This guide focuses on the standard exercises and theoretical framework provided by Nicola Fusco Paolo Marcellini Carlo Sbordone in their highly regarded series on Mathematical Analysis 2
While "77" is often associated with specific PDF page references or indexed exercise numbers in digital versions of their textbooks, this guide outlines the core content and best resources for mastering the material. Core Textbooks & Resources
The authors have published several versions of their Analysis 2 texts, ranging from comprehensive volumes to simplified editions for modern degree courses. Analisi Matematica 2 (Liguori Editore)
: A rigorous and comprehensive 704-page textbook covering the complete theoretical curriculum of Analysis 2. Elementi di Analisi Matematica 2 (Liguori Editore)
: A simplified version focusing on functions of two variables (instead of
variables) and basic differential equations to aid students in modern degree programs. Esercitazioni di Matematica (Volume 2, Parts 1 & 2)
: These are the primary exercise manuals. They are designed to be "self-sufficient," including theory summaries followed by a wide range of solved exercises.
: Focuses on sequences/series of functions and metric spaces.
: Covers differential equations and multi-variable calculus. Key Topics in Analysis 2 Exercises
Exercises in these manuals typically follow a progression from basic application to complex exam-style problems: Lezioni di analisi matematica due - Zanichelli This guide focuses on the standard exercises and
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The authors are known for their rigor and for bridging theory with subtle counterexamples. Page 77 likely falls within the chapters on limits and continuity in multiple variables, or differential calculus for vector functions, possibly the initial part of implicit functions or double integrals.
Below is a representative exercise — thematically plausible for that page — followed by a full analytical solution.
The inclusion of "Pdf" in the query highlights the modern student's reliance on digital formats. The physical textbook is a staple in Italian university bookstores (often published by Zanichelli or Editori Riuniti), but the PDF format is sought after for several reasons:
The Esercizi book is distinct because it does not just provide answers; it provides methodology.
Let ( v = (\cos\theta, \sin\theta) ). Then:
[ D_v f(0,0) = \lim_t \to 0 \fracf(t\cos\theta, t\sin\theta) - f(0,0)t = \lim_t \to 0 \fract^3(\cos^3\theta + \sin^3\theta)/t^2t = \lim_t \to 0 \fract(\cos^3\theta + \sin^3\theta)t = \cos^3\theta + \sin^3\theta. ]
Notice: For ( \theta=0 ), we get ( 1 ) (matches ( f_x )), for ( \theta=\pi/2 ) we get ( 1 ) (matches ( f_y )), but generally ( D_v f(0,0) ) equals ( v \cdot \nabla f(0,0) ) only if ( \cos^3\theta+\sin^3\theta = \cos\theta+\sin\theta ), which is false for most ( \theta ) (e.g., ( \theta=45^\circ ): LHS ( \sqrt2/2 ), RHS ( \sqrt2 )).
This confirms non-differentiability (directional derivative is not linear in ( v )).
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. ] 2. Partial derivatives at (0
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 ).
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. ]
So ( f_x(0,0) = 1, \ f_y(0,0) = 1 ).