Mathematical Tools for Physics - Department of Physics - University

9—Vector Calculus 1 231

The Kronecker delta is either one or zero depending on whetheri=jori 6 =j, and this equation sums

up the properties of the basis in a single, compact notation. You can now write a vector in this basis as

A~=A 1 ˆe 1 +A 2 ˆe 2 +A 3 eˆ 3 =Aiˆei

The last expression uses the summation convention that a repeated index is summed over its range.
When an index is repeated in a term, you will invariably have exactly two instances of the index; if you
have three it’s a mistake.
When you add or subtract vectors, the index notation is

A~+B~=C~=Aiˆei+Biˆei= (Ai+Bi)ˆei=Ciˆei or Ai+Bi=Ci

Does it make sense to writeA~+B~=Aiˆei+Bkˆek? Yes, but it’s sort of pointless and confusing. You

can change any summed index to any label you find convenient — they’re just dummy variables.

Aiˆei=Ae=Amˆem=A 1 eˆ 1 +A 2 ˆe 2 +A 3 ˆe 3

You can sometime use this freedom to help do manipulations, but in the exampleAieˆi+Bkˆekit’s no

help at all.
Combinations such as

Ei+Fi or EkFkGi=Hi or MkD=Fk

are valid. The last is simply Eq. (7.8) for a matrix times a column matrix.

Ai+Bj=Ck or EmFmGm or Ck=AijBj

have no meaning.
You can manipulate the indices for your convenience as long as you follow the rules.

Ai=BijCj is the same as Ak=BknCn or A=BpCp

The scalar product has a simple form in index notation:

A~.B~=Aiˆei.Bjˆej=AiBjˆei.ˆej=AiBjδij=AiBi (9.51)

The final equation comes by doing one of the two sums (sayj), and only the term withj=isurvives,

producing the final expression. The result shows that the sum over the repeated index is the scalar
product in disguise.
Just as the dot product of the basis vectors is the delta-symbol, the cross product provides
another important index function, the alternating symbol.

ˆei.eˆj׈ek=ijk (9.52)

eˆ 2 ׈e 3 ≡ˆy×zˆ=ˆe 1 , so  123 =eˆ 1 .eˆ 2 ׈e 3 = 1

eˆ 2 ׈e 3 =−eˆ 3 ׈e 2 , so  132 =− 123 =− 1

ˆe 3 .eˆ 1 ׈e 2 =ˆe 3 .ˆe 3 = 1, so  312 = 123 = 1

ˆe 1 .eˆ 3 ׈e 3 =eˆ 1 .0 = 0 so  133 = 0