Understanding Closed Curves & Remark on Unique Extension

[Buri]

I have a question about the Remark on the page posted. When it says "If γ and all its derivatives take the same value at a and b, there is a unique way to extend γ to a (b − a)-periodic (smooth) curve γ : R → R^n" what does this exactly mean? I suppose that the condition that the derivatives have the same value at a and b is to guarantee that there won't be any sharp edges at a and b right? So it eliminates the possibilibilty of piece-wise smooth curves? And can someone please explain to me what they when when they say "there is a unique way to extend γ to a (b - a)-periodic (smooth) curve γ: R → R^n"?

Thanks for any help you guys can give me.

 

[Buri]

Anyone?

 

[mathwonk]

there is always a unique way to extend it as a periodic function, the condition makes the extension smooth.

e.g. look att he graph of the sine function. apparently it has the same value and same (one sided) derivatives at 0 and at 2pi, so that why it extends as a continuous smooth periodic curve to the whole line.

 

[Buri]

there is always a unique way to extend it as a periodic function, the condition makes the extension smooth.

e.g. look att he graph of the sine function. apparently it has the same value and same (one sided) derivatives at 0 and at 2pi, so that why it extends as a continuous smooth periodic curve to the whole line.

Ahhh I see. Thanks a lot for your help!

 

[Goongyae]

>"If γ and all its derivatives take the same value at a and b, there is a unique way to extend γ to a (b − a)-periodic (smooth) curve γ : R → R^n"

Note that this theorem just helps establish that the periodic result is smooth.
It doesn't at all mean there's a unique way of extending a smooth curve, or that non-smooth functions can't be extended to be periodic!

Consider the function g(x)=exp(-1/x^2) truncated to be 0 for x < 0. At x = 0, the function g and all its derivatives are equal to zero. So you can always add on copies of this function to any other function without affecting the result's smoothness.