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

Curvature of r⃗(x)=⟨x,x2,0⟩\mathbf{\vec{r}\left(x\right)} = \left\langle x, x^{2}, 0\right\rangle

The calculator will find the curvature of r⃗(x)=⟨x,x2,0⟩\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

⟨\langle
,
,
⟩\rangle
If you have an explicit function y=f(x)y = f{\left(x \right)}, enter it as xx, f(x)f{\left(x \right)}, 00. For example, the curvature of y=x2y = 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 contact us.

Your Input

Find the curvature of r⃗(x)=⟨x,x2,0⟩\mathbf{\vec{r}\left(x\right)} = \left\langle x, x^{2}, 0\right\rangle.

Solution

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

Find the magnitude of r⃗′(x)\mathbf{\vec{r}^{\prime}\left(x\right)}: ∣r⃗′(x)∣=4x2+1\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 r⃗′(x)\mathbf{\vec{r}^{\prime}\left(x\right)}: r⃗′′(x)=⟨0,2,0⟩\mathbf{\vec{r}^{\prime\prime}\left(x\right)} = \left\langle 0, 2, 0\right\rangle (for steps, see derivative calculator).

Find the cross product: r⃗′(x)×r⃗′′(x)=⟨0,0,2⟩\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 r⃗′(x)×r⃗′′(x)\mathbf{\vec{r}^{\prime}\left(x\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(x\right)}: ∣r⃗′(x)×r⃗′′(x)∣=2\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 κ(x)=∣r⃗′(x)×r⃗′′(x)∣∣r⃗′(x)∣3=2(4x2+1)32.\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 κ(x)=2(4x2+1)32\kappa\left(x\right) = \frac{2}{\left(4 x^{2} + 1\right)^{\frac{3}{2}}}A.