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”.
In many editions of Analisi Matematica 2, page 77 falls within the chapter on differential calculus for functions of several variables — specifically, exercises involving:
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).
Verify the conditions for the Implicit Function Theorem (Dini's Theorem) for the equation: $$ F(x, y) = 0 $$ Specifically, analyze the solvability with respect to $y$ (finding $y=y(x)$) or $x$ (finding $x=x(y)$), calculate the first derivative, and determine the domain of the implicit function.
(A common specific variation found in this slot is: "Studiare la risolubilità dell'equazione $y - x \sin(y) = 0$ rispetto alla variabile $y$ in prossimità dell'origine.") Most Italian universities (e
Let's solve this specific variation, which is a classic Fusco-Marcellini-Sbordone example.
“Compute ( \iint_D \fracxyx^2 + y^2 dx dy ) where D is delimited by ( y = x^2 ) and ( y = 2 - x^2 ).”
This requires polar coordinates or a savvy change of variables.
Why do students search for #77 specifically? Because problems before 50 are usually warm-ups. By 70–80, the authors introduce conceptual twists. Number 77 is famous online due to its appearance in many exam simulations (Politecnico di Milano, Università di Napoli, Roma Tre).
Once upon a time, a student named Luca was preparing for his Analisi 2 exam. He had the famous yellow-and-black Sbordone exercise book open at page 77. The problem showed a double integral over a strange domain: Exercise 77 (on that page or numbered as
[ \iint_D \frac1(x^2 + y^2)^\alpha , dx,dy ] where ( D ) was the region between two circles centered at the origin: ( 1 \le x^2 + y^2 \le 4 ), and ( \alpha ) was a real parameter.
Luca thought: “Easy, just polar coordinates.” But when he tried, he got an integral that diverged for some ( \alpha ). He was confused: the domain is bounded and away from zero – why would it diverge?
Then he remembered the lesson of page 77:
“In double integrals with radial symmetry, the convergence depends on the exponent ( \alpha ) relative to the dimension (2). But here, since the domain avoids the origin, no singularity exists inside. Wait – the book’s trick is: the outer radius is finite, so the only potential singularity is at ( r \to 0 ), but ( r \ge 1 ) here. So the integral is always finite! So why does the book ask to discuss convergence?” Once upon a time, a student named Luca
He looked closer. The real exercise on page 77 (updated version) actually had the domain ( D ) as the unbounded region outside a circle, e.g., ( x^2+y^2 \ge 1 ). Now the story changed.
In some editions of the "Lite" version, Exercise 77 may refer to a Taylor expansion problem (e.g., "Write the Taylor series of the second order for a function...").
If your exercise asks for a Taylor expansion (Sviluppo di Taylor), here is the general method:
1. Formula: $$ f(x_0, y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) + \frac12[ \dots ] $$
2. Steps: