Math 2.0

Differential geometry Definition

Regular curve

Builds on Parametrised curveTangent vector of a curve

Definition

Let I⊆ℝI \subseteq \mathbb{R} be an interval, and let γ ⁣:I→ℝn\gamma \colon I \to \mathbb{R}^n be a C1C^1 parametrised curve. The curve γ\gamma is regular if its tangent vector γ′(t)\gamma'(t) is not zero for each t∈It \in I.

Remarks

The circle t↦(cos⁡t,sin⁡t)t \mapsto (\cos t, \sin t) and each line t↦p+tvt \mapsto p + t v with v≠0v \ne 0 are regular. The curve t↦(t2,t3)t \mapsto (t^2, t^3) is not regular, since its tangent vector is zero at t=0t = 0. The curve t↦(t3,t3)t \mapsto (t^3, t^3) is not regular at t=0t = 0, but its image is a straight line. So regularity is a property of the parametrisation, not of the image. A regular curve has an arc length parametrisation.

Reading path

To understand this page from the ground up, read these 16 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…definitionRegular curve, definition, no evidence levelRegular 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.
  • 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
  16. Regular curve definition

Where this leads

26 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiable_curve, Definitions
About this page
Tag06JMR2XWXW
URIm2://06JMR2XWXW
Kinddefinition
WritingW0 stub
MSC53A04
DefinesRegular curve
UsesParametrised curve, Tangent vector of a curve

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

Results about this