1. Home
  2. Calculators
  3. Calculators: Calculus III
  4. Calculus Calculator

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

$$$\langle$$$
,
,
$$$\rangle$$$
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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

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

Answer

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