# Curvature Calculator

## Calculate curvature step by step

The calculator will find the curvature of the given explicit, parametric, or vector-valued function at the given point, with steps shown.

Related calculators: Unit Binormal Vector Calculator, Torsion Calculator

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If you have an explicit function $y = f{\left(x \right)}$, enter it as $x$, $f{\left(x \right)}$, $0$. For example, the curvature of $y = x^{2}$ can be found here.
Leave empty if you don't need the curvature at a specific point.

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Find the curvature of $\mathbf{\vec{r}\left(t\right)} = \left\langle t, 3 t + 1, t^{2} - 5\right\rangle$.

### Solution

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

Find the magnitude of $\mathbf{\vec{r}^{\prime}\left(t\right)}$: $\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert} = \sqrt{4 t^{2} + 10}$ (for steps, see magnitude calculator).

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

Find the cross product: $\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 6, -2, 0\right\rangle$ (for steps, see cross product calculator).

Find the magnitude of $\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}$: $\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert} = 2 \sqrt{10}$ (for steps, see magnitude calculator).

Finally, the curvature is $\kappa\left(t\right) = \frac{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert}^{3}} = \frac{\sqrt{5}}{\left(2 t^{2} + 5\right)^{\frac{3}{2}}}.$

The curvature is $\kappa\left(t\right) = \frac{\sqrt{5}}{\left(2 t^{2} + 5\right)^{\frac{3}{2}}}$A.