\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Manifolds}{5}{}%
\contentsline {section}{\numberline {1.1}Submanifolds}{6}{}%
\contentsline {subsection}{\numberline {1.1.1}Immersed submanifold}{7}{}%
\contentsline {subsection}{\numberline {1.1.2}Embedded (regular) submanifold}{7}{}%
\contentsline {subsection}{\numberline {1.1.3}Implicit submanifold}{7}{}%
\contentsline {section}{\numberline {1.2}Smooth manifolds}{7}{}%
\contentsline {subsection}{\numberline {1.2.1}Smooth functions}{8}{}%
\contentsline {subsubsection}{Diffeomorphism}{9}{}%
\contentsline {subsection}{\numberline {1.2.2}Smooth curves}{9}{}%
\contentsline {chapter}{\numberline {2}Tangent spaces}{11}{}%
\contentsline {section}{\numberline {2.1}Vector fields}{14}{}%
\contentsline {subsection}{\numberline {2.1.1}Vector flows}{14}{}%
\contentsline {subsection}{\numberline {2.1.2}Vector field derivation}{15}{}%
\contentsline {subsubsection}{Lie series}{16}{}%
\contentsline {section}{\numberline {2.2}Differential}{16}{}%
\contentsline {subsection}{\numberline {2.2.1}Related vector fields}{17}{}%
\contentsline {subsection}{\numberline {2.2.2}Critical points}{18}{}%
\contentsline {section}{\numberline {2.3}Lie bracket}{18}{}%
\contentsline {section}{\numberline {2.4}Tangent spaces of submanifolds}{19}{}%
\contentsline {chapter}{\numberline {3}Jet spaces}{21}{}%
\contentsline {chapter}{\numberline {4}Differential forms}{23}{}%
\contentsline {section}{\numberline {4.1}Exterior derivative}{23}{}%
\contentsline {chapter}{\numberline {5}Orientability}{25}{}%
\contentsline {chapter}{\numberline {6}Fiber bundles}{27}{}%
\contentsline {chapter}{\numberline {7}De Rham cohomology}{29}{}%
\contentsline {section}{\numberline {7.1}De Rham complex}{29}{}%
\contentsline {section}{\numberline {7.2}Stokes-Cartan theorem}{29}{}%
\contentsline {chapter}{\numberline {8}Grassmanian}{31}{}%