Proving Differentiability of f in R$^2$

  • #1
Define [tex]f(0,0) = 0[/tex] and

[tex]f(x,y) = \frac {x^3}{x^2 + y^2}[/tex] if [tex](x,y) \neq (0,0)[/tex]

a) Prove that the partial derivatives of f are bounded functions in R^2.

b) Let [itex]\mathbf{u}[/itex] be any unit vector in R^2. Show that the directional derivative [tex](D_{\mathbf{u}} f)(0,0)[/tex] exists, and its absolute value is at most 1.

c)Let \gamma be a differentiable mapping of R^1 into R^2, with \gamma(0) = (0,0) and [itex]|\gamma'(0)| > 0[/itex]. Put [itex]g(t) = f(\gamma(t))[/itex] and prove that [itex]g[/itex] is differentiable for every t \in R^1.

I can do parts a) and b). I need help with part c) at t = 0. I am not sure if I need parts a) and b) for part c).

Answers and Replies

  • #2
Start off with [tex]\gamma (t) = \gamma_1 (t)\mathbf{e_1}+\gamma_2 (t)\mathbf{e_2}[/tex] and substitute this into f(x,y) and take a derivative (deal with t=0 separately). You do not need parts a and b for part c.

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