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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

For $F=(y,z,x)$,
$$
T_{ij}=\frac{\partial F_i}{\partial x_j}
=
\begin{pmatrix}
0&1&0\\
0&0&1\\
1&0&0
\end{pmatrix}.
$$
Its antisymmetric part is
$$
\boxed{
\frac12(T-T^T)=
\begin{pmatrix}
0&1/2&-1/2\\
-1/2&0&1/2\\
1/2&-1/2&0
\end{pmatrix}}.
$$
Also,
$$
B=\nabla\times F=(-1,-1,-1).
$$
For example, $-(1/2)\epsilon_{12k}B_k=1/2$, and the same componentwise calculation gives
$$
\boxed{
\frac12(T_{ij}-T_{ji})
=-\frac12\epsilon_{ijk}B_k}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / b / Solution

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