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This story has helped countless Italian engineering and math students remember the critical exponent in double improper integrals over unbounded domains.
Most Italian universities (e.g., Università di Bologna, Sapienza, PoliTo) maintain internal Moodle/Teams channels where tutors upload corrected solutions to the most requested exercises. Ask your tutor for “Foglio soluzioni – Esercizio 77”. This story has helped countless Italian engineering and
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Topic: Local invertibility and Implicit Function Theorem.
In many editions of Analisi Matematica 2, page 77 falls within the chapter on differential calculus for functions of several variables — specifically, exercises involving: Most Italian universities (e
Exercise 77 (on that page or numbered as 77) is often cited online as a challenging problem requiring careful application of theorems (e.g., Schwarz’s theorem, chain rule for vector functions). Students hunting for a solution PDF often include “upd” to indicate they want an updated solution set — possibly one corrected for typos or aligned with the latest edition (e.g., 2020/2023 reprints).
Let [ I(\alpha) = \iint_x^2+y^2 \ge 1 \frac1(x^2+y^2)^\alpha , dx,dy. ]
Using polar coordinates: [ I(\alpha) = \int_\theta=0^2\pi \int_r=1^\infty \frac1r^2\alpha \cdot r , dr, d\theta = 2\pi \int_1^\infty r^1-2\alpha , dr. ] d\theta = 2\pi \int_1^\infty r^1-2\alpha
Now Luca saw the convergence condition: [ \int_1^\infty r^1-2\alpha dr \text converges \iff 1-2\alpha < -1 \iff -2\alpha < -2 \iff \alpha > 1. ]
Conclusion:
This is the classic “convergence of improper double integrals at infinity” example, and it’s exactly what Fusco–Marcellini–Sbordone highlight on many editions’ page 77.