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 $x$ 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 $𝐫 (t)$ be a parametric smooth curve. The tangent vector is given by $𝐫^{'} (t)$ provided it exists and provided $𝐫^{'} (t) \neq 𝟎$ , 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 $𝐓 (t) = \frac{𝐫^{'} (t)}{| 𝐫^{'} (t) |} .$

Example

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

Contravariance

If $𝐫 (t)$ is given parametrically in the n-dimensional coordinate system $x i$ (here we have used superscripts as an index instead of the usual subscript) by $𝐫 (t) = (x^{1} (t), x^{2} (t), \dots, x^{n} (t))$ or $𝐫 = x^{i} = x^{i} (t), a \leq t \leq b,$ then the tangent vector field $𝐓 = T^{i}$ is given by $T^{i} = \frac{d x^{i}}{d t} .$ Under a change of coordinates $u^{i} = u^{i} (x^{1}, x^{2}, \dots, x^{n}), 1 \leq i \leq n$ the tangent vector $\bar{𝐓} = {\bar{T}}^{i}$ in the $u i$ -coordinate system is given by ${\bar{T}}^{i} = \frac{d u^{i}}{d t} = \frac{\partial u^{i}}{\partial x^{s}} \frac{d x^{s}}{d t} = T^{s} \frac{\partial u^{i}}{\partial x^{s}}$ 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 $f : ℝ^{n} \to ℝ$ 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 $𝐱 \in ℝ^{n}$ by $\nabla_{𝐯} f (𝐱) = {\frac{d}{d t} f (𝐱 + t 𝐯) |}_{t = 0} = \sum_{i = 1}^{n} v_{i} \frac{\partial f}{\partial x_{i}} (𝐱) .$ The tangent vector at the point $𝐱$ may then be defined as $𝐯 (f (𝐱)) \equiv (\nabla_{𝐯} (f)) (𝐱) .$

Properties

Let $f, g : ℝ^{n} \to ℝ$ be differentiable functions, let $𝐯, 𝐰$ be tangent vectors in $ℝ^{n}$ at $𝐱 \in ℝ^{n}$ , and let $a, b \in ℝ$ . 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 on $M$ . Then the tangent vector to $M$ at a point $x$ in the manifold is given by the derivation $D_{v} : A (M) \to ℝ$ which shall be linear — i.e., for any $f, g \in A (M)$ and $a, b \in ℝ$ 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 \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.

[1] J. Stewart (2001)

[2] D. Kay (1988)

[3] A. Gray (1993)