For an oriented planar curve, the <Frenet-Serret formulas> are
$$
\partial_s
\begin{pmatrix}\mathbf t\\\mathbf n\end{pmatrix}
=
\begin{pmatrix}0&k\\-k&0\end{pmatrix}
\begin{pmatrix}\mathbf t\\\mathbf n\end{pmatrix}.
$$
The frame is orthonormal, so differentiating each scalar product gives
$$
\partial_s(\mathbf e_i\mathbin{\cdot}\mathbf e_j)=0.
$$
In matrix form this says that the connection matrix plus its transpose is zero; it must be antisymmetric.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/3/a/ii/solution