Tangent Space

The Tangent Space at a point $p$ in a differentiable manifold $M$ is a vector space that captures the directions in which a curve at $p$ can lead. It is denoted by $T_pM$ or $T_p(M)$.

The dimension of the tangent space is equal to the dimension of the manifold. For a $n$-dimensional manifold $M$,

$$\dim(T_pM) = n$$

The collection of all tangent spaces at every point in the manifold together form the Tangent Bundle $T_\ast M$, a concept central to differential geometry and manifold theory.

See also

• Tangent bundle
• Transversality
• Cotangent space
• Cotangent bundle

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