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**2.2 Index Notation for Vector and Tensor Operations**

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

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**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

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**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

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**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

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**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

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**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