THIS ISSUEβ¦
IS THE EINSTEIN SUMMATION CONVENTION WORTH IT?
E TH E TH
G I B
E TH
T EN M GU AR
NO
YES ARGUES ELLEN JOLLEY
ARGUES SOPHIE MACLEAN
The Einstein summation convention is a way to write and manipulate vector equations in many dimensions. Simply put, when you see repeated indices, you sum over them, so βππ=π£ ππ ππ is written ππ ππ for example. This debate boils down to just one question: how much of your life do you spend doing tensor algebra? Those of us who undertake a positive amount of tensor algebra or vector calculus know that the goal is to be done with it as fast as possible! Try tensor algebra for even five minutes without using the summation conventionβI promise you will tire of constantly explaining βyes, the sum still starts from π£, and yes, it still goes to π.β
Before writing this argument, I had to Google βsummation conventionβ which is all the evidence I need for why itβs just not worth it. Iβve learnt how to use the conventionβmultiple times! In fact, Iβd say itβs something Iβm able to use, yet Iβm still not sure I know exactly what it is. Some of our readers wonβt have ever heard of it (which is one strike against it). Some have heard of it but wonβt know much about it (another strike). But I guarantee none would be confident saying they can use it without making any errors (if you think you would be, youβre in denial).
Youβll scream, βAll of them! I am summing over all indices! Obviously! Whyβd I ever skip some??β If youβre confused how many youβve got, use this simple guide: physicists use four; fluid dynamicists use three; and Italian plumbers use two. Wouldnβt it be nice to avoid saying this in every equation?
We donβt even have need for the convention! We already have a suitable way to notate summation: β
You may cry that itβs easier to make mistakes with the convention; but for applied mathematicians, the joy comes in speeding ahead to the answer by any meansβtime spent on accuracy and proof is time wasted. And as the great mathematician Bob Ross said: there are no mistakes, just happy little accidents!
Itβs taught to schoolkids. There is no ambiguity. And itβs so much less pretentious. Yes, the summation convention is fractionally faster to write out, but mathematicians are famed for being lazy and aloofβmaybe dispensing with it is all we need to break that stereotype! 27
spring 2021
