vector identities

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 identities

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)

divergence

see hyperlinks for more information.

(

∇

⋅

B

→

)

(2)

(

∇

⁡

b

)

(3)

Is div applied to a scalar. This is the

gradient

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

∇

⁡

a

⁢

b

=

a

⁢

∇

⁡

b

+

b

⁢

∇

⁡

a

(4)

Starting with

∇

⁡

f

=

i

→

⁢

∂

⁡

f

∂

⁡

x

1

+

j

→

⁢

∂

⁡

f

∂

⁡

x

2

+

k

→

⁢

∂

⁡

f

∂

⁡

x

3

(5)

Replace f with ab

∇

⁡

a

⁢

b

=

i

→

⁢

∂

⁡

a

⁢

b

∂

⁡

x

1

+

j

→

⁢

∂

⁡

a

⁢

b

∂

⁡

x

2

+

k

→

⁢

∂

⁡

a

⁢

b

∂

⁡

x

3

(6)

we cannot make assertions about the scalar functions a b, so we are forced to invoke the

product rule

assuming the most generality
For the first

dimension

it becomes

i

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

+

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(7)

and likewise

j

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

+

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(8)

k

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

+

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(9)

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

i

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

]

⟷

i

→

⁢

[

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(10)

and likewise

j

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

]

⟷

j

→

⁢

[

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(11)

k

→

⁢

[

a

⁢

∂

⁡

b

∂

⁡

x

1

]

⟷

k

→

⁢

[

∂

⁡

a

∂

⁡

x

1

⁢

b

]

(12)

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

a

⁢

[

i

→

⁢

∂

⁡

b

∂

⁡

x

1

+

j

→

⁢

∂

⁡

b

∂

⁡

x

2

+

k

→

⁢

∂

⁡

b

∂

⁡

x

3

]

+

[

i

→

⁢

∂

⁡

a

∂

⁡

x

1

+

j

→

⁢

∂

⁡

a

∂

⁡

x

2

+

k

→

⁢

∂

⁡

a

∂

⁡

x

3

]

⁢

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

structure

∇

⁡

a

⁢

b

=

a

⁢

∇

⁡

b

+

b

⁢

∇

⁡

a

(14)

4

Second Proof

∇

⁡

(

A

→

⋅

B

→

)

=

[

(

A

→

⋅

∇

)

⁢

B

→

]

+

[

(

B

→

⋅

∇

)

⁢

A

→

]

+

[

A

→

×

(

∇

×

B

→

)

]

+

[

B

→

×

(

∇

×

A

→

)

]

(15)

This is the second vector identity to prove.

∇

⋅

(

A

×

B

)

=

[

B

⋅

(

∇

×

A

)

]

-

[

A

⋅

(

∇

×

B

)

]

(16)

∂

⁡

(

a

2

⁢

b

3

-

b

2

⁢

a

3

)

∂

⁡

x

1

(17)

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

∂

⁡

(

a

2

⁢

b

3

)

∂

⁡

x

1

-

∂

⁡

(

b

2

⁢

a

3

)

∂

⁡

x

1

(18)

Each of these must get

expanded

according to the product rule.

a

2

⁢

∂

⁡

b

3

∂

⁡

x

1

+

b

3

⁢

∂

⁡

a

2

∂

⁡

x

1

-

[

a

3

⁢

∂

⁡

b

2

∂

⁡

x

1

+

b

2

⁢

∂

⁡

a

3

∂

⁡

x

1

]

(19)

-

[

a

1

⁢

∂

⁡

b

3

∂

⁡

x

2

+

b

3

⁢

∂

⁡

a

1

∂

⁡

x

2

-

[

a

3

⁢

∂

⁡

b

1

∂

⁡

x

2

+

b

1

⁢

∂

⁡

a

3

∂

⁡

x

2

]

]

(20)

a

1

⁢

∂

⁡

b

2

∂

⁡

x

3

+

b

2

⁢

∂

⁡

a

1

∂

⁡

x

3

-

[

a

2

⁢

∂

⁡

b

1

∂

⁡

x

3

+

b

1

⁢

∂

⁡

a

2

∂

⁡

x

3

]

(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

conclusions

with much more alacrity.

𝐗

=

(

a

2

⁢

∂

⁡

b

1

3

b

3

⁢

∂

⁡

a

1

2

-

a

3

⁢

∂

⁡

b

1

2

-

b

2

⁢

∂

⁡

a

1

3

-

a

1

⁢

∂

⁡

b

2

3

-

b

3

⁢

∂

⁡

a

2

1

a

3

⁢

∂

⁡

b

2

1

b

1

⁢

∂

⁡

a

2

3

a

1

⁢

∂

⁡

b

3

2

b

2

⁢

∂

⁡

a

3

1

-

a

2

⁢

∂

⁡

b

3

1

-

b

1

⁢

∂

⁡

a

3

2

)

(22)

This

transformation

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

a

1

terms:

a

1

⁢

∂

⁡

b

3

2

-

a

1

⁢

∂

⁡

b

2

3

(23)

grabbing the

a

2

terms:

-

[

a

2

⁢

∂

⁡

b

3

1

-

a

2

⁢

∂

⁡

b

1

3

]

(24)

grabbing the

a

3

terms:

a

3

⁢

∂

⁡

b

2

1

-

a

3

⁢

∂

⁡

b

1

2

(25)

grabbing the

b

1

terms:

-

b

1

⁢

∂

⁡

a

3

2

+

b

1

⁢

∂

⁡

a

2

3

(26)

grabbing the

b

2

terms:

-

[

-

[

b

2

⁢

∂

⁡

a

3

1

-

b

2

⁢

∂

⁡

a

1

3

]

]

(27)

grabbing the

b

3

terms:

-

b

3

⁢

∂

⁡

a

2

1

+

b

3

⁢

∂

⁡

a

1

2

(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 product

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

a

1

terms:

(

a

1

)

⁢

(

∂

⁡

b

2

∂

⁡

x

3

-

∂

⁡

b

3

∂

⁡

x

2

)

(29)

grabbing the

a

2

terms:

(

a

2

)

⁢

(

-

[

∂

⁡

b

1

∂

⁡

x

3

-

∂

⁡

b

3

∂

⁡

x

1

]

)

(30)

grabbing the

a

3

terms:

(

a

3

)

⁢

(

∂

⁡

b

1

∂

⁡

x

2

-

∂

⁡

b

2

∂

⁡

x

1

)

(31)

Equation 19 cannot contain

x

1

(”x”

component

)
Equation 20 cannot contain

x

2

(”y” component) and must be negative

cofactor

Equation 21 cannot contain

x

3

(”z”component)
gluing these together as a vector we see after factoring out another -1,

-

A

⋅

(

∇

×

B

)

(32)

working through the

similar

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

Jacobian

can be found in Ramo and Whinnery, ”Fields and Waves in Modern Radio”

5

Third Proof

∇

×

(

∇

×

A

)

=

∇

⁡

(

∇

⋅

A

)

-

∇

2

⁡

A

(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

=

[

3

2

-

2

3

1

]

⁢

[

3

1

-

1

3

-

2

]

⁢

[

2

1

-

1

2

3

]

(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

)

=

[

[

2

1

-

1

2

3

]

2

-

[

3

1

-

1

3

-

2

]

3

1

]

⁢

[

[

2

1

-

1

2

3

]

1

-

[

3

2

-

2

3

1

]

3

-

2

]

⁢

[

[

3

1

-

1

3

-

2

]

1

-

[

3

2

-

2

3

1

]

2

3

]

(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 derivative

with respect to the ”numerator”
The

operation

of taking the partial derivative first with respect to the first dimension and them with respect to the second dimension, is notated by

∂

⁡

[

∂

⁡

f

2

∂

⁡

x

1

]

∂

⁡

x

2

=

∂

2

⁡

f

2

∂

⁡

x

2

⁢

∂

⁡

x

1

(37)

∂

⁡

[

∂

⁡

f

1

∂

⁡

x

2

]

∂

⁡

x

2

=

∂

2

⁡

f

1

∂

⁡

x

2

2

(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

)

=

[

[

∂

2

⁡

f

2

∂

⁡

x

2

⁢

∂

⁡

x

1

-

∂

2

⁡

f

1

∂

⁡

x

2

2

]

-

[

3

1

-

1

3

-

2

]

3

1

]

⁢

[

[

2

1

-

1

2

3

]

1

-

[

3

2

-

2

3

1

]

3

-

2

]

⁢

[

[

3

1

-

1

3

-

2

]

1

-

[

3

2

-

2

3

1

]

2

3

]

(39)

What is seen is a

pattern

emerging were some of the items in this

collection

are

positive

and some are negative. None of the positive items are of the form:

∂

2

⁡

f

1

∂

⁡

x

2

2

(40)

In fact these all correspond to the

expansion

of the term:

-

∇

2

⁡

A

(41)

Given the

products

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

∇

2

⁡

A

(42)

is inferred to be

∇

2

A

=

(

→

i

)

∇

2

C

1

+

(

→

j

)

∇

2

C

2

+

(

→

k

)

∇

2

C

3

(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:

∂

2

⁡

f

1

∂

⁡

x

1

2

(44)

And so we must resort to trickery using an

[

(

→

i

)

∂

2

⁡

C

1

∂

⁡

x

1

2

+

(

→

j

)

∂

2

⁡

C

2

∂

⁡

x

2

2

+

(

→

k

)

∂

2

⁡

C

3

∂

⁡

x

3

2

]

-

[

(

→

i

)

∂

2

⁡

C

1

∂

⁡

x

1

2

+

(

→

j

)

∂

2

⁡

C

2

∂

⁡

x

2

2

+

(

→

k

)

∂

2

⁡

C

3

∂

⁡

x

3

2

]

(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)

References

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

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