Question

Find the curvature \(K\) of the curve, where \(s\) is the arc length parameter. $$ \mathbf{r}(s)=\left(1+\frac{\sqrt{2}}{2} s\right) \mathbf{i}+\left(1-\frac{\sqrt{2}}{2} s\right) \mathbf{j} $$

Short answer

The curvature \( K \) of the curve \( \mathbf{r}(s) \) as given is 0.

Step-by-step solution

Compute the First Derivative

The first task is to compute the first derivative of the curve function with respect to \(s\). We differentiate the following equations component-wise: \[\mathbf{r}_s = \frac{d}{ds}\mathbf{r}(s)\]Doing so we get: \[\mathbf{r}_s = \frac{\sqrt{2}}{2} \mathbf{i} - \frac{\sqrt{2}}{2} \mathbf{j}\]

Compute the Second Derivative

Next, differentiate the first derivative \( \mathbf{r}_s \) with respect to \( s \) once more to obtain the second derivative \( \mathbf{r}_{ss} \): \[\mathbf{r}_{ss} = \frac{d}{ds}\mathbf{r}_s\]This operation results in: \[\mathbf{r}_{ss} = 0 \mathbf{i} - 0 \mathbf{j}\]

Obtain the Curvature

The curvature of a curve is obtained by simply finding the magnitude of second derivative resulting from step 2. Using the formula: \[K=||\mathbf{r}_{ss}|| = 0 \]Thus the curvature of the given curve is 0.