Suppose f(x, y, z) is a smooth curve in space defined by the
parameter t as follows
x = x(t), y = y(t), z = z(t)
Now by the Pythagorean
theorem ds = √[(dx)2
+ (dy)2 + (dz)2]
where ds represents the differential (arc) length
along the curve , s , in space.
b b
So ∫ f(x, y, z) ds = ∫ f(x, y, z) (ds/dt) dt =
line integral of f(x,y,z) along
a a the
curve, s
b
∫ f(x, y, z)
√[(dx/dt)2
+ (dy/dt)2 +
(dz/dt)2]
dt
a
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