# Curvature of $\mathbf{\vec{r}\left(x\right)} = \left\langle x, x^{2}, 0\right\rangle$

The calculator will find the curvature of $\mathbf{\vec{r}\left(x\right)} = \left\langle x, x^{2}, 0\right\rangle$, 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.
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Find the curvature of $\mathbf{\vec{r}\left(x\right)} = \left\langle x, x^{2}, 0\right\rangle$.

### Solution

Find the derivative of $\mathbf{\vec{r}\left(x\right)}$: $\mathbf{\vec{r}^{\prime}\left(x\right)} = \left\langle 1, 2 x, 0\right\rangle$ (for steps, see derivative calculator).

Find the magnitude of $\mathbf{\vec{r}^{\prime}\left(x\right)}$: $\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(x\right)}\right\rvert} = \sqrt{4 x^{2} + 1}$ (for steps, see magnitude calculator).

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

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

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

Finally, the curvature is $\kappa\left(x\right) = \frac{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(x\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(x\right)}\right\rvert}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(x\right)}\right\rvert}^{3}} = \frac{2}{\left(4 x^{2} + 1\right)^{\frac{3}{2}}}.$

The curvature is $\kappa\left(x\right) = \frac{2}{\left(4 x^{2} + 1\right)^{\frac{3}{2}}}$A.