General Mechanics/Index Notation

Summation conventionEdit

If we label the axes as 1,2, and 3 we can write the dot product as a sum


If we number the elements of a matrix similarly,


we can write similar expressions for matrix multiplications


Notice that in each case we are summing over the repeated index. Since this is so common, it is now conventional to omit the summation sign.

Instead we simply write


We can then also number the unit vectors, êi, and write


which can be convenient in a rotating coordinate system.

Kronecker deltaEdit

The Kronecker delta is


This is the standard way of writing the identity matrix.

Levi-Civita (Alternating) symbolEdit

Another useful quantity can be defined by


With this definition it turns out that




This will let us write many formulae more compactly.