What exactly is the point of the Kronecker delta?

Question

Isn't it just a fancy way of doing an equality test?

Answer 1

The point of it is precisely to provide you a simpler way to write equality tests into your equations.



Without it, you either have to invent your own function that does the same thing, or write something like "if x and y are equal, then f is such and such, but if they're unequal, f is 0."



With the Kronecker delta, you can then just say "f = delta(x, y) * (such and such)." As I'm sure you already know, anything multiplied by zero is zero.



The Kronecker delta is not some deep, mysterious object. Sure it was intended for off-putting crap like tensor analysis and differential geometry, but you can use it any time you need to nullify a step based on the absence of a given equality.

Answer 2

To simplify, or at least compress, what would otherwise be rather long definitions. For example, if you look at the definition of the Moebius function, it usually takes three lines to define it, one for each of its three possible values. Using negative one, you can use just two lines. And using the Kronecker delta, you knock it down to one line.



Yes, the Kronecker delta is nothing more than an equality test. But of course there is a tendency to make the Kronecker delta seem way more complicated and mysterious than it really is: http://oeis.org/w/index.php?title=Kronecker_delta&diff=1446101&oldid=1446080

Answer 3

It motivates all encompassing notation, rather than having to define things piece-wise. It shows up everywhere and is a compact, convenient way of consolidating all results into one equation. Whether you are writing out the results of an inner product of orthogonality, defining a matrix/tensor as a sum of elements presented in indicial notation, in quantum mechanics, and so forth.



I do not know what an equality test is, but you will see many uses of it in your later education. It is essentially the discrete version of the Dirac delta function. The Dirac delta function furnishes modeling and calculations in many arenas, and I am not sure what we would do without it. In quantum mechanics, I am not sure what we would do without the Kronecker delta function. It is a simple advent, but facilitates so much from a grass roots level.



What you will find if you pursue further education is that while results are results, some forms are more transparent than others. Different qualities of an expression or result may be more readily seen in one notation but not apparent in another (equivalent) one (e.g. sin(x)/x at x = 0 is 1, this is very obvious when looking at the taylor series representation, but impossible to see when looking at it in terms of elementary functions). While it may seem like just bookkeeping, the ability to recast something in a different form is a very powerful tool in solving problems. The Kronecker delta function is one such function that enables this flexibility in representation. It is important to have all the tricks we can. It may seem difficult to gauge its usefulness just from me speaking obliquely about it. But, you will see in time though I am sure.



Edit: Jennifer, what is so complicated about putting a delta symbol with appropriate subscripts? That is compact, and makes things look simpler than they are. The notation and techniques developed in previous centuries are as good as ever. Math does not usually become "outdated." I now wonder of your particular level with math-oriented education. Because, people do not typically find these things complicated.

Answer 4

There is no point to it, other than making things look more complicated than they really are. Most things like that come from 20th century mathematics, but a few such things come from the 19th century, and this is one example.

Answer 5

make things easier and faster