Tangent Space

Let $x$ be a point in an $n$-dimensional Compact Manifold $M$, and attach at $x$ a copy of $\Bbb{R}^n$ tangential to $M$. The resulting structure is called the Tangent Space of $M$ at $x$ and is denoted $T_xM$. If $\gamma$ is a smooth curve passing through $x$, then the derivative of $\gamma$ at $x$ is a Vector in $T_xM$.

See also Tangent, Tangent Bundle, Tangent Plane, Tangent Vector