Tangent vector
Encyclopedia
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
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
or surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
at a given point.
Tangent vectors are described in the differential geometry of curves
Differential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....
in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space
Tangent space
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 pointing from one to the other....
of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
.