Tangent space: Difference between revisions
imported>Yi Zhe Wu (hah, it's math!) |
imported>Natalie Watson No edit summary |
||
Line 7: | Line 7: | ||
==Definition== | ==Definition== | ||
Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. | Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. There are various ways in which a tangent space can be defined, the most intuitive of which is in terms of ''directional derivatives''; the space <math>T_pM</math> is the space identified with directional derivatives at p. | ||
===Directional derivative=== | ===Directional derivative=== | ||
A ''curve'' on the manifold is defined as a [[differentiable]] map <math>\scriptstyle \gamma: (a,b) \rightarrow M</math>. Let <math>\scriptstyle \gamma(t_0) \, = \, p</math>. If one defines <math>\scriptstyle \mathcal{F}_p</math> to be all the functions <math>\scriptstyle f:M \rightarrow \mathbb{R}^n</math> that are differentiable at the point p, then one can interpret <math> | A ''curve'' on the manifold is defined as a [[differentiable]] map <math>\scriptstyle \gamma: (a,b) \rightarrow M</math>. Let <math>\scriptstyle \gamma(t_0) \, = \, p</math>. If one defines <math>\scriptstyle \mathcal{F}_p</math> to be all the functions <math>\scriptstyle f:M \rightarrow \mathbb{R}^n</math> that are differentiable at the point p, then one can interpret | ||
:<math> \gamma'(t_0)(f) = (f \circ \gamma)'(t_0) | :<math>\gamma'(t_0): \, \mathcal{F}_p \rightarrow \mathbb{R}</math> | ||
to be an operator such that | |||
:<math> \gamma'(t_0)(f) = (f \circ \gamma)'(t_0) = \lim_{h \rightarrow 0} \frac{f(\gamma(t_0+h)) - f(\gamma(t_0))}{h} </math> | |||
and is a '''directional derivative''' of f in the direction of the curve <math>\scriptstyle \gamma</math>. This operator can be interpreted as a ''tangent vector''. | |||
The tangent space is then the set of all directional derivatives of curves at the point p. | |||
[[category:CZ Live]] | [[category:CZ Live]] | ||
[[category:Mathematics Workgroup]] | [[category:Mathematics Workgroup]] |
Revision as of 17:32, 21 July 2007
The tangent space of a differentiable manifold M is a vector space at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted .
The tangent space is necessary for a manifold because it offers a way in which tangent vectors at different points on the manifold can be compared (via an affine connection). If the manifold is a submanifold of , then the tangent space at a point can be thought of as an n-dimensional hyperplane at that point. However, this ambient Euclidean space is unnecessary to the definition of the tangent space.
The tangent space at a point has the same dimension as the manifold, and the union of all the tangent spaces of a manifold is called the tangent bundle, which itself is a manifold of twice the dimension of M.
Definition
Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. There are various ways in which a tangent space can be defined, the most intuitive of which is in terms of directional derivatives; the space is the space identified with directional derivatives at p.
Directional derivative
A curve on the manifold is defined as a differentiable map . Let . If one defines to be all the functions that are differentiable at the point p, then one can interpret
to be an operator such that
and is a directional derivative of f in the direction of the curve . This operator can be interpreted as a tangent vector.
The tangent space is then the set of all directional derivatives of curves at the point p.