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Curvature of a curve in space

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The curvature of a curve in space r(t) is given by
k(t) = | r'(t) à? r''(t) | / | r'(t) |^3 .

Consider now the curve

r(u) = r(sigma(u)),

given by the reparametrization t = sigma(u) of the initial curve. Show that the curvature k of the curve r is given by

k(u) = k(sigma(u)),
where k is the curvature of the initial curve r.

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Solution Summary

The solution determines the curvature of a curve in space.

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Instead of sigma I will just write t=t(u) for simplicity of notation.

Note the following:

(a)
dr/du = dr/dt * dt/du, and notice that ...

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