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Torsion Calculator

Calculate torsion step by step

The calculator will find the torsion of the given vector-valued function at the given point, with steps shown.

Related calculator: Curvature Calculator

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Your Input

Find the torsion of r⃗(t)=⟨t2,t3,t⟩\mathbf{\vec{r}\left(t\right)} = \left\langle t^{2}, t^{3}, t\right\rangle.

Solution

Find the derivative of r⃗(t)\mathbf{\vec{r}\left(t\right)}: r⃗′(t)=⟨2t,3t2,1⟩\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle 2 t, 3 t^{2}, 1\right\rangle (for steps, see derivative calculator).

Find the derivative of r⃗′(t)\mathbf{\vec{r}^{\prime}\left(t\right)}: r⃗′′(t)=⟨2,6t,0⟩\mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 2, 6 t, 0\right\rangle (for steps, see derivative calculator).

Find the cross product: r⃗′(t)×r⃗′′(t)=⟨−6t,2,6t2⟩\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle - 6 t, 2, 6 t^{2}\right\rangle (for steps, see cross product calculator).

Find the magnitude of r⃗′(t)×r⃗′′(t)\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}: ∣r⃗′(t)×r⃗′′(t)∣=29t4+9t2+1\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert} = 2 \sqrt{9 t^{4} + 9 t^{2} + 1} (for steps, see magnitude calculator).

Find the derivative of r⃗′′(t)\mathbf{\vec{r}^{\prime\prime}\left(t\right)}: r⃗′′′(t)=⟨0,6,0⟩\mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)} = \left\langle 0, 6, 0\right\rangle (for steps, see derivative calculator).

Find the dot product: (r⃗′(t)×r⃗′′(t))⋅r⃗′′′(t)=12\left(\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right)\cdot \mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)} = 12 (for steps, see dot product calculator).

Finally, the torsion is τ(t)=(r⃗′(t)×r⃗′′(t))⋅r⃗′′′(t)∣r⃗′(t)×r⃗′′(t)∣2=39t4+9t2+1.\tau\left(t\right) = \frac{\left(\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right)\cdot \mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert}^{2}} = \frac{3}{9 t^{4} + 9 t^{2} + 1}.

Answer

The torsion is τ(t)=39t4+9t2+1\tau\left(t\right) = \frac{3}{9 t^{4} + 9 t^{2} + 1}A.