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...

setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are...

the tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M...

the tangent space at x {\displaystyle x} are called the tangent vectors at x {\displaystyle x} . This is a generalization of the notion of a vector, based...

defined as follows: T is the unit vector tangent to the curve, pointing in the direction of motion. N is the normal unit vector, the derivative of T with respect...

is a tangent vector – 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 tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

continuous tangent vector 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...

}}(t)\right|_{t=t_{0}}} is the tangent vector at the point P = γ(t0). Generally speaking, the tangent vector may be zero. The tangent vector's magnitude ‖ γ ′ ( t...

right tangent lines have equation x = 0. In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors...

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...

velocity vector at p. The collection of all tangent vectors at p forms a vector space: the tangent space to M at p, denoted TpM. If X is a tangent vector at...

ordinary n-tuples can be used as well. Definitions of tangent vectors as ordinary vectors A tangent vector at a point p may be defined, here specialized to...

M} ) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X {\displaystyle X} is:...

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,...

T_{\varphi (x)}N} . Hence it can be used to push tangent vectors on M {\displaystyle M} forward to tangent vectors on N {\displaystyle N} . The differential...

the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of...

out. One thus says that covectors are dual to vectors. Thus, to summarize: A vector or tangent vector, has components that contra-vary with a change...

product between the Killing vector and the geodesic tangent vector. Along an affinely parametrized geodesic with tangent vector U a {\displaystyle U^{a}}...

tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p, the covariant derivative ∇YX is a certain tangent vector to S...

All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same...

is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors 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...

covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they...

TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv:[0,1] → M satisfying γv(0) = p with initial tangent vector γ′v(0) = v...

of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to...

values of k). In differential geometry, a k-vector is a vector in the exterior algebra of the tangent vector space; that is, it is an antisymmetric tensor...

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...

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 tangent vectors to some function at each point. Splitting the tangent vectors into directional derivatives...