This exercise highlights a classic subtlety: . 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.
( “This exercise was mistakenly proposed as trivial in 1987. Three students solved it correctly. The fourth became a professor.” ) This exercise highlights a classic subtlety:
This topic is notoriously difficult for students because it requires a synthesis of previous knowledge (parametric curves, partial derivatives, and integration). Students searching specifically for this chapter are often preparing for a challenging exam section or looking for specific solved exercises regarding Green's Theorem or the independence of path. ( “This exercise was mistakenly proposed as trivial
: In alternative "Elementi" versions, page 77 may fall within Chapter 3, which introduces Ordinary Differential Equations (ODEs) and the Cauchy problem . Structure of the "Esercitazioni" (Exercise Books) Students searching specifically for this chapter are often
: Exploring the behavior of gradients and directional derivatives.