Math 2.0

Differential geometry Definition

Parametrised curve

Builds on Vector-valued functionContinuity in R^nC^k function characterised by continuous derivatives

Definition

Let I⊆ℝI \subseteq \mathbb{R} be an interval with more than one point. A parametrised curve in ℝn\mathbb{R}^n is a continuous map γ ⁣:I→ℝn\gamma \colon I \to \mathbb{R}^n, γ(t)=(γ1(t),…,γn(t))\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t)). Its image, or trace, is the set γ(I)={γ(t):t∈I}\gamma(I) = \{\gamma(t) : t \in I\}. For r≥1r \ge 1, the curve is a CrC^r curve if each component γi\gamma_i is rr times continuously differentiable on II. When I=[a,b]I = [a, b], the points γ(a)\gamma(a) and γ(b)\gamma(b) are the initial point and the end point of γ\gamma.

Remarks

A curve is different from its image. The maps t↦(cos⁡t,sin⁡t)t \mapsto (\cos t, \sin t) on [0,2π][0, 2\pi] and t↦(cos⁡2t,sin⁡2t)t \mapsto (\cos 2t, \sin 2t) on [0,2π][0, 2\pi] have the same image, the unit circle, but the second goes around it twice. The helix γ(t)=(cos⁡t,sin⁡t,t)\gamma(t) = (\cos t, \sin t, t), t∈ℝt \in \mathbb{R}, is a C∞C^\infty curve in ℝ3\mathbb{R}^3. The curve is simple if γ\gamma is injective, and it is closed if I=[a,b]I = [a, b] and γ(a)=γ(b)\gamma(a) = \gamma(b).

Reading path

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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 curvedefinition
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.
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  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

Where this leads

83 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiable_curve, Definitions
  2. Web https://en.wikipedia.org/wiki/Parametric_equation, Parametric plane curves
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Tag0JF1XXGWB4
URIm2://0JF1XXGWB4
Kinddefinition
WritingW0 stub
MSC53A04
DefinesParametrised curve
UsesVector-valued function, Continuity in R^n, C^k function characterised by continuous derivatives

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