3. Index Notation Flashcards
Component Form
Definition
(a1, a2, a3)
Component Form
Advantages and Disadvantages
Advantages -simple -uses ordinary algebra Disadvantages -tedious for long expressions -does not make the coordinate-invariance obvious
Gibbs Notation
Definition
|a, |a . |b , etc.
Gibbs Notation
Advantages and Disadvantages
Advantages -compact and elegant -explicitly coordinate-invariant Disadvantages -can be difficult/ambiguous to manipulate some expressions involving operators
Index Notation
Advantages and DIsadvantages
- 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
Index Notation
Basics
-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
Free Index
- 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
Scalar Product
Index Notation
-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
Einstein Summation Convention
-whenever an index is repeated within a term, summation from 1 to 3 is implied/assumed
Dummy Index
- 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
How many times can an index appear within a term?
- no index can appear more than twice within the same term
- but repeated indices can be resued in different terms
Scalar Product
Order of Terms
(|a . |b) (|c . |d) = aj ck bj dk
-the index shows which factors belong to which scalar product, not the position of the terms
Tensors
- 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
Second-Rank Tensor
- 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
Matrix Multiplication in Index Notation
[||A ||B}ij = (k=1->3)ΣAik Bkj
-using Einsteins summation convention, this becomes:
Aik Bjk
Inner Product
- 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
The Kronecker Delta
-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
Properties of the Kronecker Delta
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
The Vector Product
Index Notation
|a x |b = εijk aj bk
The Alternating Tensor
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
Properties of εijk
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
Scalar Triple Product
Index Notation
|a . (|b x |c) = εijk ai bj ck
Inner Product of Two Alternating Tensors
εijk εilm = 𝛿jl 𝛿km - 𝛿jm 𝛿kl
How to remember the inner product of two alternating tensors
-the repeated index is the first
-the deltas combine:
(2nd index with 2nd)(3rd with 3rd) - (2nd with 3rd)(3rd with 2nd)
The Vector Triple Product Rule
|a x (|b x |c) = (|a . |c)|b - (|a . |b)|c
Del
Index Notation
(|∇)i = ∂/∂xi = ∂i
where x1 = x, x2 = y, x3 = z
Grad
Index Notation
(grad φ)i = ∂φ/∂xi
Divergence
Index Notation
div |F = |∇ . |F = ∂Fi/∂xi
Curl
Index Notation
(curl |F)i = (|∇ x |F)i = εijk ∂/∂xj Fk
Index Notation and Differential Operators
- while order of factors doesn’t matter for algebraic products, it DOES for differential operators
i. e. |∇ . |F = ∂i Fi ≠ Fi ∂i
Partial Derivative
Index Notation
∂xj/∂xi = 𝛿ij
Identities
div(g |F)
div(g |F) = |F . grad g + g div |F
Identities
curl(|F g)
curl(|F g ) = g curl |F - |F x grad g
Laplacian
Index Notation
∇² = ∂² / ∂xi ∂xi
-where the repeated index indicates notation
Identities
div (grad f)
div (grad f) = ∇² f
Identities
curl (grad f)
curl (grad f ) = - curl (grad f )
-therefore, curl (grad f ) = 0
Identities
curl (curl |F )
curl (curl |F) = grad (div |F ) - ∇² |F
