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# Use the Frenct-Serret formulas to prove cach of the fol-lowing. (Primes denote derivatives with respect to $t$ . Startas in the proof of Theorem \$10 . )$$\begin{array}{l}{\text { (a) } \mathbf{r}^{\prime \prime}=s^{\prime \prime} \mathbf{T}+\kappa\left(s^{\prime}\right)^{2} \mathbf{N}} \\ {\text { (b) } \mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}=\kappa\left(s^{\prime}\right)^{3} \mathbf{B}} \\ {\text { (c) } \mathbf{r}^{\prime \prime \prime}=\left[s^{\prime \prime \prime}-\kappa^{2}\left(s^{\prime}\right)^{3}\right] \mathbf{T}+\left[3 \kappa s^{\prime} s^{\prime \prime}+\kappa^{\prime}\left(s^{\prime}\right)^{2}\right] \mathbf{N}} \\ {+\kappa \tau\left(s^{\prime}\right)^{3} \mathbf{B}}\end{array}$$$$(d) \tau=\frac{\left(\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right) \cdot \mathbf{r}^{\prime \prime \prime}}{\left|\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right|^{2}}$$

## $$\tau=\frac{\left[\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right] \vec{r}^{\prime \prime \prime}}{\left|\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right|^{2}}$$

Vectors

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Campbell University

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Vectors

Vector Functions

##### Lily A.

Johns Hopkins University

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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Join Bootcamp