3. Index Notation

Question 1

Component Form

Definition

Answer 1

(a1, a2, a3)

Question 2

Component Form

Advantages and Disadvantages

Answer 2

Advantages
-simple
-uses ordinary algebra
Disadvantages
-tedious for long expressions
-does not make the coordinate-invariance obvious

Question 3

Gibbs Notation

Definition

Answer 3

|a, |a . |b , etc.

Question 4

Gibbs Notation

Advantages and Disadvantages

Answer 4

Advantages
-compact and elegant
-explicitly coordinate-invariant
Disadvantages
-can be difficult/ambiguous to manipulate some expressions involving operators

Question 5

Index Notation

Advantages and DIsadvantages

Answer 5

• it is compact and exploits coordinate-invariance, but is also unambiguous using ordinary algebraic operations
• it can be difficult at first
• also known as suffix notation

Question 6

Index Notation

Basics

Answer 6

-consider a simple vector addition:
->Gibbs Notation: |a + |b = |c
->Components: a1+b1=c1 , a2+b2=c2 , a3+b3=c3
-in index notation this is written:
ai + bi = ci
-where implicitly the free index i takes values 1, 2, 3

Question 7

Free Index

Answer 7

• for ai + bi = ci , i is called a free index
• in a sum all terms must have the same free index, ai+bi=ci is a valid equation but aj+bk=cj is not
• we can use any index letter so, ak+bk=ck is valid

Question 8

Scalar Product

Index Notation

Answer 8

-consider a scalar product:
|a . |b = a1b1 + a2b2 + a3b3
= Σajbj between j=1 and j=3
-in index notation we drop the sum according to the Einstein summation convention giving:
|a . |b = aj bj

Question 9

Einstein Summation Convention

Answer 9

-whenever an index is repeated within a term, summation from 1 to 3 is implied/assumed

Question 10

Dummy Index

Answer 10

• an index of summation (a repeated index) is called a dummy index
• the choice of letter is arbitrary and different terms may use different dummy indices

Question 11

How many times can an index appear within a term?

Answer 11

• no index can appear more than twice within the same term

- but repeated indices can be resued in different terms

Question 12

Scalar Product

Order of Terms

Answer 12

(|a . |b) (|c . |d) = aj ck bj dk

-the index shows which factors belong to which scalar product, not the position of the terms

Question 13

Tensors

Answer 13

• in index notation a scalar quantity has no free indices, while a vector quantity has one free index
• the concept of a coordinate-invariant quantity can be generalised to objects with an arbitrary number of indices, known as tensors

Question 14

Second-Rank Tensor

Answer 14

• a second rank tensor is a tensor with two indices
• it can be represented as a 3x3 matrix
• the rank of a tensor is determined by the number of indices it has

Question 15

Matrix Multiplication in Index Notation

Answer 15

[||A ||B}ij = (k=1->3)ΣAik Bkj
-using Einsteins summation convention, this becomes:
Aik Bjk

Question 16

Inner Product

Answer 16

• any product of two vectors or tensors that share a dummy index e.g. aij uj is called an inner product
• a generalisation of the scalar product

Question 17

The Kronecker Delta

Answer 17

-the Kronecker delta is a special tensor defined as
𝛿ij = {1 if i=j, 0 if i≠j}
-represented as a matrix it corresponds to the identity matrix

Question 18

Properties of the Kronecker Delta

Answer 18

1) it is symmetric 𝛿ij=𝛿ji
2) if i or j is a free index, 𝛿ij acts as a substitution operator:
𝛿ij aj = ai
3) 𝛿ij can be used to represent the scalar product 𝛿ij ai bj = ajbj = |a.|b

Question 19

The Vector Product

Index Notation

Answer 19

|a x |b = εijk aj bk

Question 20

The Alternating Tensor

Answer 20

εijk = 1 for (i,j,k) = (1,2,3) or (2,3,1) or (3,1,2)
εijk = -1 for (i,j,k) = (3,2,1) or (1,3,2) or (2,1,3)
εijk = 0 otherwise

Question 21

Properties of εijk

Answer 21

1) εijk = εkij = εjki
- ε is symmetric under cyclic permutation
2) εijk = -εjik
- anti-symmetric under exchange of indices (for any two indices)
3) εiik = 0
4) εijk is called the ‘alternating tensor’ or the ‘Levi-Civita’ symbol

Question 22

Scalar Triple Product

Index Notation

Answer 22

|a . (|b x |c) = εijk ai bj ck

Question 23

Inner Product of Two Alternating Tensors

Answer 23

εijk εilm = 𝛿jl 𝛿km - 𝛿jm 𝛿kl

Question 24

How to remember the inner product of two alternating tensors

Answer 24

-the repeated index is the first
-the deltas combine:
(2nd index with 2nd)(3rd with 3rd) - (2nd with 3rd)(3rd with 2nd)

Question 25

The Vector Triple Product Rule

Answer 25

|a x (|b x |c) = (|a . |c)|b - (|a . |b)|c

Question 26

Del

Index Notation

Answer 26

(|∇)i = ∂/∂xi = ∂i

where x1 = x, x2 = y, x3 = z

Question 27

Grad

Index Notation

Answer 27

(grad φ)i = ∂φ/∂xi

Question 28

Divergence

Index Notation

Answer 28

div |F = |∇ . |F = ∂Fi/∂xi

Question 29

Curl

Index Notation

Answer 29

(curl |F)i = (|∇ x |F)i = εijk ∂/∂xj Fk

Question 30

Index Notation and Differential Operators

Answer 30

• while order of factors doesn’t matter for algebraic products, it DOES for differential operators
  i. e. |∇ . |F = ∂i Fi ≠ Fi ∂i

Question 31

Partial Derivative

Index Notation

Answer 31

∂xj/∂xi = 𝛿ij

Question 32

Identities

div(g |F)

Answer 32

div(g |F) = |F . grad g + g div |F

Question 33

Identities

curl(|F g)

Answer 33

curl(|F g ) = g curl |F - |F x grad g

Question 34

Laplacian

Index Notation

Answer 34

∇² = ∂² / ∂xi ∂xi

-where the repeated index indicates notation

Question 35

Identities

div (grad f)

Answer 35

div (grad f) = ∇² f

Question 36

Identities

curl (grad f)

Answer 36

curl (grad f ) = - curl (grad f )

-therefore, curl (grad f ) = 0

Question 37

Identities

curl (curl |F )

Answer 37

curl (curl |F) = grad (div |F ) - ∇² |F