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Chapter 10: Problem 21

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

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

Step by step solution

01

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: \[\mathbf{r}_s = \frac{\sqrt{2}}{2} \mathbf{i} - \frac{\sqrt{2}}{2} \mathbf{j}\]
02

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: \[\mathbf{r}_{ss} = 0 \mathbf{i} - 0 \mathbf{j}\]
03

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.

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