Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 3 / a / ii

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- Solution ii

Solution

For an oriented planar curve, the Frenet-Serret formulas are

\partial_{s} (t n) = (0 - k k 0) (t n) .

The frame is orthonormal, so differentiating each scalar product gives

\partial_{s} (e_{i} \cdot e_{j}) = 0.

In matrix form this says that the connection matrix plus its transpose is zero; it must be antisymmetric.

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