{fullhash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \strng{bibnamehash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \strng{authorbibnamehash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \strng{authornamehash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \strng{authorfullhash}{8d7ebf70d21bdd08061e63dfb55c2a0a} + \field{labelalpha}{Mer19} + \field{sortinit}{M} + \field{sortinithash}{cfd219b90152c06204fab207bc6c7cab} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{note}{Unpublished lecture notes} + \field{title}{Lecture notes on Differential Geometry} + \field{year}{2019} + \verb{urlraw} + \verb https://www.merry.io/courses/differential-geometry/ + \endverb + \verb{url} + \verb https://www.merry.io/courses/differential-geometry/ + \endverb + \endentry + \entry{book:munkres:topology}{book}{} + \name{author}{1}{}{% + {{hash=9357bb87897cb34de20366e6cca786c3}{% + family={Munkres}, + familyi={M\bibinitperiod}, + given={J.R.}, + giveni={J\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Prentice Hall, Incorporated}% + } + \strng{namehash}{9357bb87897cb34de20366e6cca786c3} + \strng{fullhash}{9357bb87897cb34de20366e6cca786c3} + \strng{bibnamehash}{9357bb87897cb34de20366e6cca786c3} + \strng{authorbibnamehash}{9357bb87897cb34de20366e6cca786c3} + \strng{authornamehash}{9357bb87897cb34de20366e6cca786c3} + \strng{authorfullhash}{9357bb87897cb34de20366e6cca786c3} + \field{labelalpha}{Mun00} + \field{sortinit}{M} + \field{sortinithash}{cfd219b90152c06204fab207bc6c7cab} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{isbn}{9780131816299} + \field{title}{Topology} + \field{year}{2000} + \endentry + \entry{lectures:teufel}{misc}{} + \name{author}{1}{}{% + {{hash=91396cef9166c4f47cfbe40ba6179b13}{% + family={Teufel}, + familyi={T\bibinitperiod}, + given={Stefan}, + giveni={S\bibinitperiod}}}% + } + \strng{namehash}{91396cef9166c4f47cfbe40ba6179b13} + \strng{fullhash}{91396cef9166c4f47cfbe40ba6179b13} + \strng{bibnamehash}{91396cef9166c4f47cfbe40ba6179b13} + \strng{authorbibnamehash}{91396cef9166c4f47cfbe40ba6179b13} + \strng{authornamehash}{91396cef9166c4f47cfbe40ba6179b13} + \strng{authorfullhash}{91396cef9166c4f47cfbe40ba6179b13} + \field{labelalpha}{Teu13} + \field{sortinit}{T} + \field{sortinithash}{6f7aff9db9dcfeb7f95fd5bbd2f78df9} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{note}{Unpublished lecture notes} + \field{title}{Mathematische Physik: Klassische Mechanik} + \field{year}{2013} + \verb{urlraw} + \verb https://www.math.uni-tuebingen.de/de/forschung/maphy/personen/stefanteufel/skripte/skript2013.pdf + \endverb + \verb{url} + \verb https://www.math.uni-tuebingen.de/de/forschung/maphy/personen/stefanteufel/skripte/skript2013.pdf + \endverb + \endentry + \entry{book:thirring}{book}{} + \name{author}{1}{}{% + {{hash=abe7ad95d7885afdb493fee7ab36189e}{% + family={Thirring}, + familyi={T\bibinitperiod}, + given={W.}, + giveni={W\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer New York}% + } + \strng{namehash}{abe7ad95d7885afdb493fee7ab36189e} + \strng{fullhash}{abe7ad95d7885afdb493fee7ab36189e} + \strng{bibnamehash}{abe7ad95d7885afdb493fee7ab36189e} + \strng{authorbibnamehash}{abe7ad95d7885afdb493fee7ab36189e} + \strng{authornamehash}{abe7ad95d7885afdb493fee7ab36189e} + \strng{authorfullhash}{abe7ad95d7885afdb493fee7ab36189e} + \field{labelalpha}{Thi03} + \field{sortinit}{T} + \field{sortinithash}{6f7aff9db9dcfeb7f95fd5bbd2f78df9} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{edition}{3rd} + \field{isbn}{9780387406152} + \field{title}{Classical Mathematical Physics: Dynamical Systems and Field Theories} + \field{year}{2003} + \endentry + \entry{book:tu}{book}{} + \name{author}{1}{}{% + {{hash=c37642edbd89e50fd3e8867fb6e209e5}{% + family={Tu}, + familyi={T\bibinitperiod}, + given={Loring\bibnamedelima W.}, + giveni={L\bibinitperiod\bibinitdelim W\bibinitperiod}}}% + } + \list{publisher}{1}{% + {Springer, New York}% + } + \strng{namehash}{c37642edbd89e50fd3e8867fb6e209e5} + \strng{fullhash}{c37642edbd89e50fd3e8867fb6e209e5} + \strng{bibnamehash}{c37642edbd89e50fd3e8867fb6e209e5} + \strng{authorbibnamehash}{c37642edbd89e50fd3e8867fb6e209e5} + \strng{authornamehash}{c37642edbd89e50fd3e8867fb6e209e5} + \strng{authorfullhash}{c37642edbd89e50fd3e8867fb6e209e5} + \field{labelalpha}{Tu11} + \field{sortinit}{T} + \field{sortinithash}{6f7aff9db9dcfeb7f95fd5bbd2f78df9} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{edition}{2nd} + \field{title}{An Introduction to Manifolds} + \field{year}{2011} + \verb{doi} + \verb 10.1007/978-1-4419-7400-6 + \endverb + \verb{urlraw} + \verb https://link.springer.com/book/10.1007%2F978-1-4419-7400-6 + \endverb + \verb{url} + \verb https://link.springer.com/book/10.1007%2F978-1-4419-7400-6 + \endverb + \endentry + \enddatalist +\endrefsection +\endinput +