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=

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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: $\kappa(t)=\frac{1}{5}$