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Mathematical methods I course of the University of Oxford

Past exam of the mathematical methods I course of the University of Oxford

2021

Contract the <Levi-Civita symbol> identity $\varepsilon_{ijk}\varepsilon_{i\ell m}=\delta_{j\ell}\delta_{km}-\delta_{jm}\delta_{k\ell}$. Using <Einstein summation convention>,
$$
\begin{aligned}
|\mathbf a\times\mathbf b|^2
&=\varepsilon_{ijk}a_jb_k\varepsilon_{i\ell m}a_\ell b_m\\
&=(\delta_{j\ell}\delta_{km}-\delta_{jm}\delta_{k\ell})a_jb_ka_\ell b_m\\
&=|\mathbf a|^2|\mathbf b|^2-(\mathbf a\cdot\mathbf b)^2.
\end{aligned}
$$
This <Lagrange identity for the cross product> includes zero and parallel vectors without a separate division by their norms.

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2021 / 8 / a / ii / Solution

Past exam of the mathematical methods I course of the University of Oxford / 2021 / 8 / a / ii / Solution

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Past exam of the mathematical methods I course of the University of Oxford / 2021 / 8 / a / ii / Solution

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