- #1
ehrenfest
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Define f(0,0) = 0 and
f(x,y) = \frac {x^3}{x^2 + y^2} if (x,y) \neq (0,0)
a) Prove that the partial derivatives of f are bounded functions in R^2.
b) Let \mathbf{u} be any unit vector in R^2. Show that the directional derivative (D_{\mathbf{u}} f)(0,0) 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 |\gamma'(0)| > 0. Put g(t) = f(\gamma(t)) and prove that g 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).
f(x,y) = \frac {x^3}{x^2 + y^2} if (x,y) \neq (0,0)
a) Prove that the partial derivatives of f are bounded functions in R^2.
b) Let \mathbf{u} be any unit vector in R^2. Show that the directional derivative (D_{\mathbf{u}} f)(0,0) 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 |\gamma'(0)| > 0. Put g(t) = f(\gamma(t)) and prove that g 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).