Geometric Algebra and Calculus: A Deep Dive into Vector Derivatives and Maxwell Equations
Geometric Algebra
Dr Chris Doran
ARM Research
6. Geometric Calculus
The vector derivative
L6 S2
Define a vector operator that
returns the derivative in any
given direction by
Define a set of Euclidean
coordinates
This operator has the algebraic properties of a vector in a geometric
algebra, combined with the properties of a differential operator.
Basic Results
L6 S3
Extend the definition of the
dot and wedge product
The exterior derivative defined by the wedge
product is the
d
operator of differential forms
The exterior derivative applied twice
gives zero
Leibniz rule
Same for inner derivative
Where
Overdot notation very useful in practice
Two dimensions
L6 S4
Now suppose we define a ‘complex’ function
These are precisely the terms that
vanish for an analytic function – the
Cauchy-Riemann equations
Unification
The Cauchy-Riemann equations arise naturally
from the vector derivative in two dimensions.
Analytic functions
L6 S6
Any function that can be that can be
written as a function of
z
is analytic:
•
The CR equations are the same as saying a function is
independent of
z*
.
•
In 2D this guarantees we are left with a function of
z
only
•
Generates of solution of the more general equation
Three dimensions
L6 S7
Maxwell equations in vacuum
around sources and currents,
in natural units
Remove the curl term via
Find
All 4 of Maxwell’s
equations in 1!
Spacetime
L6 S8
The key differential operator
in spacetime physics
Form the relative split
Or
So
Recall Faraday bivector
So finally
Unification
The most compact formulation of
the Maxwell equations. Unifies all
four equations in one.
More than some symbolic trickery.
The vector derivative is an
invertible operator.
Directed integration
L6 S10
Start with a simple line
integral along a curve
Key concept here is the vector-
valued measure
More general form of line integral is
Surface integrals
L6 S11
Now consider a 2D surface embedded
in a larger space
Directed surface element
This extends naturally to higher dimensional surfaces
•
The surface element is a blade
•
It enters integrals via the
geometric product
Fundamental theorem
L6 S12
Left-sided version
Overdots show where the
vector derivative acts
Right-sided version
General result
L
is a multilinear
function
Divergence theorem
L6 S13
Set
Vector
Grade
n-1
Constant Grade
n
L
is a scalar-valued linear
function of
A
Scalar measure
The divergence
theorem
Cauchy integral formula
L6 S14
One of the most famous results in
19
th
century mathematics
Knowledge of an analytic function
around a curve is enough to learn
the value of the function at each
interior point
We want to understand this in
terms of Geometric Algebra
And extend it to arbitrary
dimensions!
Cauchy integral formula
L6 S15
Translate the various
terms into their GA
equivalents
Find the dependence
on the real axis drops
out of the integrand
Cauchy integral formula
L6 S16
Applying the
fundamental theorem of
geometric calculus
Scalar measure
Function is analytic
The Green’s function for the vector
derivative in the plane
Cauchy integral formula
L6 S17
1.
dz
encodes the tangent vector
2.
Complex numbers give a
geometric product
3.
The integrand includes the
Green’s function in 2D
4.
The I comes from the directed
volume element
Generalisation
L6 S18
This extends Cauchy’s integral formula to arbitrary dimensions
Unification
L6 S19
The Cauchy integral formula, the divergence theorem, Stoke’s
theorem, Green’s theorem etc. are all special cases of the
fundamental theorem of geometric calculus
Resources
L6 S20
geometry.mrao.cam.ac.uk
chris.doran@arm.com
cjld1@cam.ac.uk
@chrisjldoran
#geometricalgebra
github.com/ga
Explore the world of geometric algebra and calculus through topics such as vector derivatives, Cauchy-Riemann equations, Maxwell equations, and spacetime physics. Unify diverse mathematical concepts to gain insights into analytic functions, differential operators, and directed integration.
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Presentation Transcript
Geometric Algebra 6. Geometric Calculus Dr Chris Doran ARM Research
L6 S2 The vector derivative Define a vector operator that returns the derivative in any given direction by Define a set of Euclidean coordinates This operator has the algebraic properties of a vector in a geometric algebra, combined with the properties of a differential operator.
L6 S3 Basic Results Extend the definition of the dot and wedge product The exterior derivative defined by the wedge product is the d operator of differential forms The exterior derivative applied twice gives zero Leibniz rule Where Overdot notation very useful in practice Same for inner derivative
L6 S4 Two dimensions Now suppose we define a complex function These are precisely the terms that vanish for an analytic function the Cauchy-Riemann equations
Unification The Cauchy-Riemann equations arise naturally from the vector derivative in two dimensions.
L6 S6 Analytic functions Any function that can be that can be written as a function of z is analytic: The CR equations are the same as saying a function is independent of z*. In 2D this guarantees we are left with a function of z only Generates of solution of the more general equation
L6 S7 Three dimensions Maxwell equations in vacuum around sources and currents, in natural units Remove the curl term via Find All 4 of Maxwell s equations in 1!
L6 S8 Spacetime The key differential operator in spacetime physics Form the relative split Recall Faraday bivector Or So So finally
Unification The most compact formulation of the Maxwell equations. Unifies all four equations in one. More than some symbolic trickery. The vector derivative is an invertible operator.
L6 S10 Directed integration Start with a simple line integral along a curve Key concept here is the vector- valued measure More general form of line integral is
L6 S11 Surface integrals Now consider a 2D surface embedded in a larger space Directed surface element This extends naturally to higher dimensional surfaces The surface element is a blade It enters integrals via the geometric product
L6 S12 Fundamental theorem Overdots show where the vector derivative acts Left-sided version Right-sided version L is a multilinear function General result
L6 S13 Divergence theorem L is a scalar-valued linear function of A Set Vector Grade n-1 Constant Grade n Scalar measure The divergence theorem
L6 S14 Cauchy integral formula One of the most famous results in 19th century mathematics We want to understand this in terms of Geometric Algebra Knowledge of an analytic function around a curve is enough to learn the value of the function at each interior point And extend it to arbitrary dimensions!
L6 S15 Cauchy integral formula Translate the various terms into their GA equivalents Find the dependence on the real axis drops out of the integrand
L6 S16 Cauchy integral formula Applying the fundamental theorem of geometric calculus Scalar measure Function is analytic The Green s function for the vector derivative in the plane
L6 S17 Cauchy integral formula 1. 2. dz encodes the tangent vector Complex numbers give a geometric product The integrand includes the Green s function in 2D The I comes from the directed volume element 3. 4.
L6 S18 Generalisation This extends Cauchy s integral formula to arbitrary dimensions
L6 S19 Unification The Cauchy integral formula, the divergence theorem, Stoke s theorem, Green s theorem etc. are all special cases of the fundamental theorem of geometric calculus
L6 S20 Resources geometry.mrao.cam.ac.uk chris.doran@arm.com cjld1@cam.ac.uk @chrisjldoran #geometricalgebra github.com/ga
