This is a sample question from a multivariate calculus class. Any insight would be appreciated.
Suppose the curve $\mathbf{r} = \mathbf{r}(s)$ is parametrized by a natural parameter and lies on the unit sphere centered at the origin.
Show that its curvature satisfies $$\kappa = \sqrt{ 1 + \left(\mathbf{r}'' \cdot \left(\mathbf{r} \times \mathbf{r}' \right) \right)^2}.$$
Below is what I'm familiar with and what I've tried to use, but I can't seem to connect the ideas together.
The unit sphere at the origin can be represented as $x^2 + y^2 + z^2 = 1$. If $\mathbf{r} = \langle\ x(s),\ y(s),\ z(s) \ \rangle$ lies on the sphere, then $\mathbf{r}$ will intersect the sphere at any point such that $[x(s)]^2 + [y(s)]^2 + [z(s)]^2 = 1.$
From this I gather that $\| {\mathbf{r}} \| = 1.$ Since the norm of $\mathbf{r}$ is constant, then $\mathbf{r} \cdot \mathbf{r}' = 0$. Therefore $\mathbf{r}$ and $\mathbf{r}'$ are orthogonal to one another. But we know that $\mathbf{r}'$ is tangent to our curve, and $\mathbf{r}''$ would be normal to our curve. I suppose we could use unit vectors and then the Frenet-Serret equations may come into play, but I don't see it.
I'm familiar with the various curvature formulas, and I'd like to believe $\kappa = \| \mathbf{r}''(s) \|$ will be the one that works for us.
Thank you very much!
Best Answer
I'm going to use the "dot" notation for derivatives with respect to $s$, so that $\dot{\mathbf r}(s) = \mathbf r'(s)$, etc. Then, assuming that the "natural parameter" $s$ is the arc-length along $\mathbf r(s)$, we have:
Since $\mathbf r(s)$ is contained in the unit sphere,
$\mathbf r(s) \cdot \mathbf r(s) = 1, \tag{1}$
from which it readily follows upon differentiation with respect to the arc-length parameter $s$, that
$\mathbf r(s) \cdot \dot{\mathbf r}(s) = 0; \tag{2}$
furthermore we see that
$\dot{\mathbf r}(s) \cdot \dot{\mathbf r}(s) = 1 \tag{3}$
by virtue of the fact that $s$ is the arc-length along $\mathbf r(s)$; $\dot {\mathbf r}(s) = \mathbf T(s)$ is the unit tangent to $\mathbf r(s)$, and a member of the Frenet-Serret apparatus (or frame) of $\mathbf r(s)$. It then follows from (1)-(3) and the elementary properties of the vector cross product that $\mathbf r(s) \times \dot{\mathbf r}(s)$ is itself a unit vector and is orthogonal to both the unit vectors $\mathbf r(s)$ and $\dot{\mathbf r}(s)$, i.e.,
$(\mathbf r(s) \times \dot{\mathbf r}(s)) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s)) = 1, \tag{4}$
$(\mathbf r(s) \times \dot{\mathbf r}(s)) \cdot \mathbf r(s) = (\mathbf r(s) \times \dot{\mathbf r}(s)) \cdot \dot{\mathbf r}(s) = 0. \tag{5}$
(1)-(5) show that $\mathbf r(s)$, $\dot{\mathbf r}(s)$, and $\mathbf r(s) \times \dot{\mathbf r}(s)$ themselves form an orthonormal frame at each point of $\mathbf r(s)$; we expand the Frenet-Serret normal
$\mathbf N(s) = (1 / \kappa)\dot{\mathbf T}(s) = (1 / \kappa) \ddot {\mathbf r}(s) \tag{6}$
in terms of this frame, starting by noting that differentiation of (2) yields
$\dot{\mathbf r}(s) \cdot \dot{\mathbf r}(s) + \mathbf r(s) \cdot \ddot {\mathbf r}(s) = 0, \tag{7}$
which by virtue of (3) and (6) reads
$1 + \kappa \mathbf N(s) \cdot \mathbf r(s) = 0. \tag{8}$
(7) also has the advantage of showing that $\ddot {\mathbf r}(s) \ne 0$, so that $\kappa = \Vert \ddot {\mathbf r}(s) \Vert \ne 0$ and $\mathbf N(s)$ is well-defined at all points of $\mathbf r(s)$. (8) shows that the component of $\mathbf N(s)$ along the unit vector $\mathbf r(s)$ is in fact
$\mathbf N(s) \cdot \mathbf r(s) = -\dfrac{1}{\kappa}. \tag{9}$
The $\dot{\mathbf r}(s)$ component of $\mathbf N(s)$ vanishes for all $s$, since
$\mathbf N(s) \cdot \dot{\mathbf r}(s) = \mathbf N(s) \cdot \mathbf T(s) = 0, \tag{10}$
and finally the component of $\mathbf N(s)$ along $\mathbf r(s) \times \dot{\mathbf r}(s)$ is
$\mathbf N(s) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s)) = \dfrac{1}{\kappa}\ddot{\mathbf r}(s) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s)), \tag{11}$
so that $\mathbf N(s)$ may in fact be written
$\mathbf N(s) = -\dfrac{1}{\kappa} \mathbf r(s) + \dfrac{1}{\kappa}\ddot{\mathbf r}(s) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s))(\mathbf r(s) \times \dot{\mathbf r}(s)), \tag{12}$
and using (12) to compute $\mathbf N(s) \cdot \mathbf N(s) = 1$ yields
$\dfrac{1}{\kappa^2}(1 + (\ddot{\mathbf r}(s) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s)))^2 = 1, \tag{13}$
which is easily re-arranged:
$\kappa = \sqrt{1 + (\ddot{\mathbf r}(s) \cdot (\mathbf r(s) \times \dot{\mathbf r}(s)))^2}, \tag{14}$
the requisite result. QED.
Hope this helps. Cheerio,
and as always,
Fiat Lux!!!