Tangent vectors

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rⁿ. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point ${\displaystyle x}$ is a linear derivation of the algebra defined by the set of germs at ${\displaystyle x}$.

Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.

Let ${\displaystyle \mathbf {r} (t)}$ be a parametric smooth curve. The tangent vector is given by ${\displaystyle \mathbf {r} ^{\prime }(t)}$, where we have used the a prime instead of the usual dot to indicate differentiation with respect to parameter t. The unit tangent vector is given by

...
Wikipedia