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# GEOMETRIC ALGEBRAS IN E.M. PowerPoint PPT Presentation

GEOMETRIC ALGEBRAS IN E.M. Manuel Berrondo Brigham Young University Provo, UT, 84097 berrondo@byu.edu. Physical Applications:. Electro-magnetostatics Dispersion and diffraction E.M. Examples of geometric algebras . Complex Numbers G 0,1 = C Hamilton Quaternions G 0,2 = H

GEOMETRIC ALGEBRAS IN E.M.

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### GEOMETRIC ALGEBRAS IN E.M.

Manuel Berrondo

Brigham Young University

Provo, UT, 84097

berrondo@byu.edu

### Physical Applications:

• Electro-magnetostatics

• Dispersion and diffraction E.M.

### Examples of geometric algebras

• Complex Numbers G0,1= C

• Hamilton Quaternions G0,2= H

• Pauli Algebra G3

• Dirac Algebra G1,3

Extension of the vector space

• inverse of a vector:

• reflections, rotations, Lorentz transform.

• integrals: Cauchy (≥ 2 d), Stokes’ theorem

Based on the idea of MULTIVECTORS:

and including lengths and angles:

+

closure

commutativity

associativity

zero

negative

closure

commutativity

associativity

unit

reciprocal

and + distributivity

### Algebraic Properties of R

Vector spaces :linear combinations

### Algebras

• include metrics:

• define geometricproduct

FIRST STEP: Inverse of a vector a≠ 0

a-1

0

2

a

1

3d:ê1 ê2 ê3

êi •êk = i k

êi-1= êi

Euclidian

4d: 0 1 2 3

μ•ν= gμν (1,-1,-1,-1)

Minkowski

### Orthonormal basis

Example: (ê1+ ê2)-1 = (ê1+ ê2)/2

1

### Electrostatics: method of images

w

r

- q

ql

q

q

Plane: charge -q atcê1, image (-q) at -cê1

Sphere radius a, charge q atbê1, find image (?)

Choose scales for r and w:

### SECOND STEP: EULER FORMULAS

In 3-d:

given that

In general, rotating A about θ k:

q

B = B k

### THIRD STEP: ANTISYM. PRODUCT

b

a

sweep

b

sweep

a

• anticommutative

• associative

• distributive

• absolute value => area

### Geometric or matrix product

• non commutative

• associative

• distributive

• closure:

• extend vector space

• unit = 1

• inverse (conditional)

### Examples of Clifford algebras

G2:

R

R2

I R

C = even algebra = spinors

I

e2

• z

a

C isomorphic to R2

1

e1

• z*

z = ê1a

ê1z = a

Reflection: z  z*

a a’ = ê1z* = ê1a ê1

In general, a’ = n a n, with n2 = 1

a’

e2

φ

Rotations in R2

Euler:

a

e1

### Inverse of multivector in G2Conjugate:

Generalizing

G3

R3

iR

R

H = R + i R3=

= even algebra = spinors

iR3

### Geometric product in G3 :

Rotations:

e P/2 generates rotations with respect to

planedefined by bivector P

### Inverse of multivector in G3defining Clifford conjugate:

• Generalizing:

### Geometric Calculus (3d)

• is a vector differential operator acting on:

 (r) – scalar field

E(r) – vector field

### First order Green Functions

Euclidian spaces :

is solution of

### Maxwell’s Equation

• Maxwell’s multivector F = E +i c B

• current density paravector J = (ε0c)-1(cρ + j)

• Maxwell’s equation:

### Electrostatics

Gauss’s law. Solution using first

order Green’s function:

Explicitly:

### Magnetostatics

Ampère’s law. Solution using first

order Green’s function:

Explicitly:

### Fundamental Theorem of Calculus

• Euclidian case :

dkx = oriented volume, e.g. ê1 ê2 ê3 = i

dk-1x = oriented surface, e.g. ê1 ê2 = iê3

F (x)– multivector field, V – boundary of V

### Example: divergence theorem

• d3x = i d, where d = |d3 x|

• d2 x = i nda , where da = |d2 x|

• F = v(r)

### Green’s Theorem (Euclidian case )

• where the first order Green function is:

### Cauchy’s theorem in n dimensions

• particular case : F = f y f = 0

where f (x) is a monogenic multivectorial

field: f = 0

### Inverse of differential paravectors:

Helmholtz:

Spherical wave (outgoing or incoming)

### Electromagnetic diffraction

• First order Helmholtz equation:

• Exact Huygens’ principle:

In the homogeneous case J = 0

### Special relativity and paravectors in G3

• Paravector p = p0 + p

Examples of paravectors:

### Lorentz transformations

Transform the paravector

p = p0 + p = p0 + p=+ p┴ into:

B p B = B2 (p0 + p=) + p┴

R p R† = p0 + p= + R2 p┴