Math 2.0

Differential geometry Definition

Tangent vector of a curve

Builds on Parametrised curveVector-valued function

Definition

Let I⊆ℝI \subseteq \mathbb{R} be an interval, let γ ⁣:I→ℝn\gamma \colon I \to \mathbb{R}^n be a parametrised curve, and let t0∈It_0 \in I. If the limit γ′(t0)=lim⁡h→0γ(t0+h)−γ(t0)h=(γ1′(t0),…,γn′(t0))\gamma'(t_0) = \lim_{h \to 0} \frac{\gamma(t_0 + h) - \gamma(t_0)}{h} = \big( \gamma_1'(t_0), \dots, \gamma_n'(t_0) \big) exists, it is the tangent vector, or velocity vector, of γ\gamma at t0t_0. Its norm ∥γ′(t0)∥\|\gamma'(t_0)\| is the speed of γ\gamma at t0t_0. If γ′(t0)≠0\gamma'(t_0) \ne 0, the unit tangent vector is T(t0)=γ′(t0)/∥γ′(t0)∥T(t_0) = \gamma'(t_0) / \|\gamma'(t_0)\|, and the tangent line at γ(t0)\gamma(t_0) is {γ(t0)+s γ′(t0):s∈ℝ}\{\gamma(t_0) + s \, \gamma'(t_0) : s \in \mathbb{R}\}.

Remarks

At an end point of II, the limit is one-sided. For the circle γ(t)=(cos⁡t,sin⁡t)\gamma(t) = (\cos t, \sin t), γ′(t)=(−sin⁡t,cos⁡t)\gamma'(t) = (-\sin t, \cos t). It is orthogonal to γ(t)\gamma(t) and has length 11. The tangent vector can be zero. For γ(t)=(t2,t3)\gamma(t) = (t^2, t^3), γ′(0)=(0,0)\gamma'(0) = (0, 0), and the image has a cusp at the origin. The tangent vector depends on the parametrisation, but for a regular curve its direction does not change under an increasing change of parameter.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Parametrised curve definition
  15. Tangent vector of a curve definition

Where this leads

27 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiable_curve, Tangent vector
About this page
Tag0D98771SM0
URIm2://0D98771SM0
Kinddefinition
WritingW0 stub
MSC53A04
DefinesTangent vector of a curve
UsesParametrised curve, Vector-valued function

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by