This tag is for questions regarding to the tangent space, the linear space that best approximates an object at a given point. Intuitively, the tangent space $ T_p(M)$ at a point $ p$ on an $ n$-dimensional manifold $ M$ is an $ n$-dimensional hyperplane in $ {\mathbb{R}}^m$ that best approximates $ M$ around $ p$, when the hyperplane origin is translated to $ p$.

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

**Definition :** Let $~M~$ be a manifold, $~p\in M~$. The tangent space $~T_pM~$ is the set of all linear maps $~v : C^\infty (M)\to\mathbb R~$ of the form
$$v(f) = \left[\dfrac d{dt} \right]_{t=0} f(g(t))$$
for some smooth curve $~\gamma\in C^\infty (J,M)~$ with $~g(0) = p~$.

The elements $~v\in T_pM~$ are called the tangent vectors to $~M~$ at $~p~$.

For more details you may visit

$1.~$ https://en.wikipedia.org/wiki/Tangent_space

$2.~$ http://www.math.toronto.edu/mgualt/courses/18-367/docs/DiffGeomNotes-8.pdf

$3.~$ https://projecteuclid.org/download/pdf_1/euclid.lnms/1215540658

$4.~$ http://planning.cs.uiuc.edu/node386.html

$5.~$ http://www.math.caltech.edu/~2014-15/3term/ma001c-an/week3.pdf