6 Differential Geometry
In differential geometry, it is convention to generalize tangent vectors as to a differential operator.
6.1 Elementary Example
6.1.1 Simple
At a point \(P\), tangent vectors are expanded in a finite basis of the tangent space.
\[ T_{P}M = \operatorname{span}\{ e_{1},\ e_{2} \} \]
\[ v = v^{1} e_{1} + v^{2} e_{2} \]
where
- \(T_{P}M\) is the tangent space at \(P\).
- \(e_{1}, e_{2}\) are basis tangent vectors.
- \(v^{1}, v^{2}\) are components of \(v\).
6.1.2 General
In three dimensions the tangent space is spanned by three coordinate directions, and \(v\) has three components.
\[ T_{P}M = \operatorname{span}\{ e_{1},\ e_{2},\ e_{3} \} \]
\[ v = v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3} \]
where
- \(\dim T_{P}M = 3\).
- \(e_{i}\) may be written \(\partial / \partial x^{i}\big|_{P}\) in local coordinates.
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