Concept

Tangent vector

Summary
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 x is a linear derivation of the algebra defined by the set of germs at x. Motivation Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties. Calculus Let \mathbf{r}(t) be a parametric smooth curve. The tangent vector is given by \mathbf{r}'(t), where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter t. The unit tangent vector is given by \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{
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