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Arc Length Function of a curve, 's', is defined by:

s(t) = [inte]|r'(u)|du =

[inte][squ]((dx/du)^2 + (dy/du)^2 + (dz/du)^2)du

integrate both sides and you get ds/dt = |r'(t)|.

Arc length is independent of the parameterization that's used, is that why there appears to be no rhyme or reason to the interchangeability of 'u' and 't' in the equations above?

Also along the same lines, curvature of a curve is defined as:

k = |dT/ds| where T is the unit tangent vector.

The curvature is easier to compute if it is expressed in terms of the parameter 't' instead of 's', so using the Chain Rule the book gets:

dT/ds = (dT/ds)(ds/dt)??????

I guess I'm a little confused by the parameterization or the chain rule because I'm not seeing how this came about.

Can someone please explain?

Thanks much.