Lectures on Vector Calculus 2025/26 Update, 100% Guaranteed Pass | Comprehensive Lecture Notes

Vector Calculus
Lectures on Vector Calculus 2025/26 Update, 100% Guaranteed Pass

| Comprehensive Lecture Notes




 Vector Calculus

 vector algebra

 Orthonormality and the Kronecker Delta

 Comprehensive Lecture Notes

,1 Vector Algebra and Index Notation

1.1 Orthonormality and the Kronecker Delta
We begin with three dimensional Euclidean space R3. In R3 we can define
three special coordinate vectors ê 1 , ê 2 , and eˆ 3 . 1 We choose these vectors to
be orthonormal, which is to say, both orthogonal and normalized (to unity).
We may express these conditions mathematically by means of the dot product
or scalar product as follows:


ê 1 · ê 2 = ê 2 · ê 1 = 0
ê 2 · ê 3 = ê 3 · ê 2 = 0 (orthogonality) (1.1)
ê 1 · ê 3 = ê 3 · ê 1 = 0


and
ê 1 · ê 1 = ê 2 · ê 2 = ê 3 · ê 3 = 1 (normalization). (1.2)

To save writing, we will abbreviate these equations using dummy indices
instead. (They are called ‘indices’ because they index something, and they
are called ‘dummy’ because the exact letter used is irrelevant.) In index
notation, then, I claim that the conditions (1.1) and (1.2) may be written


ê i · ê j = δij. (1.3)


How are we to understand this equation? Well, for starters, this equation
is really nine equations rolled into one! The index i can assume the values 1,
2, or 3, so we say “i runs from 1 to 3”, and similarly for j. The equation is
1These vectors
are also denoted ŝ , (ˆ, and ĥ , or x̂ , yˆ and xˆ. We will use all three
notations interchangeably.




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,valid for all possible choices of values for the indices. So, if we pick,
say, i = 1 and j = 2, (1.3) would read


ê 1 · ê 2 = δ12. (1.4)


Or, if we chose i = 3 and j = 1, (1.3) would read


ê 3 · ê 1 = δ31. (1.5)


Clearly, then, as i and j each run from 1 to 3, there are nine possible choices
for the values of the index pair i and j on each side, hence nine equations.
The object on the right hand side of (1.3) is called the Kronecker delta.
It is defined as follows:

1 if i = j,
δij = (1.6)
0 otherwise.

The Kronecker delta assumes nine possible values, depending on the
choices for i and j. For example, if i = 1 and j = 2 we have δ12 = 0,
because i and j are not equal. If i = 2 and j = 2, then we get δ22 = 1, and
so on. A convenient way of remembering the definition (1.6) is to imagine
the Kronecker delta as a 3 by 3 matrix, where the first index represents the
row number and the second index represents the column number. Then we
could write (abusing notation slightly)
,
1 0 0
δij = 0 1 0 . (1.7)
0 0 1


2

, Finally, then, we can understand Equation (1.3): it is just a shorthand
way of writing the nine equations (1.1) and (1.2). For example, if we choose
i = 2 and j = 3 in (1.3), we get


ê 2 · ê 3 = 0, (1.8)


(because δ23 = 0 by definition of the Kronecker delta). This is just one of
the equations in (1.1). Letting i and j run from 1 to 3, we get all the nine
orthornormality conditions on the basis vectors ê 1 , eˆ 2 and eˆ 3 .

Remark. It is easy to see from the definition (1.6) or from (1.7) that the
Kronecker delta is what we call symmetric. That is


δij = δji. (1.9)


Hence we could have written Equation (1.3) as


ê i · ê j = δji. (1.10)


(In general, you must pay careful attention to the order in which the indices appear
in an equation.)


Remark. We could have written Equation (1.3) as

ê a · ê b = δab, (1.11)


which employs the letters a and b instead of i and j. The meaning of the equation
is exactly the same as before. The only difference is in the labels of the indices.
This is why they are called ‘dummy’ indices.



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