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} x is a linear derivation of the algebra defined by the set of germs 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 \( {\mathbf {r}}(t) \) be a parametric smooth curve. The tangent vector is given by \( {\mathbf {r}}^{\prime }(t) \), 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
\( {\mathbf {T}}(t)={\frac {{\mathbf {r}}^{\prime }(t)}{|{\mathbf {r}}^{\prime }(t)|}}\,. \)
Example
Given the curve
\( {\displaystyle \mathbf {r} (t)=\{(1+t^{2},e^{2t},\cos {t})|\ t\in \mathbb {R} \}} \)
in \( \mathbb {R} ^{3} \), the unit tangent vector at t = 0 {\displaystyle t=0} t=0 is given by
\( {\displaystyle \mathbf {T} (0)={\frac {\mathbf {r} ^{\prime }(0)}{\|\mathbf {r} ^{\prime }(0)\|}}=\left.{\frac {(2t,2e^{2t},\ -\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.}\)
Contravariance
If \( {\mathbf {r}}(t) \) is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by \( {\mathbf {r}}(t)=(x^{1}(t),x^{2}(t),\ldots ,x^{n}(t)) \) or
\( {\mathbf {r}}=x^{i}=x^{i}(t),\quad a\leq t\leq b\,, \)
then the tangent vector field \( {\mathbf {T}}=T^{i} \)is given by
\( T^{i}={\frac {dx^{i}}{dt}}\,. \)
Under a change of coordinates
\( u^{i}=u^{i}(x^{1},x^{2},\ldots ,x^{n}),\quad 1\leq i\leq n \)
the tangent vector \( {\bar {{\mathbf {T}}}}={\bar {T}}^{i} \) in the ui-coordinate system is given by
\( {\bar {T}}^{i}={\frac {du^{i}}{dt}}={\frac {\partial u^{i}}{\partial x^{s}}}{\frac {dx^{s}}{dt}}=T^{s}{\frac {\partial u^{i}}{\partial x^{s}}}\)
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]
Definition
Let \( f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}} \) be a differentiable function and let \) \mathbf {v} \) be a vector in \( \mathbb {R} ^{n}\). We define the directional derivative in the \( \mathbf {v} \) direction at a point \( {\mathbf {x}}\in {\mathbb {R}}^{n} \) by
\( D_{{\mathbf {v}}}f({\mathbf {x}})=\left.{\frac {d}{dt}}f({\mathbf {x}}+t{\mathbf {v}})\right|_{{t=0}}=\sum _{{i=1}}^{{n}}v_{i}{\frac {\partial f}{\partial x_{i}}}({\mathbf {x}})\,.\)
The tangent vector at the point \( \mathbf {x} \) may then be defined[3] as
\( {\displaystyle \mathbf {v} (f(\mathbf {x} ))\equiv (D_{\mathbf {v} }(f))(\mathbf {x} )\,.}\)
Properties
Let \( f,g:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}} \) be differentiable functions, let \( {\mathbf {v}},{\mathbf {w}}\) be tangent vectors in \( \mathbb {R} ^{n} \) at x ∈ \( {\mathbf {x}}\in {\mathbb {R}}^{n}\) , and let \( a,b\in {\mathbb {R}}\). Then
\( (a{\mathbf {v}}+b{\mathbf {w}})(f)=a{\mathbf {v}}(f)+b{\mathbf {w}}(f)\)
\( {\mathbf {v}}(af+bg)=a{\mathbf {v}}(f)+b{\mathbf {v}}(g)\)
\( {\mathbf {v}}(fg)=f({\mathbf {x}}){\mathbf {v}}(g)+g({\mathbf {x}}){\mathbf {v}}(f)\,..\)
Tangent vector on manifolds
Let M be a differentiable manifold and let A(M) be the algebra of real-valued differentiable functions on M. Then the tangent vector to M at a point x {\displaystyle x} x in the manifold is given by the derivation \( D_{v}:A(M)\rightarrow {\mathbb {R}} \) which shall be linear — i.e., for any f , g ∈ A ( M ) {\displaystyle f,g\in A(M)} f,g\in A(M) and a , b ∈ R {\displaystyle a,b\in \mathbb {R} } a,b\in {\mathbb {R}} we have
\( D_{v}(af+bg)=aD_{v}(f)+bD_{v}(g)\,.
Note that the derivation will by definition have the Leibniz property
\( {\displaystyle D_{v}(f\cdot g)(x)=D_{v}(f)(x)\cdot g(x)+f(x)\cdot D_{v}(g)(x)\,.}
References
J. Stewart (2001)
D. Kay (1988)
A. Gray (1993)
Bibliography
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.
Hellenica World - Scientific Library
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