Alphabeta MathArc Length and Rectifiable Curves: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
A line segment has length equal to the distance between its endpoints, and a finitely piecewise-linear path has length equal to the sum of its edge lengths
Example
For every θ≥0, the unit-circle path t↦(cos t,sin t) on [0,θ] has length θ
Counterexample
The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable
Counterexample
The Koch curve is a uniform limit of polygonal paths of lengths (4/3)ⁿ but is not rectifiable
Example
γ(t)=(t,|t|) on [-1,1] is rectifiable of length 2√2 but is not differentiable at 0
Counterexample
Two paths can have the same trace and endpoints but different lengths: one traverses [0,1] once and another traverses it forward, backward, and forward
Counterexample
Sawtooth paths converge uniformly to a line segment while every sawtooth has length √2 and the limit has length 1
Example