How twisty is it?

Calculus Level 3

Evaluate the curvature of the point (0,0,0)(0,0,0) on a twisted cubic curve r(t)=t i+t2 j+t3 k\mathbf{r}(t) = t\ \mathbf{i} + t^2\ \mathbf{j} + t^3\ \mathbf{k}.

Details and assumptions

The curvature of a curve given by a vector function r can be found as follows: κ(t)=r(1)(t)×r(2)(t)r(1)(t)3\kappa(t) = \frac{|\mathbf{r}^{(1)}(t)\times\mathbf{r}^{(2)}(t)|}{|\mathbf{r}^{(1)}(t)|^3}

where r(1)(t)\mathbf{r}^{(1)}(t) denotes the first derivative of r(t)\mathbf{r}(t), r(2)(t)\mathbf{r}^{(2)}(t) denotes the second derivative of r(t)\mathbf{r}(t) and a×b\mathbf{a} \times \mathbf{b} denotes the cross product of vectors a\mathbf{a} and b\mathbf{b}.


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