2.2 Index Notation for Vector and Tensor Operations

 

 

 

Operations on Cartesian components of vectors and tensors may be expressed very efficiently and clearly using index notation.

 

2.1. Vector and tensor components.

 

Let x be a (three dimensional) vector and let S be a second order tensor.   Let  be a Cartesian basis. Denote the components of x in this basis by , and denote the components of S by

Using index notation, we would express x and S as

 

2.2. Conventions and special symbols for index notation

 

 Range Convention: Lower case Latin subscripts (i, j, k…) have the range .  The symbol  denotes three components of a vector  and .  The symbol  denotes nine components of a second order tensor,

 

 Summation convention (Einstein convention): If an index is repeated in a product of vectors or tensors, summation is implied over the repeated index.  Thus

 

In the last two equations, ,  and  denote the  component matrices of A, B and C.

 

 The Kronecker Delta:  The symbol  is known as the Kronecker delta, and has the properties

thus

You can also think of  as the components of the identity tensor, or a  identity matrix.  Observe the following useful results

 

 The Permutation Symbol: The symbol  has properties

thus

Note that

 

2.3. Rules of index notation

 

1. The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors.  Thus

are valid, but

are meaningless

 

 

 

 

2. Free indices on each term of an equation must agree.  Thus

are valid, but

are meaningless.

 

3.  Free and dummy indices may be changed without altering the meaning of an expression, provided that rules 1 and 2 are not violated. Thus

 

2.4. Vector operations expressed using index notation

 

 Addition.  

 

 Dot Product 

 

 Vector Product

 

 Dyadic Product  

 

 Change of Basis.  Let a be a vector. Let  be a Cartesian basis, and denote the components of a in this basis by .  Let  be a second basis, and denote the components of a in this basis by .  Then, define

where  denotes the angle between the unit vectors   and .  Then

 

2.5. Tensor operations expressed using index notation.

 

 Addition.  

 

 Transpose 

 

 Scalar Products

 

 Product of a tensor and a vector

 

 Product of two tensors 

* Determinant

* Inverse

 Change of Basis.  Let A be a second order tensor. Let  be a Cartesian basis, and denote the components of A in this basis by .  Let  be a second basis, and denote the components of A in this basis by .  Then, define

where  denotes the angle between the unit vectors   and .  Then

 

 

2.6. Calculus using index notation

 

The derivative  can be deduced by noting that  and  .  Therefore

                                                                 

The same argument can be used for higher order tensors

                                                              

 

 

2.7. Examples of algebraic manipulations using index notation

 

1. Let a, b, c, d be vectors.  Prove that

 

Express the left hand side of the equation using index notation (check the rules for cross products and dot products of vectors to see how this is done)

Recall the identity

so

Multiply out, and note that

(multiplying by a Kronecker delta has the effect of switching indices…) so

Finally, note that

and similarly for other products with the same index, so that

 

2. The stress—strain relation for linear elasticity may be expressed as

where  and  are the components of the stress and strain tensor, and  and  denote Young’s modulus and Poisson’s ratio.  Find an expression for strain in terms of stress.

 

Set i=j to see that

Recall that , and notice that we can replace the remaining ii by kk

Now, substitute for  in the given stress—strain relation

 

3. Solve the equation

 

for  in terms of  and

 

Multiply both sides by  to see that

Substitute back into the equation given for  to see that

 

4. Let .  Calculate

 

We can just apply the usual chain and product rules of differentiation

 

5. Let .  Calculate

 

Using the product rule