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'classical differential geometry'
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| Title of object: |
classical differential geometry |
| Canonical Name: |
ClassicalDifferentialGeometry |
| Type: |
Topic |
| Created on: |
2005-05-18 14:00:11 |
| Modified on: |
2007-10-22 18:02:53 |
| Classification: |
msc:53A04, msc:53A05 |
Preamble:
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% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{amsthm}
\usepackage{mathrsfs}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{xypic}
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\newtheorem{thm}{Theorem}
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Content:
\subsection*{Curves in $\R^2$}
\begin{itemize}
\item inflexion point
\item curvature (plane curve)
\item circle of curvature
\item curvature determines the curve
\item curvature of Nielsen's spiral
\item orthogonal curves
\item parallel curves
\item properties of parallel curves
\item evolute
\item \PMlinkname{Serret-Frenet equations in $\R^2$}{SerretFrenetEquationsInMathbbR2}
\item Famous curves in the plane
\item arc-parametrizations
\item envelope
\item determining envelope
\end{itemize}
\subsection*{Curves in $\R^3$}
\begin{itemize}
\item \PMlinkname{Serret-Frenet equations in $\R^3$}{SerretFrenetFormulas}
\item space curve
\item \PMlinkname{curvature}{CurvatureOfACurve} and \PMlinkname{torsion}{Torsion} of a space curve
\item moving trihedron
\end{itemize}
\subsection*{Surfaces in $\R^3$}
\begin{itemize}
\item surface of revolution
\item surface normal
\item normal section
\item normal curvatures
\item Meusnier's theorem
\item mean curvature at surface point
\item \PMlinkname{first fundamental form}{FirstFundamentalForm}
\item second fundamental form
\item sphere map and shape operator
\item Gaussian curvature and mean curvature
\item geodesic
\item variational calculus
\item Gauss-Bonnet theorem
\item standard connection in $\R^3$
\item Gauss equation
\end{itemize}
\subsection*{The space $\R^3$}
\begin{itemize}
\item ortho-normal frame fields in $\R^3$ (or non constant ortho-normal triples of vector fields)
\item rate of rotation of an o.f.f.
\item euclidean spin connection
\item $\R^3$ Cartan's estructural equations I,II
\end{itemize}
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