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The Fusco, Marcellini, Sbordone textbook series, specifically Analisi Matematica 2, is a cornerstone of undergraduate mathematics in Italy. The "Pdf 77" likely refers to page 77 or Exercise 77 within the associated exercise volumes, which typically cover multivariable functions or differentiability in that section of the curriculum. Overview of the Series
This collection is divided between theoretical "Lezioni" and practical "Esercitazioni," known for their rigorous approach and clarity.
Theory (Lezioni): Authored by Nicola Fusco, Paolo Marcellini, and Carlo Sbordone.
Exercises (Esercitazioni): Usually published in two volumes by Zanichelli or Liguori, providing fully solved problems that mirror exam structures. Focus: Page 77 and Key Topics
In many editions of the Analisi 2 exercises, the material around page 77 focuses on the transition from basic topology to calculus of several variables: Lezioni di analisi matematica due - Zanichelli
We use polar coordinates: ( x = r\cos\theta, y = r\sin\theta, r = \sqrtx^2+y^2 ). Utilizing Esercizi Pdf 77 If "Esercizi Pdf 77"
[ f(r\cos\theta, r\sin\theta) = \fracr^3(\cos^3\theta + \sin^3\theta)r^2 = r(\cos^3\theta + \sin^3\theta). ]
Thus ( |f(x,y)-f(0,0)| \leq r(|\cos^3\theta| + |\sin^3\theta|) \leq 2r \to 0 ) as ( r \to 0 ).
So ( f ) is continuous at the origin.
By definition:
[ f_x(0,0) = \lim_h \to 0 \fracf(h,0) - f(0,0)h = \lim_h \to 0 \frach^3/h^2h = \lim_h \to 0 \frachh = 1. ]
[ f_y(0,0) = \lim_k \to 0 \fracf(0,k) - f(0,0)k = \lim_k \to 0 \frack^3/k^2k = 1. ] y) ) for ( (x
So ( f_x(0,0) = 1, \ f_y(0,0) = 1 ).
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The search query "Fusco Marcellini Sbordone Analisi Matematica 2 Esercizi Pdf 77" refers to a very specific niche within Italian academic literature. It points to one of the most widely used university textbooks for mathematical analysis and indicates a user looking for digital solutions for a specific chapter—Chapter 7, often titled "Curve piane e integrali di linea" (Plane Curves and Line Integrals) or "Forme differenziali" (Differential Forms), depending on the edition. y) \neq (0
Below is a detailed breakdown of the text, the significance of the specific chapter referenced ("77"), and the academic context of this resource.
We compute ( f_x(x,y) ) for ( (x,y) \neq (0,0) ):
[ f_x(x,y) = \frac\partial\partial x \left( \fracx^3 + y^3x^2 + y^2 \right) = \frac3x^2(x^2+y^2) - (x^3+y^3)(2x)(x^2+y^2)^2. ]
Simplify: ( \frac3x^4 + 3x^2y^2 - 2x^4 - 2xy^3(x^2+y^2)^2 = \fracx^4 + 3x^2y^2 - 2xy^3(x^2+y^2)^2 ).
Along the line ( y = x ):
[ f_x(x,x) = \fracx^4 + 3x^4 - 2x^4(2x^2)^2 = \frac2x^44x^4 = \frac12. ]
But ( f_x(0,0) = 1 ). So ( f_x ) is not continuous at ( (0,0) ). Similarly for ( f_y ).