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Question:(a) Define the curvature function $\kappa$ of a plane curve.

The curvature of $\kappa$ of a plane curve is the amount of turning in the osculating plane. In other words it decribes the speed of rotation.

Is how I defined it okay? I feel like I need more or something.

Question:(b) Determine the curvature function of the circle $\alpha(t)=(r\cos t,r\sin t)$. To find our curvature we must find the derivative of $\alpha$;

$\alpha'(t)=(-r\sin t, r\cos t)$.

Now to find our constant we do the following:

$|\alpha'(t)|= \sqrt{r^2\sin^2 t+r^2\cos^2 t}= \sqrt{r^2(\cos^2 t+\sin^2 t)}= \sqrt{r^2}= r.$

Thus, we get the arc-length as,

$s= \int_{0}^{t}r dt= rt \bigg |^t_0= rt.$

So, we can write $t= \dfrac{s}{r}$. Using the value of $t$ we substitute it into $\beta$ such that it is a curve parametrized by arc length,

$\beta(s)=(r\cos\dfrac{s}{r}, r\sin\dfrac{s}{r}).$

Using this we find the derivative to be,

$\beta'(s)=(-\sin\dfrac{s}{r}, \cos\dfrac{s}{r}).$

Looking for our constant we get,

$|\beta'(s)|=\sqrt{(-\sin\dfrac{s}{r})^2+\cos^2\dfrac{s}{r}}= \sqrt{1}=1 ,\ \ \ \forall s.$

Finding our second derivative we get,

$\beta"(s)=(-\dfrac{1}{r}\cos\dfrac{s}{r}, -\dfrac{1}{r}\sin\dfrac{s}{r}).$

$|\beta"(s)|=\sqrt{(-\dfrac{1}{r}\cos\dfrac{s}{r})^2+ (-\dfrac{1}{r}\sin\dfrac{s}{r})^2}= \sqrt{\dfrac{1}{r^2}}= \dfrac{1}{r}$

Therefore, we see that the curvature $\kappa$ of the circle is a constant,$\dfrac{1}{r}$.

Is my work okay? I just want to make sure.

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For the most part this is fine, although I have a few items of feedback:

  1. Curvature is often defined as a positive quantity, as you have here. For several reasons (uniqueness of a curves up to rigid motions given curvature; extensibility to higher dimensions, etc) the signed curvatures is often more useful. You should check which of these your professor is expecting for this problem (if the curvature is signed, then the curvature of a circle will depend on both its radius and orientation.)

  2. Your definition is a bit off: mathematical definitions must be very precise; what precisely do you mean by "amount of turning"?

  3. Several times you say you are "finding a constant" when you derive the arc length of a curve. I don't understand what you mean by that -- arc length is not constant, nor, generally, is the norm of the derivative of a curve constant (that it is a constant for the particular parameterization of the circle provided by your professor is a happy accident). Otherwise your work in reparameterizing the curve is correct, though.

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