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Path Integrals in Three-Space
Text If c(t) = (x(t),y(t),z(t)) is a path in three-space 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 = ∫abf(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 Three-Space
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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.
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Exercises 1. While path integrals can be carried out in two-space and three-space, 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.
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