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Tangent Space
Undergraduate
Definition
Vector space of all possible "directions" at a point on manifold
Formulas
Tₚ M = γ'(0) : γ(0) = p
Dimension: dim Tₚ M = dim M
Basis: ∂/∂xⁱ |ₚ
Examples
Example 1
Tangent space of plane = ℝ²
Example 2
Tangent plane of sphere
Applications
Vector fields
Differential forms
Physics
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