Expressing the magnitude of a vector difference in indicial notation

Question

I'm trying to express the following relation in indicial notation $$ |\vec{u} - \vec{v}_p| \, . $$

The only way I found out is replacing the difference above by $$ \vec{u} - \vec{v}_p = \vec{v}_r \, , $$ then one can write $$ |\vec{u} - \vec{v}_p| = |\vec{v}_r| = (v_{r,i}v_{r,i})^{1/2} \, . $$

Is there anyway to express the vector difference directly in the indicial notation?

Adrian Keister · Accepted Answer · 2018-04-27T13:12:44.373

0

You could always do $$|\vec{u}-\vec{v}_p|=\sqrt{\sum_{i=1}^{n}(u_i-v_{p,i})^2},$$ so that you don't need the helper vector $\vec{v}_r$. Another option is to use the Einstein summation convention, and just write $$|\vec{u}-\vec{v}_p|=\sqrt{(u_i-v_{p,i})(u_i-v_{p,i})},$$ but you'd have to make sure your readers were aware that you were invoking the convention.

edited Apr 27 '18 at 13:12

answered Apr 26 '18 at 15:33

Adrian Keister

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I agree with the first suggestion (index at the summation is $i$ instead of $j$), and i think it's clear. I've also thought of your second suggestion but i don't clearly see the summation invoking in this form. – Diego Venturi Apr 27 '18 at 11:22
Well, I see it now! Opening up the terms it becomes clear. Thanks. – Diego Venturi Apr 27 '18 at 11:25
Thanks for catching the typo - I've fixed it. – Adrian Keister Apr 27 '18 at 13:13

Expressing the magnitude of a vector difference in indicial notation

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