Expand/Contract All

Vector Differentiation

 
 

Vector Function

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}

Derivative Of A Vector Function

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.

Differentiation Results

If #~u and #~v are vector functions which are differentiable on a given interval, and λ and &mu. are scalars then

  • (λ #~u + &mu. ~#v)#' _ = _ λ #~u#' + &mu. #~v#'
  • (#~u &dot. ~#v)#' _ = _ #~u &dot. #~v#' + #~u#' &dot. #~v
  • (#~u # ~#v)#' _ = _ #~u # #~v#' + #~u#' # #~v

It follows that

  • | #~u |^2 _ = _ (#~u &dot. ~#u)#' _ = _ #~u &dot. #~u#' + #~u#' &dot. #~u _ = _ 2 ( #~u &dot. #~u#' )

If &phi. is a differentiable (scalar) function into the domain of ~#v, then,

  • (#~v(&phi.))#' _ = _ &phi.#' #~v#'(&phi.)

Note

  • #0 _ = _ (( #~v # #~u ) &dot. #~v )#' _ = _ (( #~v # #~u ) &dot. #~v#' ) + (( #~v # #~u )#' &dot. #~v )