Tangent vector

For a more general, but more technical, treatment of tangent vectors, see Tangent space.

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

Example

Given the curve $$\mathbf{r}(t)=\{(1+t^2,e^{2t},\cos t)\mid t\in \mathbb{ℝ}\}$$ in ${\mathbb{ℝ}}^3$, the unit tangent vector at $t=0$ is given by $$\mathbf{T}(0)=\frac{{\mathbf{r}}^{\prime }(0)}{\Vert {\mathbf{r}}^{\prime }(0)\Vert }={\frac{(2t,2e^{2t},-\sin t)}{\sqrt{4t^2+4e^{4t}+{\sin }^{2}t}}|}_{t=0}=(0,1,0).$$

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 $\mathbf{r}(t)=(x^1(t),x^2(t),\dots ,x^n(t))$ or $$\mathbf{r}=x^i=x^i(t),a\le t\le b,$$ then the tangent vector field $\mathbf{T}=T^i$ is given by $$T^i=\frac{d{x}^i}{dt}.$$ Under a change of coordinates $$u^i=u^i(x^1,x^2,\dots ,x^n),1\le i\le n$$ the tangent vector $\overline{\mathbf{T}}={\overline{T}}^i$ in the uⁱ-coordinate system is given by $${\overline{T}}^i=\frac{d{u}^i}{dt}=\frac{\partial u^i}{\partial x^s}\frac{d{x}^s}{dt}=T^s\frac{\partial u^i}{\partial x^s}$$ 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.

Definition

Let $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ 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 $${\mathrm{\nabla }}_{\mathbf{v}}f(\mathbf{x})={\frac{d}{dt}f(\mathbf{x}+t\mathbf{v})|}_{t=0}=\sum _{i=1}^n{v}_i\frac{\partial f}{\partial x_i}(\mathbf{x}).$$ The tangent vector at the point $\mathbf{x}$ may then be defined as $$\mathbf{v}(f(\mathbf{x}))\equiv ({\mathrm{\nabla }}_{\mathbf{v}}(f))(\mathbf{x}).$$

Properties

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

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

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 $x$ in the manifold is given by the derivation $D_v:A(M)\to \mathbb{ℝ}$ which shall be linear — i.e., for any $f,g\in A(M)$ and $a,b\in \mathbb{ℝ}$ we have

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

Note that the derivation will by definition have the Leibniz property

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

See also

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

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.