Arc length with vector functions - three dimensional space, Mathematics

Arc Length with Vector Functions

In this part we will recast an old formula into terms of vector functions.  We wish to find out the length of a vector function,

r (t) = {f (t), g(t) , h (t)}

on the interval a ≤ t ≤ b .

in fact we already know how to do this.  Remind that we can write the vector function into the parametric form,

 x = f (t)

 y = g(t)

z = h (t)

As well, remind that with two dimensional parametric curves the arc length is illustrated by,

L = ∫ba √ [f' (t)]2 + [g' (t)]2 dt

Here is a natural extension of this to three dimensions. Thus, the length of the curve r ?t ? on the interval a ≤ t ≤ b is,

L = ∫ba √ [f' (t)]2 + [g' (t)]2 + [h' (t)] dt

There is a good simplification which we can make for this.

Note: The integrand that is the function we're integrating is nothing much more than the magnitude of the tangent vector,

1226_Arc Length with Vector Functions - Three Dimensional Space.png

 Hence, the arc length can be written as,

L = ∫ba || r' (t)|| dt

Posted Date: 4/13/2013 1:31:33 AM | Location : United States







Related Discussions:- Arc length with vector functions - three dimensional space, Assignment Help, Ask Question on Arc length with vector functions - three dimensional space, Get Answer, Expert's Help, Arc length with vector functions - three dimensional space Discussions

Write discussion on Arc length with vector functions - three dimensional space
Your posts are moderated
Related Questions
the value of y for which x=-1.5

Mimi is filling a tennis ball can along with water. She wants to know the volume of the cylinder shaped can. Which formula will she use? The volume of a cylinder is π times the

Erin is painting a bathroom along with four walls each measuring 8 ft through 5.5 ft. Ignoring the doors or windows, what is the area to be painted? The area of the room is the

how to find the indicated term?

Can anybody provide me the solution of the following example? You are specified the universal set as T = {1, 2, 3, 4, 5, 6, 7, 8} And the given subjects of the universal s

Suppose a fluid (say, water) occupies a domain D? R^(3 ) and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and will be transported by the fluid

Let a 0 , a 1 ::: be the series recursively defined by a 0 = 1, and an = 3 + a n-1 for n ≥ 1. (a) Compute a 1 , a 2 , a 3 and a 4 . (b) Compute a formula for an, n ≥ 0.

Product Rule: (f g)′ = f ′ g + f g′ As with above the Power Rule, so the Product Rule can be proved either through using the definition of the derivative or this can be proved

∫1/sin2x dx = ∫cosec2x dx = 1/2 log[cosec2x - cot2x] + c = 1/2 log[tan x] + c Detailed derivation of ∫cosec x dx = ∫cosec x(cosec x - cot x)/(cosec x - cot x) dx = ∫(cosec 2 x

advanteges of duality