Chapter 13.1, Problem 42E

(a)

To determine

Prove that $\frac{d}{dt}\left(u\cdot v\times w\right)=\left(\frac{du}{dt}\cdot v\times w\right)+\left(u\cdot \frac{dv}{dt}\times w\right)+\left(u\cdot v\times \frac{dw}{dt}\right)$ where u, v, and w are differentiable vector functions.

(b)

To determine

Prove that the vector functions $\frac{d}{dt}\left(r\cdot \frac{dr}{dt}\times \frac{d{r}^2}{d{t}^2}\right)=r\cdot \left(\frac{dr}{dt}\times \frac{d^{3}r}{d{t}^3}\right)$.