Continuity of (x^2 + y^2)sin(1/(x^2+y^2)) Derivative at 0

[brydustin]

obviously x^2sin(1/x) isn't continuous in its derivative at zero. But it seems if you take
(x^2 + y^2) sin(1/(x^2+y^2)) and switch to polar coordinates then the derivative is continuous at zero... but that can't be right... can it?

 

[Office_Shredder]

The second function doesn't restrict to the first function on the x-axis, so I'm not sure why you would be surprised their differentiability properties would be different. On the other hand the function

$$(x^2+y^2) \sin( \frac{1}{\sqrt{x^2+y^2}} )$$
is the radial extension of your original function to the x-y plane, and when you switch to polar coordinates you get that it is not differentiable at zero (since it's the exact same function once you do that)

 

[Whovian]

Please be more specific. What are your calculations, for instance?

Also, the derivative in polar coordinates aren't that simple. If we have r=f(θ), then f'(θ) isn't necessarily the slope of the line tangent to f at θ.

 

[brydustin]

The second function doesn't restrict to the first function on the x-axis, so I'm not sure why you would be surprised their differentiability properties would be different. On the other hand the function

$$(x^2+y^2) \sin( \frac{1}{\sqrt{x^2+y^2}} )$$
is the radial extension of your original function to the x-y plane, and when you switch to polar coordinates you get that it is not differentiable at zero (since it's the exact same function once you do that)

g'(r,alpha) = (for all alpha) = z2r*sin(1/r*sqrt(z)) - r^2/sqrt(r*z) cos(1/r*sqrt(z)) is what I get. where z = cos^2(alpha) + sin^2(alpha). And the limit should exist as r goes to zero by the squeeze theorem., ... right?

 

[brydustin]

Please be more specific. What are your calculations, for instance?

Also, the derivative in polar coordinates aren't that simple. If we have r=f(θ), then f'(θ) isn't necessarily the slope of the line tangent to f at θ.

OH, no... I made a mistake. I made a bad substitution as I forgot that r = sqrt(x^2+y^2) and then its really trivial. Funny, that I had this idea but did the transformation wrong, still.