Index Notation 1

Question 1

What are the 4 key points to note for index notation

Answer 1

• every sum goes with an index which is repeated twice
• non-repeated indices are not summed
• do not write the summation symbol as a repeated index implies summation
• an index may not appear more than twice on one side of an equality

Question 2

how would you write x.y, [Ax]i and [AB]ij in index notation

Answer 2

• x.y = xi*yi
• [Ax]i = Aij*xj
• [AB]ij = Aik*Bkj

Question 3

What does the i in [Ax]i or ij in [AB]ij tell you

Answer 3

• that these are the indices staying ‘constant’ and not being summed over
• so it is the index not being repeated

Question 4

what does the kronecker delta δij represent and how is it defined

Answer 4

• the kronecker delta represents the identity matrix I
• δij = {1 for i = j, 0 for i != j

Question 5

the same way Iy = y, what does δij*yj equal and why

Answer 5

• δij*yj = yi
• because of the rule that the index ‘attached’ to the expression is the one that stays constant and is not being summed over
• basically the reverse logic of what the i in [Ax]i represents

Question 6

how do you express the cross product in index notation

Answer 6

• the permutation symbol ε_ijk

Question 7

what is the permutation symbol ε_ijk defined as (three points)

Answer 7

• ε_ijk = 1 if (ijk) is an even permutation of (1,2,3)
• ε_ijk = -1 if (ijk) is an odd permutation of (1,2,3)
• ε_ijk = 0 otherwise

Question 8

why does ε_312 = 1

Answer 8

• because to obtain 312 from 123, you need to perform two swaps
• swap 3 and 1 to get 321, then 2 and 1 to get 312
• two swaps = even permutation = 1

Question 9

why does ε_112 = 0

Answer 9

• because there are no number of swaps you can do to get 123 to 112

Question 10

how would you write the cross product [x x(cross) y]i using the permutation symbol

Answer 10

• [x cross y]i = ε_ijkxjyk
• again, the i does not get repeated on the RHS

Question 11

after seeing a bunch of examples, what does the advantage of index notation seem to be

Answer 11

• we can turn vector calculus into simpler multiplication problems

Question 12

what does cyclic permutation mean in how ε_ijk can be represented

Answer 12

• ε_ijk = ε_kij = ε_jki
• you can move the front index to the back and the permutation symbol is equal

Question 13

what are ε_ijk = ε_kij = ε_jki individually equal to and what is the trick to know how

Answer 13

• ε_ijk = -ε_jik
• ε_kij = -ε_ikj
• ε_jki = -ε_kji
• the trick is to swap the first two indices then the ε becomes negative

Question 14

what is the contracted epsilon identity relating ε_ijk and δ_ij and how can you remember this

Answer 14

• ε_ijkε_klm = δ_ilδ_jm - δ_im*δ_jl
• the RHS will have no k’s
• the first δ will have the first ‘valid’ indices of the first and second ε
• the second δ will have the second ‘valid’ ones
• the third and fourth δs are just the first two with their second index swapped

Question 15

for the expression ε_ijk*ε_klm, the 2 k’s mean that this is being summed over k. say i=1, j=2, l=1 and m=3. what would this expression ‘expanded’ actually look like and equal and why

Answer 15

• ‘summed over k’ means that there will be 3 versions of the expression that are summed for k=1, 2 and 3
• ε_ijkε_klm = ε_121ε_113 + ε_122ε_213 + ε_123ε_313 = 0 + 0 + 0 = 0

Question 16

what is [a x (b x c)]i equivalent to in dot product form

Answer 16

• [a x (b x c)]i = [b . (a . c) - c . (a . b)]i
• “bac-cab” trick