The big argument: Is the Einstein summation convention worth it?

We hear arguments for and against this controversial convention


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 $\sum_{i=1}^N a_i b_i$ is written $a_i b_i$ for example.

Yes: worth it, argues Ellen Jolley

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 even five minutes without using the summation convention—I promise you will tire of constantly explaining “yes, the sum still starts from $1$, and yes, it still goes to $N$.”

You’ll scream, “All of them! I am summing over all indices! Obviously! Why’d I ever skip some??” If you’re confused how many you’ve got, use this simple guide: physicists use four; fluid dynamicists use three; and Italian plumbers use two. Wouldn’t it be nice to avoid saying this in every equation?

You may cry that it’s easier to make mistakes with the convention; but for applied mathematicians, the joy comes in speeding ahead to the answer by any means—time spent on accuracy and proof is time wasted. And as the great mathematician Bob Ross said: there are no mistakes, just happy little accidents!

No: not worth it, argues Sophie Maclean

Before writing this argument, I had to Google ‘summation convention’ which is all the evidence I need for why it’s just not worth it. I’ve learnt how to use the convention—multiple times! In fact, I’d say it’s something I’m able to use, yet I’m still not sure I know exactly what it is.

Some of our readers won’t have ever heard of it (which is one strike against it). Some have heard of it but won’t 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, you’re in denial).

We don’t even have need for the convention! We already have a suitable way to notate summation:
It’s taught to schoolkids. There is no ambiguity. And it’s so much less pretentious. Yes, the summation convention is fractionally faster to write out, but mathematicians are famed for being lazy and aloof—maybe dispensing with it is all we need to break that stereotype!

Ellen is a PhD student at UCL studying fluid mechanics. She specialises in the flow around droplets and ice particles.

Sophie Maclean is a recent maths graduate from the University of Cambridge and very much misses her degree. She has no free time—she is a Chalkdust editor.

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