Inmathematics, atangent vector is avector that istangent to acurve orsurface at a given point. Tangent vectors are described in thedifferential geometry of curves in the context of curves inRⁿ. More generally, tangent vectors are elements of atangent space of adifferentiable manifold. Tangent vectors can also be described in terms ofgerms. Formally, a tangent vector at the point${\displaystyle x}$ is a linearderivation of the algebra defined by the set of germs at${\displaystyle x}$.

Before proceeding to a general definition of the tangent vector, we discuss its use incalculus and itstensor properties.

Let${\displaystyle \mathbf{r}(t)}$ be a parametricsmooth curve. The tangent vector is given by${\displaystyle {\mathbf{r}}^{\prime }(t)}$ provided it exists and provided${\displaystyle {\mathbf{r}}^{\prime }(t)\ne \mathbf{0}}$, where we have used a prime instead of the usual dot to indicate differentiation with respect to parametert.^[1] The unit tangent vector is given by$${\displaystyle \mathbf{T}(t)=\frac{{\mathbf{r}}^{\prime }(t)}{|{\mathbf{r}}^{\prime }(t)|}.}$$

Given the curve$${\displaystyle \mathbf{r}(t)=\left\{\left(1+t^2,e^{2t},\cos t\right)\mid t\in \mathbb{R}\right\}}$$in${\displaystyle {\mathbb{R}}^3}$, the unit tangent vector at${\displaystyle t=0}$ is given by$${\displaystyle \mathbf{T}(0)=\frac{{\mathbf{r}}^{\prime }(0)}{\Vert {\mathbf{r}}^{\prime }(0)\Vert }={\frac{(2t,2e^{2t},-\sin t)}{\sqrt{4t^2+4e^{4t}+{\sin }^{2}t}}|}_{t=0}=(0,1,0).}$$Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to${\displaystyle t}$.

If${\displaystyle \mathbf{r}(t)}$ is given parametrically in then-dimensional coordinate systemxⁱ (here we have used superscripts as an index instead of the usual subscript) by${\displaystyle \mathbf{r}(t)=(x^1(t),x^2(t),\dots ,x^n(t))}$ or$${\displaystyle \mathbf{r}=x^i=x^i(t),a\le t\le b,}$$then the tangent vector field${\displaystyle \mathbf{T}=T^i}$ is given by$${\displaystyle T^i=\frac{d{x}^i}{dt}.}$$Under a change of coordinates$${\displaystyle u^i=u^i(x^1,x^2,\dots ,x^n),1\le i\le n}$$the tangent vector${\displaystyle \overline{\mathbf{T}}={\overline{T}}^i}$ in theuⁱ-coordinate system is given by$${\displaystyle {\overline{T}}^i=\frac{d{u}^i}{dt}=\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^s}\frac{d{x}^s}{dt}=T^s\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^s}}$$where we have used theEinstein summation convention. Therefore, a tangent vector of a smooth curve will transform as acontravariant tensor of order one under a change of coordinates.^[2]

Let${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ be a differentiable function and let${\displaystyle \mathbf{v}}$ be a vector in${\displaystyle {\mathbb{R}}^n}$. We define thedirectional derivative in the${\displaystyle \mathbf{v}}$ direction at a point${\displaystyle \mathbf{x}\in {\mathbb{R}}^n}$ by$${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f(\mathbf{x})={\frac{d}{dt}f(\mathbf{x}+t\mathbf{v})|}_{t=0}=\sum _{i=1}^n{v}_i\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}(\mathbf{x}).}$$The tangent vector at the point${\displaystyle \mathbf{x}}$ may then be defined^[3] as$${\displaystyle \mathbf{v}(f(\mathbf{x}))\equiv ({\mathrm{\nabla }}_{\mathbf{v}}(f))(\mathbf{x}).}$$

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

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

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

${\displaystyle D_v(af+bg)=a{D}_v(f)+b{D}_v(g).}$

Note that the derivation will by definition have the Leibniz property

${\displaystyle D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x).}$

• 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.

• Vectors (mathematics and physics)

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