Tangent vector - Alchetron, The Free Social Encyclopedia

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. In other words, a tangent vector at the point x is a linear derivation of the algebra defined by the set of terms 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 r ( t ) be a parametric smooth curve. The tangent vector is given by r ′ ( 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

T ( t ) = r ′ ( t ) | r ′ ( t ) | .

Example

Given the curve

r ( t ) = { ( 1 + t 2 , e 2 t , cos ⁡ t ) | t ∈ R }

in R 3 , the unit tangent vector at time t = 0 is given by

T ( 0 ) = r ′ ( 0 ) | r ′ ( 0 ) | = ( 2 t , 2 e 2 t , − s i n t ) 4 t 2 + 4 e 4 t + sin 2 ⁡ t | t = 0 = ( 0 , 1 , 0 ) .

Contravariance

If r ( t ) is given parametrically in the n-dimensional coordinate system xⁱ (here we have used superscripts as an index instead of the usual subscript) by r ( t ) = ( x 1 ( t ) , x 2 ( t ) , … , x n ( t ) ) or

r = x i = x i ( t ) , a ≤ t ≤ b ,

then the tangent vector field T = T i is given by

T i = d x i d t .

Under a change of coordinates

u i = u i ( x 1 , x 2 , … , x n ) , 1 ≤ i ≤ n

the tangent vector T ¯ = T ¯ i in the uⁱ-coordinate system is given by

T ¯ i = d u i d t = ∂ u i ∂ x s d x s d t = T s ∂ u i ∂ x s

where we have used the Einstein summation convention. Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates.

Definition

Let f : R n → R be a differentiable function and let v be a vector in R n . We define the directional derivative in the v direction at a point x ∈ R n by

D v f ( x ) = d d t f ( x + t v ) | t = 0 = ∑ i = 1 n v i ∂ f ∂ x i ( x ) .

The tangent vector at the point x may then be defined as

v ( f ( x ) ) ≡ D v ( f ( x ) ) .

Properties

Let f , g : R n → R be differentiable functions, let v , w be tangent vectors in R n at x ∈ R n , and let a , b ∈ R . Then

( a v + b w ) ( f ) = a v ( f ) + b w ( f )
v ( a f + b g ) = a v ( f ) + b v ( g )
v ( f g ) = f ( x ) v ( g ) + g ( x ) v ( f ) .

Tangent vector on manifolds

Let M be a differentiable manifold and let A ( M ) be the algebra of real-valued differentiable functions M . Then the tangent vector to M at a point x in the manifold is given by the derivation D v : A ( M ) → R which shall be linear — i.e., for any f , g ∈ A ( M ) and a , b ∈ R we have

D v ( a f + b g ) = a D v ( f ) + b D v ( g ) .

Note that the derivation will by definition have the Leibniz property

D v ( f ⋅ g ) = D v ( f ) ⋅ g ( x ) + f ( x ) ⋅ D v ( g ) .

References

Tangent vector Wikipedia

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