- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.7 Exercises
- Suppose a curve $C$ is described by two equivalent functions $X$ and $Y,$ where $Y(t) = X[u(t)].$ Prove that at each point of $C$ the velocity vectors associated with $X$ and $Y$ are parallel, but that the corresponding acceleration vectors need not be parallel.
- Assuming that $u'(t)$ is never zero on the interval where $C$ is defined, and if $X'[u(t)]$ is also nonzero on the same interval, then $Y'(t)$ is parallel to $X'[u(t)]$ with $$ Y'(t) = X'[u(t)]u'(t) $$ But, if we integrate again, we get $$ Y''(t) = X''[u(t)](u'(t))^2 + X'[u(t)]u''(t) $$ Which implies that $Y''$ is parallel to $X''$ if and only if $u''(t) = 0$ for all $t$ in $I. \quad \blacksquare$