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} .

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} '(t)} provided it exists and provided r ′ ( t ) ≠ 0 {\displaystyle \mathbf {r} '(t)\neq \mathbf {0} } , where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter t.^[1] The unit tangent vector is given by T ( t ) = r ′ ( t ) | r ′ ( t ) | . {\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|}}\,.}

Given the curve r ( t ) = { ( 1 + t 2 , e 2 t , cos ⁡ t ) ∣ t ∈ R } {\displaystyle \mathbf {r} (t)=\left\{\left(1+t^{2},e^{2t},\cos {t}\right)\mid t\in \mathbb {R} \right\}} in R 3 {\displaystyle \mathbb {R} ^{3}} , the unit tangent vector at t = 0 {\displaystyle t=0} is given by T ( 0 ) = r ′ ( 0 ) ‖ r ′ ( 0 ) ‖ = ( 2 t , 2 e 2 t , − sin ⁡ t ) 4 t 2 + 4 e 4 t + sin 2 ⁡ t | t = 0 = ( 0 , 1 , 0 ) . {\displaystyle \mathbf {T} (0)={\frac {\mathbf {r} '(0)}{\|\mathbf {r} '(0)\|}}=\left.{\frac {(2t,2e^{2t},-\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.} Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to t {\displaystyle t} .

If r ( t ) {\displaystyle \mathbf {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 ) ) {\displaystyle \mathbf {r} (t)=(x^{1}(t),x^{2}(t),\ldots ,x^{n}(t))} or r = x i = x i ( t ) , a ≤ t ≤ b , {\displaystyle \mathbf {r} =x^{i}=x^{i}(t),\quad a\leq t\leq b\,,} then the tangent vector field T = T i {\displaystyle \mathbf {T} =T^{i}} is given by T i = d x i d t . {\displaystyle T^{i}={\frac {dx^{i}}{dt}}\,.} Under a change of coordinates u i = u i ( x 1 , x 2 , … , x n ) , 1 ≤ i ≤ n {\displaystyle u^{i}=u^{i}(x^{1},x^{2},\ldots ,x^{n}),\quad 1\leq i\leq n} the tangent vector T ¯ = T ¯ i {\displaystyle {\bar {\mathbf {T} }}={\bar {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 {\displaystyle {\bar {T}}^{i}={\frac {du^{i}}{dt}}={\frac {\partial u^{i}}{\partial x^{s}}}{\frac {dx^{s}}{dt}}=T^{s}{\frac {\partial u^{i}}{\partial 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.^[2]

Let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a differentiable function and let v {\displaystyle \mathbf {v} } be a vector in R n {\displaystyle \mathbb {R} ^{n}} . We define the directional derivative in the v {\displaystyle \mathbf {v} } direction at a point x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} by ∇ v f ( x ) = d d t f ( x + t v ) | t = 0 = ∑ i = 1 n v i ∂ f ∂ x i ( x ) . {\displaystyle \nabla _{\mathbf {v} }f(\mathbf {x} )=\left.{\frac {d}{dt}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}=\sum _{i=1}^{n}v_{i}{\frac {\partial f}{\partial x_{i}}}(\mathbf {x} )\,.} The tangent vector at the point x {\displaystyle \mathbf {x} } may then be defined^[3] as v ( f ( x ) ) ≡ ( ∇ v ( f ) ) ( x ) . {\displaystyle \mathbf {v} (f(\mathbf {x} ))\equiv (\nabla _{\mathbf {v} }(f))(\mathbf {x} )\,.}

Let f , g : R n → R {\displaystyle f,g:\mathbb {R} ^{n}\to \mathbb {R} } be differentiable functions, let v , w {\displaystyle \mathbf {v} ,\mathbf {w} } be tangent vectors in R n {\displaystyle \mathbb {R} ^{n}} at x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} , and let a , b ∈ R {\displaystyle a,b\in \mathbb {R} } . Then

1. ( a v + b w ) ( f ) = a v ( f ) + b w ( f ) {\displaystyle (a\mathbf {v} +b\mathbf {w} )(f)=a\mathbf {v} (f)+b\mathbf {w} (f)}
2. v ( a f + b g ) = a v ( f ) + b v ( g ) {\displaystyle \mathbf {v} (af+bg)=a\mathbf {v} (f)+b\mathbf {v} (g)}
3. v ( f g ) = f ( x ) v ( g ) + g ( x ) v ( f ) . {\displaystyle \mathbf {v} (fg)=f(\mathbf {x} )\mathbf {v} (g)+g(\mathbf {x} )\mathbf {v} (f)\,.}

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

D v ( a f + b g ) = a D v ( f ) + b D v ( g ) . {\displaystyle D_{v}(af+bg)=aD_{v}(f)+bD_{v}(g)\,.}

Note that the derivation will by definition have the Leibniz property

D v ( f ⋅ g ) ( x ) = D v ( f ) ( x ) ⋅ g ( x ) + f ( x ) ⋅ D v ( g ) ( x ) . {\displaystyle D_{v}(f\cdot g)(x)=D_{v}(f)(x)\cdot g(x)+f(x)\cdot D_{v}(g)(x)\,.}

• Differentiable curve § Tangent vector
• Differentiable surface § Tangent plane and normal vector

• Gray, Alfred (1993), Modern Differential Geometry of Curves and Surfaces, Boca Raton: CRC Press.
• Stewart, James (2001), Calculus: Concepts and Contexts, Australia: Thomson/Brooks/Cole.
• Kay, David (1988), Schaums Outline of Theory and Problems of Tensor Calculus, New York: McGraw-Hill.