Path Integrals in ThreeSpace
Text If c(t) = (x(t),y(t),z(t)) is a path in threespace and f(x,y,z) is function defined over c, then the path integral of f along c is given by ∫_{c} f(x,y,z) ds = ∫_{a}^{b}f(x(t),y(t),z(t))s'(t)dt. where s'(t) = √(x'(t)^{2} + y'(t)^{2} + z'(t)^{2}).
If f(x,y,z) = 1 then the path integral is simply the arc length of c.
Demos
Path Integrals in ThreeSpace
 
This demo graphs a path (x(t), y(t), z(t)) and the path integral of a function f(x, y, z) along that path.

Exercises 1. While path integrals can be carried out in twospace and threespace, they could still be considered to be part of single variable calculus. Why is this?
2. Find the arc length of the curve (cos(t), sin(t), sin(2t)), 0 ≤ t ≤ 2π.
