/sci/ - Science & Math

>>8750278

>not recognizing a stale meme

>>8586654

>analysis is trivial and boring

>not stating available assumptions

>>8423572
>>8423612

I'm sorry, I just saw this problem, and I have always found ricci calculus to be exceptionally fun, so forgive me, but I couldn't resist. Clearly you already understand it, but this is for me.

lets define [math]\textbf{a}=a^{i}\textbf{e}_{i}[/math], [math]\textbf{b}=b^{i}\textbf{e}_{i}[/math], [math]\textbf{c}=c^{i}\textbf{e}_{i}[/math], and [math]\textbf{d}=d^{i}\textbf{e}_{i}[/math]

the the cross products are [math]\textbf{a}\times\textbf{b}=E^{ijk}\textbf{e}_{i}a_{j}b_{k}[/math], and [math]\textbf{c}\times\textbf{d}=E^{ijk}\textbf{e}_{i}c_{j}d_{k}[/math].

now there is a trivial equivalence theorem giving us [math]E^{ijk}\textbf{e}_{i}c_{j}d_{k}=E_{ijk}\textbf{e}^{i}c^{j}d^{k}[/math], and the dot product is then procured by contracting on the vector fields free indices

and we have
[math]E^{ijk}a_{j}b_{k}E_{ilm}c^{l}d^{m}=\delta^{ijk}_{ilm}a_{j}b_{k}c^{l}d^{m}=\delta^{j}_{l}\delta^{k}_{m} a_{j}b_{k}c^{l}d^{m} - \delta^{j}_{m}\delta^{k}_{l} a_{j}b_{k}c^{l}d^{m}=(a_{l}b_{m}c^{l}d^{m})-(a_{m}b_{l}c^{l}d^{m})=(\textbf{a}\cdot\textbf{c})(\textbf{b}\cdot\textbf{d})-(\textbf{a}\cdot\textbf{d})(\textbf{b}\cdot\textbf{c})[/math]

good lord that felt good. I hope latex doesn't go totally gay on me.

>undergrad jerkoff thread

>"You should be able to do this"-thread
>page 1 rudin
>freshman/HS math

man, step it up op, atleast pick a non highschool problem.

plus your phrasing of the problem was actually quite poor.

>>8204029

probably a girl

>>8203903

>actually taking notes from wildburger

I did.