Math 2.0

Differential geometry Definition

Closed curve

Builds on Parametrised curve

Definition

Let a<ba < b, and let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a parametrised curve. The curve γ\gamma is closed if γ(a)=γ(b)\gamma(a) = \gamma(b). A closed curve is simple if, in addition, γ(s)≠γ(t)\gamma(s) \ne \gamma(t) for all s<ts < t in [a,b][a, b] other than the pair s=as = a, t=bt = b.

Remarks

The unit circle γ(t)=(cos⁡t,sin⁡t)\gamma(t) = (\cos t, \sin t), t∈[0,2π]t \in [0, 2\pi], is a simple closed curve. The curve t↦(sin⁡2t,sin⁡t)t \mapsto (\sin 2t, \sin t), t∈[0,2π]t \in [0, 2\pi], is closed but not simple: it passes through the origin at t=0t = 0, t=πt = \pi and t=2πt = 2\pi. The line integral of a vector field around a closed curve is often written ∮γF⋅d𝐫\oint_\gamma F \cdot d\mathbf{r}. It is the circulation of FF around γ\gamma. A simple closed curve in the plane bounds a region. This is the setting of Green's theorem.

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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 curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinition
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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
  15. Closed curve definition

Where this leads

31 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Curve, Topological curve
  2. Web https://en.wikipedia.org/wiki/Differentiable_curve, Definitions
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WritingW1 stated
MSC53A04
DefinesClosed curve
UsesParametrised curve

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