Vector tangent to a curve or surface at a given point
For a more general, but more technical, treatment of tangent vectors, see Tangent space.
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 is a linear derivation of the algebra defined by the set of germs at .
In mathematics, a tangentvector is a vector that is tangent to a curve or surface at a given point. Tangentvectors are described in the differential...
the tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangentvectors in M...
setting, a vector field gives a tangentvector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are...
the tangent space at x {\displaystyle x} are called the tangentvectors at x {\displaystyle x} . This is a generalization of the notion of a vector, based...
continuous tangentvector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3...
right tangent lines have equation x = 0. In mathematics, a tangentvector is a vector that is tangent to a curve or surface at a given point. Tangent vectors...
is a tangentvector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors. They...
nearby tangent spaces, so it permits tangentvector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
M} ) then the nonzero tangentvectors at each point in the manifold can be classified into three disjoint types. A tangentvector X {\displaystyle X} is:...
velocity vector at p. The collection of all tangentvectors at p forms a vector space: the tangent space to M at p, denoted TpM. If X is a tangentvector at...
}}(t)\right|_{t=t_{0}}} is the tangentvector at the point P = γ(t0). Generally speaking, the tangentvector may be zero. The tangentvector's magnitude ‖ γ ′ ( t...
ordinary n-tuples can be used as well. Definitions of tangentvectors as ordinary vectors A tangentvector at a point p may be defined, here specialized to...
The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity...
manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. Vector bundles are almost always required to be locally trivial,...
the covariant derivative is a way of specifying a derivative along tangentvectors of a manifold. Alternatively, the covariant derivative is a way of...
All curves through point p have a tangentvector, not only world lines. The sum of two vectors is again a tangentvector to some other curve and the same...
tangential vector fields. Given a tangential vector field X and a tangentvector Y to S at p, the covariant derivative ∇YX is a certain tangentvector to S...
product between the Killing vector and the geodesic tangentvector. Along an affinely parametrized geodesic with tangentvector U a {\displaystyle U^{a}}...
covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they...
is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangentvectors to real numbers), and a metric field...
mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the...
transformation rule. A vector v, and local tangent basis vectors {ex, ey} and {er, eφ} . Coordinate representations of v. In the shown example, a vector v = ∑ i ∈...
field is a map to the tangent space, it represents the tangentvectors to some function at each point. Splitting the tangentvectors into directional derivatives...
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase...
the cotangent bundle of the manifold. The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore...
of an affine connection, a geodesic is defined to be a curve whose tangentvectors remain parallel if they are transported along it. Applying this to...