vector identities

1 Preface

In all works that I have read, that describe electromagnetic fields from a technical point of view, the authors use vector identitiesMathworldPlanetmath to establish proof. These vector identities,for example, are used to establish the veracity of the poynting vector or establish the wave equation. We have no intristic reason to believe these identities are true, however the proofs of which can be tedious. Nonwithstanding, doing so can have rewards as we gain insight into the nature of combinatorics and the constraints of permutations. A list of these vector identities is provided and for each one also is provided a proof of the identity.

2 What are curl and div anyway?

curl see hyperlinks for more information.

(×B) (1)

divergenceMathworldPlanetmath see hyperlinks for more information.

(B) (2)
(b) (3)

Is div applied to a scalar. This is the gradientMathworldPlanetmath or grad. curl takes a vector and gives back a vector; div takes a vector and gives back a scalar. grad takes a scalar and gives back a vector.

3 First Proof

ab=ab+ba (4)

Starting with

f=ifx1+jfx2+kfx3 (5)

Replace f with ab

ab=iabx1+jabx2+kabx3 (6)

we cannot make assertions about the scalar functions a b, so we are forced to invoke the product ruleMathworldPlanetmath assuming the most generality For the first dimensionMathworldPlanetmathPlanetmathPlanetmath it becomes

i[abx1+ax1b] (7)

and likewise

j[abx1+ax1b] (8)
k[abx1+ax1b] (9)

Instead of viewing this as a single vector we split the vector into two vectors.

i[abx1]i[ax1b] (10)

and likewise

j[abx1]j[ax1b] (11)
k[abx1]k[ax1b] (12)

factor the a scalar from the first vector and b scalar from the second vector

a[ibx1+jbx2+kbx3]+[iax1+jax2+kax3]b (13)

Now the defintion of the gradient can be seen to occur twice in the last equation, once for b and once for a and it is shown by imposing a structureMathworldPlanetmath

ab=ab+ba (14)

4 Second Proof

(AB)=[(A)B]+[(B)A]+[A×(×B)]+[B×(×A)] (15)

This is the second vector identity to prove.

(A×B)=[B(×A)]-[A(×B)] (16)
(a2b3-b2a3)x1 (17)

Structurally, this means we have to apply the product rule twice: Once on the first term and again on the second term.

(a2b3)x1-(b2a3)x1 (18)

Each of these must get expanded according to the product rule.

a2b3x1+b3a2x1-[a3b2x1+b2a3x1] (19)
-[a1b3x2+b3a1x2-[a3b1x2+b1a3x2]] (20)
a1b2x3+b2a1x3-[a2b1x3+b1a2x3] (21)

Ok. Now we start to look at this as an array. The importance of this cannot be underestimated to build up what I call a Noetherian viewpoint. If we insist on imposing a structure on something that is abelian, we will be able to draw conclusionsMathworldPlanetmath with much more alacrity.

𝐗=(a2b13b3a12-a3b12-b2a13-a1b23-b3a21a3b21b1a23a1b32b2a31-a2b31-b1a32) (22)

This transformationMathworldPlanetmath now allows us to count that 3x4 or 12 terms is the same as 2x6 terms. grabbing the a1 terms:

a1b32-a1b23 (23)

grabbing the a2 terms:

-[a2b31-a2b13] (24)

grabbing the a3 terms:

a3b21-a3b12 (25)

grabbing the b1 terms:

-b1a32+b1a23 (26)

grabbing the b2 terms:

-[-[b2a31-b2a13]] (27)

grabbing the b3 terms:

-b3a21+b3a12 (28)

Note that the Sign of the second group of six is Inverted from the first group. All of these have an identical form and it remains to extract the Dot productMathworldPlanetmath of the A vector with something that can be made to look like the curl of B. Factoring and transforming back to the partial deriviative form: grabbing the a1 terms:

(a1)(b2x3-b3x2) (29)

grabbing the a2 terms:

(a2)(-[b1x3-b3x1]) (30)

grabbing the a3 terms:

(a3)(b1x2-b2x1) (31)

Equation 19 cannot contain x1(”x” componentPlanetmathPlanetmath) Equation 20 cannot contain x2(”y” component) and must be negative cofactorPlanetmathPlanetmath Equation 21 cannot contain x3(”z”component) gluing these together as a vector we see after factoring out another -1,

-A(×B) (32)

working through the similarMathworldPlanetmathPlanetmathPlanetmath logic for the other 6 terms, we see

B(×A) (33)

Together, the result is proven. This is a lot of work, but because we organize it as systematically as we do,we are able to manage the tedium, and gain some insight into why this can only work because of the factorization of 12 and the fact that curl is intrinsically 3 dimensional. Note also that this identity only works with the cartesian coordinate systems. Variation based on the JacobianMathworldPlanetmathPlanetmath can be found in Ramo and Whinnery, ”Fields and Waves in Modern Radio”

5 Third Proof

×(×A)=(A)-2A (34)

In this proof, I have resorted to an even more barebones representation. It is really the only way you can get to see the view from 10,000 feet and still be able to elicit forth the regroupings. Looking at the expanded curl operator

×A=[32-231][31-13-2][21-123] (35)

The denominators are not really denominators, but the dimensions of the space. The second one is made negative to remind all of the cofactor, lest it get inadvertently dropped.

×(×A)=[[21-123]2-[31-13-2]31][[21-123]1-[32-231]3-2][[31-13-2]1-[32-231]23] (36)

It is important to be sure of these relationships because it would appear for certain it will deal with the mixed partial derivatives. For example, the last equation, the ”denominator” when there are two ”fractions” separated by subtraction, is not one of dimension, but is in fact the partial derivativeMathworldPlanetmath with respect to the ”numerator” The operationMathworldPlanetmath of taking the partial derivative first with respect to the first dimension and them with respect to the second dimension, is notated by

[f2x1]x2=2f2x2x1 (37)
[f1x2]x2=2f1x22 (38)

The notation, barebones or not must keep track of the order of mixed partials, in case we have to

explain why the order matters(though it shouldn’t, but at this point who knows?) Back substition

×(×A)=[[2f2x2x1-2f1x22]-[31-13-2]31][[21-123]1-[32-231]3-2][[31-13-2]1-[32-231]23] (39)

What is seen is a pattern emerging were some of the items in this collectionMathworldPlanetmath are positive and some are negative. None of the positive items are of the form:

2f1x22 (40)

In fact these all correspond to the expansionMathworldPlanetmath of the term:

-2A (41)

Given the productsPlanetmathPlanetmathPlanetmath that appear on the left hand side of the above equation the meaning of

2A (42)

is inferred to be

2A=(i)2C1+(j)2C2+(k)2C3 (43)

This will give us an ”array” 3 by 3 that has overcounted the items on the ”main diagonal” the curl of the curl has no terms of the form:

2f1x12 (44)

And so we must resort to trickery using an

[(i)2C1x12+(j)2C2x22+(k)2C3x32]-[(i)2C1x12+(j)2C2x22+(k)2C3x32] (45)

As a substituion for zero. What remains is to show that the positive part of the above expression together with the remaining part of the curl is indeed

(A) (46)


  • RW Ramo, Simon, and Whinnery, John R. (1944). Fields and Waves in Modern Radio. General Electric Company.
  • KC Krause, John D. and Carver, Keith R. (1973). Electromagnetics Second Edition. McGraw Hill Book Company
Title vector identities
Canonical name VectorIdentities
Date of creation 2013-03-22 18:07:43
Last modified on 2013-03-22 18:07:43
Owner mark_t314159 (20778)
Last modified by mark_t314159 (20778)
Numerical id 34
Author mark_t314159 (20778)
Entry type Definition
Classification msc 15A72