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Mathematics for Physics

191 Tangent Space

A vector space of all tangent directions at a point that is used to approximate a curved domain by a flat vector space at that point.

definition [d] (Tangent Space = Tangent Plane) The vector space of all vectors tangent to a manifold \(M\) at a point \(P\):

  • \(T_{P}(M) = \{ \text{tangent vectors to } M \text{ at } P \}\) .

where

  • \(M\) is a smooth manifold.
  • \(P\) is a point of \(M\).
  • \(T_{P}(M)\) is the tangent space at \(P\).

Note:

  • \(\dim T_{P}(M) = \dim M\).
  • the union of all tangent spaces is the tangent bundle \(T(M)\).
  • tangent plane is common for surfaces and local pictures.

definition [d] (Tangent Space = Tangent Plane) The \(n\)-dimensional vector space \(T_{P}M\) spanned by the coordinate basis at \(P\):

  • \(T_{P}M = \operatorname{span}\left\{ \displaystyle \dfrac{\partial}{\partial x^{1}}\bigg|_{P},\, \ldots,\, \dfrac{\partial}{\partial x^{n}}\bigg|_{P} \right\}\) .

where

  • \(T_{P}M\) is the tangent space at \(P\).
  • \(n\) is the dimension of the manifold.
  • \((x^{1},\ldots,x^{n})\) are local coordinates near \(P\).
  • \(\partial / \partial x^{i}\big|_{P}\) are the coordinate basis vectors at \(P\).

Note:

  • any tangent vector is \(v = v^{i}\, \partial_{i}\big|_{P}\).
  • metric components are \(g_{ij} = \langle \partial_{i}, \partial_{j} \rangle\).

191.1 Elementary Example

191.1.1 Simple

The tangent space at a point is spanned by a finite list of tangent basis vectors.

\[ T_{P}M = \{ e_{1},\ e_{2} \} \]

\[ v = 2 e_{1} - e_{2} \]

where

  • \(T_{P}M\) is the tangent space at the point \(P\).
  • \(v\) is a tangent vector at \(P\).

191.1.2 General

If \(\dim M = 3\), then \(T_{P}M\) is three-dimensional with coordinate basis \(\partial_{i}\big|_{P}\).

\[ T_{P}M = \operatorname{span}\left\{ \dfrac{\partial}{\partial x^{1}}\Big|_{P},\, \dfrac{\partial}{\partial x^{2}}\Big|_{P},\, \dfrac{\partial}{\partial x^{3}}\Big|_{P} \right\} \]

\[ v = v^{1} \partial_{1}\big|_{P} + v^{2} \partial_{2}\big|_{P} + v^{3} \partial_{3}\big|_{P} \]

where

  • \(\partial_{i}\big|_{P}\) are the coordinate basis vectors at \(P\).
  • \(v^{i}\) are the components of \(v\).

191.2 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — tangent space \(T_{P}(M)\); tangent bundle \(T(M)\).
  2. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — tangent vectors; coordinate basis \(\partial_{i}\).
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — basis vectors and components in curvilinear coordinates.