Vector Calculus – Exercises V

Posted on March 17, 2024 by ateixeira
Exercise 18

Write {\vec{a}\times \vec{b}+(\vec{a}\cdot \vec{d})\vec{c}=\vec{e}} in suffix notation.

As always we will write the equation in suffix notation for the {i} coordinate:

\displaystyle (\vec{a}\times \vec{b})_i+((\vec{a}\cdot \vec{d})\vec{c})_i=e_i \Leftrightarrow \epsilon_{ijk}a_j b_k + a_l d_l = e_i

Exercise 19

Write {\delta _{ij}c_j+\epsilon_{kji}a_k b_j = d_l e_m c_i b_l c_m} in vector notation.

{\begin{aligned} \delta _{ij}c_j+\epsilon_{kji}a_k b_j &= d_l e_m c_i b_l c_m \\ c_i + (\vec{a}\times \vec{b})_i &= d_l b_l e_m c_m c_i \\ c_i + (\vec{a}\times \vec{b})_i &= (\vec{d}\cdot \vec{b})(\vec{e}\cdot \vec{c})c_i \\ \vec{c} + \vec{a}\times \vec{b} &= (\vec{d}\cdot \vec{b})(\vec{e}\cdot \vec{c})\vec{c} \end{aligned}}

Exercise 20

Show that {\vec{a}\times \vec{b} = -\vec{b}\times \vec{a} } using suffix notation.

{\begin{aligned} \vec{a}\times \vec{b} &= \epsilon_{ijk}a_j b_k \\ &= \epsilon_{ijk} b_k a_j \\ &= - \epsilon_{ikj} b_k a_j \\ &= -\vec{b}\times \vec{a} \end{aligned}}

Exercise 21

Let {A} and {B} be {N\times N} matrices. Show that {(AB)^T = B^T A^T}.

It is

\displaystyle C_{ij}= A_{ik}B_{kj}

Then it follows

\displaystyle C^T= (C_{ij})^T = C_{ji}=A_{ki}B_{jk}=B_{jk}A_{ki}=B^T A^T

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