Tangent Space
Overview
Classical Definition - Tangent Vector
Let {% S %} be a surface in {% \mathbb{R}^3 %}. Given a point {% p \in S %}.
and a curve
{% \vec{\gamma}:(-\epsilon, \epsilon) \mapsto \mathbb{R}^3 %}
with {% \vec{\gamma}(0) = p %}. A tangent vector to {% S %} at {% p %}
is any vector that can be expressed as {% \vec{\gamma}'(0) %} for some curve {% \vec{\gamma} %}.
Directional Derivative
The classical definition above is sufficient for surfaces that are embedded in some higher dimensional space. However,
a more abstract definition of a tangent vector exists that does not require the maninfold to be an embedding.
Given a curve
{% \vec{\gamma}:(-\epsilon, \epsilon) \mapsto \mathbb{R}^3 %}
and for any real-valued function {% f %}, the directional derivative of {% f %}
along {% \gamma %} is given by
{% D_{\gamma}(f) = \frac{d}{dt} f(\gamma(t))|_{t=0} %}
Note the the directional derivative is defined to be an operator. The set of directional derivatives at a point
of a manifold is then defined to be the tangent space at that point.
Topics
