Fusco Marcellini Sbordone Analisi Matematica 2 Esercizi Pdf 77 Today

While the search for a "Pdf" version is common, it is important to note the legal context.

Let ( f: \mathbbR^2 \to \mathbbR ) be defined as:

[ f(x,y) = \begincases \dfracx^3 + y^3x^2 + y^2, & (x,y) \neq (0,0) \ 0, & (x,y) = (0,0) \endcases ] While the search for a "Pdf" version is

We use polar coordinates: ( x = r\cos\theta, y = r\sin\theta, r = \sqrtx^2+y^2 ).

[ f(r\cos\theta, r\sin\theta) = \fracr^3(\cos^3\theta + \sin^3\theta)r^2 = r(\cos^3\theta + \sin^3\theta). ] We use polar coordinates: ( x = r\cos\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.

For students or individuals studying from "Analisi Matematica 2" by Fusco, Marcellini, and Sbordone, exercises are crucial for understanding and mastering the concepts. Here are some steps you can take: y = r\sin\theta

This exercise highlights a classic subtlety: existence of partial derivatives + continuity ≠ differentiability. The failure of differentiability is due to the term ( -hk(h+k)/(h^2+k^2)^3/2 ) with angular dependence. The directional derivative formula ( D_v f = v \cdot \nabla f ) fails because the derivative is not a linear map — a key warning for students moving from single to multivariable calculus.

Such exercises on page 77 would serve as a bridge to later chapters on differentiability theorems, C¹ implies differentiability (here ( f \notin C^1 )), and Schwarz theorem (here mixed partials at origin: compute ( f_xy(0,0) ) would show symmetry? Actually ( f_xy ) and ( f_yx ) differ — another typical advanced exercise).