Alphabeta Math

Fubini and Change of Variables: Examples and Counterexamples

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✓ 13 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Coxeter Groups
Linear Algebra
Number Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

An integrable function on the unit square with one Dirichlet section and only one defined order of ordinary iteration

Counterexample

One existing iterated integral does not imply multiple Riemann integrability

Example

A Riemann-integrable Thomae-type function whose x-sections are nonintegrable at every rational height

Example

Fubini computes ∫₀¹∫₋₁¹ x exp(xy) dx dy by reversing the order

Example

A coordinate shear preserves Jordan content by translating every section

Example

The content of a concrete three-dimensional parallelepiped computed from its spanning matrix

Example

Polar change of variables on a compact annular sector gives the Jacobian factor r and its area

Example

Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box

Example

Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam

Example

The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers

Counterexample

Polar coordinates on a full closed angular period are not injective and are singular at radius zero

Counterexample

Omitting the absolute value from the Jacobian gives negative length under the reflection x↦1-x

Counterexample

Dropping injectivity double-counts under x↦ x² on two disjoint intervals

Sources