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

 

and

 

This will let us write many formulae more compactly.