A function _ #~v #: [ ~t_1 , ~t_2 ] -> V_3 _ where [ ~t_1 , ~t_2 ] is a closed interval in &reals., is called a ~#{vector function}.
The function _ #~v _ is ~#{continuous} at _ ~t_0 &in. [ ~t_1 , ~t_2 ], if given &epsilon. ( &epsilon. > 0 ) then &exist. &eta. ( &eta. > 0 ) such that
| #~v(~t_0) - #~v(~t) | < &epsilon. _ whenever _ | ~t_0 - ~t | < &eta. .
The function _ #~v _ is said to #~{tend to a limit}, _ #~l, _ as ~t tends to ~t_0 ( ~t_0 &in. [ ~t_1 , ~t_2 ] ), if given &epsilon. ( &epsilon. > 0 ) then &exist. &eta. ( &eta. > 0 ) such that
| #~l - #~v(~t) | < &epsilon. _ whenever _ | ~t_0 - ~t | < &eta. .
In which case we write:
#~l _ _ #:= _ _ lim{#~v(~t)&br. _ ,~t -> ~t_0}
If _ #~v #: [ ~t_1 , ~t_2 ] -> V_3 _ for some given point ~t_0, consider the function
fract{&Delta.#~v,&Delta.~t} _ #:= _ fract{#~v(~t) - ~#v(~t_0),(~t - ~t_0)}
Now for fixed ~t_0, _ &Delta.#~v ./ &Delta.~t _ is a function of ~t. What happens as ~t tends to ~t_0? If &Delta.#~v ./ &Delta.~t tends to a limit, then we define:
fract{d#~v,d~t}(~t_0) _ #:= _ lim{,~t -> ~t_0}fract{#~v(~t) - ~#v(~t_0),(~t - ~t_0)}
This is called the #~{derivative} of #~v at ~t_0.
Note that the derivative can exist at one point but not at another. If the derivative of #~v exists at every point in the interval [ ~t_1 , ~t_2 ] then ~#v is said to be #~{differentiable} on the interval. The derivative is then a vector function on the interval. We will use the notation #~v#' to denote the derivative of #~v.
If #~u and #~v are vector functions which are differentiable on a given interval, and λ and &mu. are scalars then
It follows that
If &phi. is a differentiable (scalar) function into the domain of ~#v, then,
Note