# Curvature Calculator

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

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mathbf{vec{r}(t)}= (, , )

If you need to find the curvature of a parametric function, form the vector (x(t),y(t),0).
If you don't have the third coordinate, set it to 0.

Calculate at t=

Leave empty, if you don't need the curvature at a specific point.

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## Solution

Your input: find the curvature of $$\mathbf{\vec{r}(t)}=\left(5 \sin{\left(t \right)}, 5 \cos{\left(t \right)}, 7\right)$$$The formula for the curvature is $$\kappa(t)=\frac{\lVert\mathbf{\vec{r}^{\prime}(t)}\times\mathbf{\vec{r}^{\prime\prime}(t)}\rVert}{\left(\lVert\mathbf{\vec{r}^{\prime}(t)}\rVert\right)^3}$$$

Find the first and second derivatives.

The first derivative is $$\mathbf{\vec{r}^{\prime}(t)}=\left(5 \cos{\left(t \right)}, - 5 \sin{\left(t \right)}, 0\right)$$$The second derivative is $$\mathbf{\vec{r}^{\prime\prime}(t)}=\left(- 5 \sin{\left(t \right)}, - 5 \cos{\left(t \right)}, 0\right)$$$

Note. For steps in finding derivatives, see derivative calculator.

Now, find the norm (length) of $$\mathbf{\vec{r}^{\prime}(t)}$$$: $$\lVert\mathbf{\vec{r}^{\prime}(t)}\rVert=\sqrt{\left(5 \cos{\left(t \right)}\right)^2+\left(- 5 \sin{\left(t \right)}\right)^2+\left(0\right)^2}=5$$$

Thus, $${\left(\lVert\mathbf{\vec{r}^{\prime}(t)}\rVert\right)}^3=125$$$Next, find the cross product of the first and second derivatives: $$\mathbf{\vec{r}^{\prime}(t)}\times\mathbf{\vec{r}^{\prime\prime}(t)}=\left(0,0,-25\right)$$$ (steps can be seen here).

Now, find the norm (length) of $$\mathbf{\vec{r}^{\prime}(t)}\times\mathbf{\vec{r}^{\prime\prime}(t)}$$$: $$\lVert\mathbf{\vec{r}^{\prime}(t)}\times\mathbf{\vec{r}^{\prime\prime}(t)}\rVert=\sqrt{\left(0\right)^2+\left(0\right)^2+\left(-25\right)^2}=25$$$

Finally, the curvature is $$\kappa(t)=\frac{1}{5}$$$Answer: $$\kappa(t)=\frac{1}{5}$$$