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Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.

Assume that the vector space we're in is $\Re^{3}$. Prove that

\begin{eqnarray*} (1) &\;\;\;\;\;\;\;\;& (\vec{\mathbf{\tau}} \cdot \vec{\mathbf{\beta}} \cdot \vec{\mathbf{\beta^{'}}}) &=& \kappa , \\ (2)&&(\vec{\mathbf{\beta}} \cdot \vec{\mathbf{\beta^{'}}} \cdot \vec{\mathbf{\beta^{''}}}) &=& \kappa^{2}(k / \kappa)^{'} ,\\ (3)&&(\vec{\mathbf{\tau}} \cdot \vec{\mathbf{\tau^{'}}} \cdot \vec{\mathbf{\tau^{''}}})&=& k^{3}(\kappa/k)^{'} , \end{eqnarray*}

where $\tau$ is the unit tangent vector, $\beta$ is the binormal vector, $\kappa$ is torsion, and $k$ is curvature. I started to attempt these proofs by starting from the vector form of the curve $\vec{r}(t) = x(t)\vec{i} +y(t)\vec{j} +z(t)\vec{k}$ and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?

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    @J.M.: You are correct (also credit to Dylan). It is the scalar triple product (mixed or box product). I am discovering that the text for my class (ISBN 0817643842) uses different notation than most other differential geometry books. Also, it has tons of typos.2012-02-13

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You may consider the curve parametrized for arc length. The formulas for curvature and torsion become

k(t)=||r''(t)||, \ \tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)

You can substitute these in the right hand of the equation and obtain the left hand side. Sources: http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html http://yourhomeworkhelp.org/math-tests/geometry-tests/