Part A:  Indicial Notation

2A1   Summation Convention, Dummy Indices  

Consider the sum                                        

 

Thus, the repeated indices (i,j,k) are called dummy indices and the dummy indices are summed.  [Einstein's convention]

Very similar to         x andare dummies.

In this convention, an index should not be repeated more than once,

i.e.     should not be written in indicial form.

For the three dimensional Cartesian system let us assume n=3.  Then                                       

concisely represents a vector

 

Double and Triple Sum

Let us consider the following example.

and similarly, by                             

 

2A3   Indicial Notation and Vector Symbols    

        Consider a right handed, orthogonal Cartesian coordinate system.  Let be the base vectors. 

From the orthogonality of the base vectors:

              But         

        Introduce the symbol called the Kronecker Delta 

      and thus   

       We can see the matrix

         Also

  2.    In general