Tangent vector

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 {displaystyle x} is a linear derivation of the algebra defined by the set of germs at x {displaystyle x} . 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 {displaystyle x} is a linear derivation of the algebra defined by the set of germs at x {displaystyle x} . Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties. Let r ( t ) {displaystyle mathbf {r} (t)} be a parametric smooth curve. The tangent vector is given by r ′ ( t ) {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