Question

Find the curvature of each of the vector-valued functions defined in Exercises 39–44.

$\mathit{r}\left(t\right)=\left(\sin t,\cos t,\cosh t\right)$

Short answer

The curvature of the given vector-valued function defined by the point is$\kappa =\frac{2}{{\cos }^{3}ht}.$

Step-by-step solution

Step 1. Given Information. 

The given vector-valued function is$\mathit{r}\left(t\right)=\left(\sin t,\cos t,\cosh t\right).$

Step 2. Find the curvature. 

To find the curvature of the given vector-valued function, we will use the formula for the Curvature of a Space Curve$\kappa =\frac{||r\text{'}\left(t\right)\times r\text{'}\text{'}\left(t\right)||}{{||r\text{'}\left(t\right)||}^3}..$

So,

Step 3. Calculate.  

Put the values of (a) and (b) in the formula of Curvature of a space curve,

$$\begin{gathered} \kappa =\frac{2}{{\left(\cos ht\right)}^3} \\ \kappa =\frac{2}{{\cos }^{3}ht} \end{gathered}$$