Math 2.0

Real functions Definition

Rectifiable curve

Builds on Parametrised curve

Definition

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a parametrised curve. For a partition PP given by a=t0<t1<⋯<tk=ba = t_0 < t_1 < \dots < t_k = b, let L(γ,P)=∑i=1k∥γ(ti)−γ(ti−1)∥L(\gamma, P) = \sum_{i = 1}^{k} \|\gamma(t_i) - \gamma(t_{i-1})\| be the length of the inscribed polygonal path. The curve γ\gamma is rectifiable if sup⁡PL(γ,P)<∞\sup_P L(\gamma, P) < \infty, where the supremum is over all partitions of [a,b][a, b]. Then the length of γ\gamma is ℓ(γ)=sup⁡PL(γ,P)\ell(\gamma) = \sup_P L(\gamma, P).

Remarks

If γ\gamma is of class C1C^1, then it is rectifiable and ℓ(γ)=∫ab∥γ′(t)∥ dt\ell(\gamma) = \int_a^b \|\gamma'(t)\| \, dt. A refinement of PP does not decrease L(γ,P)L(\gamma, P), by the triangle inequality. The graph of f(x)=xsin⁡(1/x)f(x) = x \sin(1/x) for 0<x≤10 < x \le 1, f(0)=0f(0) = 0, is a continuous curve that is not rectifiable. The Koch curve is another example. A curve is rectifiable if and only if each component γi\gamma_i has bounded variation on [a,b][a, b].

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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 curvedefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinition
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  1. Euclidean space object
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  3. Fréchet derivative definition
  4. Function of several variables concept
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  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. Rectifiable curve definition

Where this leads

48 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Arc_length, Definition
  2. Web https://en.wikipedia.org/wiki/Arc_length, Curves with infinite length
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DefinesRectifiable curve
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