Alphabeta MathLine Integrals and the Gradient Theorem: 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
- 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
- Line Integrals and the Gradient Theorem
- 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 Derivative and the Mean Value Theorems
- The Fundamental Theorems of Calculus
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
The scalar line integral of x over the right unit semicircle equals two
Example
Scalar and vector line integrals along an affine line segment
Example
A polynomial potential evaluates work along every path by endpoints
Example
Constructing a potential on a rectangle by coordinate-segment integrals
Counterexample
The vector field (y,0) gives different integrals along two paths with the same endpoints
Counterexample
The vortex field is closed but not exact on the punctured plane
False statement
False: every closed C1 field on a connected open set is exact
False statement
False: vector line integrals are invariant under reversing a path
Example
A vector line integral around the vortex counts repeated traversals
Sources
- J. Lebl, Basic Analysis II, Example 9.2.16
- J. Lebl, Basic Analysis II, Example 9.2.18
- J. Lebl, Basic Analysis II, section 9.3
- J.-B. Campesato, Poincare Lemma, sections 1 and 2
- J. Lebl, Basic Analysis II, Example 9.3.1
- J. Lebl, Basic Analysis II, Example 9.3.7
- J.-B. Campesato, Poincare Lemma, section 1
- J. Lebl, Basic Analysis II, Proposition 9.2.15
- J. Lebl, Basic Analysis II, sections 9.2 and 9.3