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

Answer

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