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IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Exterior Algebra and the Maxwell-BoltzmannEquations
Joseph Ferrara
April 15, 2011
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Overview
Overview
1 The Space of p-Vectors
2 Exterior Differentiation
3 The Maxwell-Boltzmann Equations
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors
Let L be an n-dimensional vector space over Rα, β, γ, · · · ,∈ L
a,b,c,· · · ,∈ R
We choose a p ≤ n where each case defines a new space and call∧pL the space of p-vectors on L.
Elements in∧2 L :
∑i ai (αi ∧ βi )
Basic Properties of 2-Vectorsa1(α1 ∧ β) + a2(α2 ∧ β) = (a1α1 + a2α2) ∧ βb1(α ∧ β1) + b2(α ∧ β2) = α ∧ (b1β1 + b2β2)
α ∧ α = 0
α ∧ β = −(β ∧ α)
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors
Let L be an n-dimensional vector space over Rα, β, γ, · · · ,∈ L
a,b,c,· · · ,∈ RWe choose a p ≤ n where each case defines a new space and call∧
pL the space of p-vectors on L.
Elements in∧2 L :
∑i ai (αi ∧ βi )
Basic Properties of 2-Vectorsa1(α1 ∧ β) + a2(α2 ∧ β) = (a1α1 + a2α2) ∧ βb1(α ∧ β1) + b2(α ∧ β2) = α ∧ (b1β1 + b2β2)
α ∧ α = 0
α ∧ β = −(β ∧ α)
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors
Let L be an n-dimensional vector space over Rα, β, γ, · · · ,∈ L
a,b,c,· · · ,∈ RWe choose a p ≤ n where each case defines a new space and call∧
pL the space of p-vectors on L.
Elements in∧2 L :
∑i ai (αi ∧ βi )
Basic Properties of 2-Vectorsa1(α1 ∧ β) + a2(α2 ∧ β) = (a1α1 + a2α2) ∧ βb1(α ∧ β1) + b2(α ∧ β2) = α ∧ (b1β1 + b2β2)
α ∧ α = 0
α ∧ β = −(β ∧ α)
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors continued
Let Ψ ={σ1, σ2, · · · , σn
}be a basis of L.
Consider two vectors, α =∑n
i=1 aiσi and β =
∑nj=1 bjσ
j
Taking their exterior product yields,
α ∧ β =(∑n
i=1 aiσi)∧(∑n
j=1 bjσj)
=∑n
j=1
∑ni=1 aibj(σ
i ∧ σj)
= a1b1(σ1 ∧ σ1) + a1b2(σ1 ∧ σ2) + · · ·+ a1bn(σ1 ∧ σn)+ a2b1(σ2 ∧ σ1) + · · ·+ anb1(σn ∧ σ1)+ a2b2(σ2 ∧ σ2) + a2b3(σ2 ∧ σ3) + · · ·+ a2bn(σ2 ∧ σn)+ a3b2(σ3 ∧ σ2) + · · ·+ anb2(σn ∧ σ2)+ · · ·=∑n
i<j(aibj − ajbi )(σi ∧ σj).
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors continued
Let Ψ ={σ1, σ2, · · · , σn
}be a basis of L.
Consider two vectors, α =∑n
i=1 aiσi and β =
∑nj=1 bjσ
j
Taking their exterior product yields,
α ∧ β =(∑n
i=1 aiσi)∧(∑n
j=1 bjσj)
=∑n
j=1
∑ni=1 aibj(σ
i ∧ σj)
= a1b1(σ1 ∧ σ1) + a1b2(σ1 ∧ σ2) + · · ·+ a1bn(σ1 ∧ σn)+ a2b1(σ2 ∧ σ1) + · · ·+ anb1(σn ∧ σ1)+ a2b2(σ2 ∧ σ2) + a2b3(σ2 ∧ σ3) + · · ·+ a2bn(σ2 ∧ σn)+ a3b2(σ3 ∧ σ2) + · · ·+ anb2(σn ∧ σ2)+ · · ·=∑n
i<j(aibj − ajbi )(σi ∧ σj).
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors continued
Let Ψ ={σ1, σ2, · · · , σn
}be a basis of L.
Consider two vectors, α =∑n
i=1 aiσi and β =
∑nj=1 bjσ
j
Taking their exterior product yields,
α ∧ β =(∑n
i=1 aiσi)∧(∑n
j=1 bjσj)
=∑n
j=1
∑ni=1 aibj(σ
i ∧ σj)
= a1b1(σ1 ∧ σ1) + a1b2(σ1 ∧ σ2) + · · ·+ a1bn(σ1 ∧ σn)+ a2b1(σ2 ∧ σ1) + · · ·+ anb1(σn ∧ σ1)+ a2b2(σ2 ∧ σ2) + a2b3(σ2 ∧ σ3) + · · ·+ a2bn(σ2 ∧ σn)+ a3b2(σ3 ∧ σ2) + · · ·+ anb2(σn ∧ σ2)+ · · ·
=∑n
i<j(aibj − ajbi )(σi ∧ σj).
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of 2-Vectors continued
Let Ψ ={σ1, σ2, · · · , σn
}be a basis of L.
Consider two vectors, α =∑n
i=1 aiσi and β =
∑nj=1 bjσ
j
Taking their exterior product yields,
α ∧ β =(∑n
i=1 aiσi)∧(∑n
j=1 bjσj)
=∑n
j=1
∑ni=1 aibj(σ
i ∧ σj)
= a1b1(σ1 ∧ σ1) + a1b2(σ1 ∧ σ2) + · · ·+ a1bn(σ1 ∧ σn)+ a2b1(σ2 ∧ σ1) + · · ·+ anb1(σn ∧ σ1)+ a2b2(σ2 ∧ σ2) + a2b3(σ2 ∧ σ3) + · · ·+ a2bn(σ2 ∧ σn)+ a3b2(σ3 ∧ σ2) + · · ·+ anb2(σn ∧ σ2)+ · · ·=∑n
i<j(aibj − ajbi )(σi ∧ σj).
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of p-Vectors
The space of∧p L can be derived similarly.
Basic Properties of p-Vectors(aα + bβ) ∧ α2 · · · ∧ αp = a(α ∧ α2 ∧ · · · ∧ αp) + b(β ∧ α2 ∧ · · · ∧ αp)
α1 ∧ · · · ∧ αp = 0 if i 6= j , αi = αj
α1 ∧ · · · ∧ αp = απ(1) ∧ · · · ∧ απ(p) = sgn(π)(α1 ∧ α2 ∧ · · · ∧ αp)
ϕ ∈∧p L⇒ ϕ =
∑H aHσ
H whereH = {h1, h2, · · · , hp} and 1 5 h1 < h2 < · · · < hp 5 n
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
2-vectorsp-vectors
The Space of p-Vectors
The space of∧p L can be derived similarly.
Basic Properties of p-Vectors(aα + bβ) ∧ α2 · · · ∧ αp = a(α ∧ α2 ∧ · · · ∧ αp) + b(β ∧ α2 ∧ · · · ∧ αp)
α1 ∧ · · · ∧ αp = 0 if i 6= j , αi = αj
α1 ∧ · · · ∧ αp = απ(1) ∧ · · · ∧ απ(p) = sgn(π)(α1 ∧ α2 ∧ · · · ∧ αp)
ϕ ∈∧p L⇒ ϕ =
∑H aHσ
H whereH = {h1, h2, · · · , hp} and 1 5 h1 < h2 < · · · < hp 5 n
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
Exterior Differentiation
Properties of the Exterior Derivatived(ω + η) = dω + dη
d(λ ∧ µ) = dλ ∧ µ+ (−1)(degλ)(λ ∧ dµ)
∀ω, d(dω) = 0
∀f , df =∑
i∂f∂x i dx
i
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,
d(dω) = d(∑
i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Proof of d(dω) = 0
∀ω, d(dω) = 0
Proof. Let ω = aHdxH where aH(x1, x2, · · · , xn) is smooth and
differentiable as often as we like. Applying d to ω we obtain,d(dω) = d
(∑i∂aH∂x i dx
idxH)
=∑
i,j∂2aH∂x i∂x j dx
jdx idxH
= 12
∑i,j 2 ∂2aH
∂x i∂x j dxjdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j + ∂2aH
∂x i∂x j
)dx jdx idxH
= 12
∑i,j
(∂2aH∂x i∂x j − ∂2aH
∂x j∂x i
)dx jdx idxH
Since we assumed that all functions aH are smooth we have equality ofmixed partial derivatives, and thus conclude that ∀ω, d(dω) = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
The Maxwell-Boltzmann Equations
∇ · B = 0∇× E = −∂B
∂t∇ · E = ρ
∇× B− ∂E∂t = J
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
Magnetic Field
Traditional: B=(Bx ,By ,Bz) = Bx i + By j + BzkExterior : B = Bxdydz + Bydzdx + Bzdxdy
Electric Field
Traditional: E=(Ex ,Ey ,Ez) = Ex i + Ey j + EzkExterior : E = Exdx + Eydy + Ezdz
Unified EMF
α = B + E ∧ dt = B + Edt= Bxdydz + Bydzdx + Bzdxdy + (Exdx + Eydy + Ezdz) dt
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
Magnetic Field
Traditional: B=(Bx ,By ,Bz) = Bx i + By j + BzkExterior : B = Bxdydz + Bydzdx + Bzdxdy
Electric Field
Traditional: E=(Ex ,Ey ,Ez) = Ex i + Ey j + EzkExterior : E = Exdx + Eydy + Ezdz
Unified EMF
α = B + E ∧ dt = B + Edt= Bxdydz + Bydzdx + Bzdxdy + (Exdx + Eydy + Ezdz) dt
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
Magnetic Field
Traditional: B=(Bx ,By ,Bz) = Bx i + By j + BzkExterior : B = Bxdydz + Bydzdx + Bzdxdy
Electric Field
Traditional: E=(Ex ,Ey ,Ez) = Ex i + Ey j + EzkExterior : E = Exdx + Eydy + Ezdz
Unified EMF
α = B + E ∧ dt = B + Edt= Bxdydz + Bydzdx + Bzdxdy + (Exdx + Eydy + Ezdz) dt
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
We want to find the form of the exterior derivative on a generalspacetime differential, ω.
Let ω = ωHdxH with H = {h1, h2, · · · , hp} where ωH is any function
of spacetime.
Applying d to ω yields,dω = dωHdx
H
= ∂ωH
∂x dxdxH + ∂ωH
∂y dydxH + ∂ωH
∂z dzdxH + ∂ωH
∂t dtdxH
=∑3
i=1∂ωH
∂x i dxidxH + ∂ωH
∂t dtdxH
Thus the general form is as follows,
dω = d ′ω + ∂ω∂t dt
We will use H = {dx , dy , dz , dt}.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
We want to find the form of the exterior derivative on a generalspacetime differential, ω.
Let ω = ωHdxH with H = {h1, h2, · · · , hp} where ωH is any function
of spacetime.
Applying d to ω yields,dω = dωHdx
H
= ∂ωH
∂x dxdxH + ∂ωH
∂y dydxH + ∂ωH
∂z dzdxH + ∂ωH
∂t dtdxH
=∑3
i=1∂ωH
∂x i dxidxH + ∂ωH
∂t dtdxH
Thus the general form is as follows,
dω = d ′ω + ∂ω∂t dt
We will use H = {dx , dy , dz , dt}.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
EM EquationsGeneral Spacetime Differentials
We want to find the form of the exterior derivative on a generalspacetime differential, ω.
Let ω = ωHdxH with H = {h1, h2, · · · , hp} where ωH is any function
of spacetime.
Applying d to ω yields,dω = dωHdx
H
= ∂ωH
∂x dxdxH + ∂ωH
∂y dydxH + ∂ωH
∂z dzdxH + ∂ωH
∂t dtdxH
=∑3
i=1∂ωH
∂x i dxidxH + ∂ωH
∂t dtdxH
Thus the general form is as follows,
dω = d ′ω + ∂ω∂t dt
We will use H = {dx , dy , dz , dt}.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
∇ · B = 0
∇× E = −∂B∂t
α = B + Edt
Let’s consider the exterior derivative on α,
dα = d ′α + ∂α∂t dt
= d ′ (B + Edt) + ∂(B+Edt)∂t dt
= d ′B + d ′Edt + ∂B∂t dt + ∂E
∂t dt ∧ dt
= d ′B +[d ′E + ∂B
∂t
]dt
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
∇ · B = 0
∇× E = −∂B∂t
α = B + Edt
Let’s consider the exterior derivative on α,
dα = d ′α + ∂α∂t dt
= d ′ (B + Edt) + ∂(B+Edt)∂t dt
= d ′B + d ′Edt + ∂B∂t dt + ∂E
∂t dt ∧ dt
= d ′B +[d ′E + ∂B
∂t
]dt
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Setting dα = 0 yields the following two equations,{d ′B = 0
d ′E + ∂B∂t = 0
Notice the similarities!
d′B = 0 ⇔ ∇ · B = 0
d′E + ∂B∂t = 0 ⇔ ∇× E + ∂B
∂t = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Setting dα = 0 yields the following two equations,{d ′B = 0
d ′E + ∂B∂t = 0
Notice the similarities!
d′B = 0 ⇔ ∇ · B = 0
d′E + ∂B∂t = 0 ⇔ ∇× E + ∂B
∂t = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Let’s first consider d ′B = 0. Then,
d′B = d ′ (Bxdydz + Bydzdx + Bzdxdy)= d ′Bxdydz + d ′Bydzdx + d ′Bzdxdy
= ∂Bx
∂x dxdydz +∂By
∂y dydzdx + ∂Bz
∂z dzdxdy
=(∂Bx
∂x +∂By
∂y + ∂Bz
∂z
)dxdydz
0 = ∂Bx
∂x +∂By
∂y + ∂Bz
∂z
=(∂∂x ,
∂∂y ,
∂∂z
)· (Bx ,By ,Bz)
= ∇ · B
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Let’s first consider d ′B = 0. Then,
d′B = d ′ (Bxdydz + Bydzdx + Bzdxdy)= d ′Bxdydz + d ′Bydzdx + d ′Bzdxdy
= ∂Bx
∂x dxdydz +∂By
∂y dydzdx + ∂Bz
∂z dzdxdy
=(∂Bx
∂x +∂By
∂y + ∂Bz
∂z
)dxdydz
0 = ∂Bx
∂x +∂By
∂y + ∂Bz
∂z
=(∂∂x ,
∂∂y ,
∂∂z
)· (Bx ,By ,Bz)
= ∇ · B
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
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Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Let’s first consider d ′B = 0. Then,
d′B = d ′ (Bxdydz + Bydzdx + Bzdxdy)= d ′Bxdydz + d ′Bydzdx + d ′Bzdxdy
= ∂Bx
∂x dxdydz +∂By
∂y dydzdx + ∂Bz
∂z dzdxdy
=(∂Bx
∂x +∂By
∂y + ∂Bz
∂z
)dxdydz
0 = ∂Bx
∂x +∂By
∂y + ∂Bz
∂z
=(∂∂x ,
∂∂y ,
∂∂z
)· (Bx ,By ,Bz)
= ∇ · B
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Let’s first consider d ′B = 0. Then,
d′B = d ′ (Bxdydz + Bydzdx + Bzdxdy)= d ′Bxdydz + d ′Bydzdx + d ′Bzdxdy
= ∂Bx
∂x dxdydz +∂By
∂y dydzdx + ∂Bz
∂z dzdxdy
=(∂Bx
∂x +∂By
∂y + ∂Bz
∂z
)dxdydz
0 = ∂Bx
∂x +∂By
∂y + ∂Bz
∂z
=(∂∂x ,
∂∂y ,
∂∂z
)· (Bx ,By ,Bz)
= ∇ · B
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Secondly we consider,d ′E + ∂B
∂t =
d ′ (Exdx + Eydy + Ezdz) + ∂∂t (Bxdydz + Bydzdx + Bzdxdy)
Splitting into two parts for convenience: d ′E and ∂B∂t
First,
d′E = d ′ (Exdx + Eydy + Ezdz)= d ′Exdx + d ′Eydy + d ′Ezdz
= ∂Ex
∂y dydx + ∂Ex
∂z dzdx +∂Ey
∂x dxdy
+∂Ey
∂z dzdy + ∂Ez
∂x dxdz + ∂Ez
∂y dydz
=(∂Ey
∂x −∂Ex
∂y
)dxdy +
(∂Ez
∂y −∂Ey
∂z
)dydz +
(∂Ex
∂z −∂Ez
∂x
)dzdx
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Secondly we consider,d ′E + ∂B
∂t =
d ′ (Exdx + Eydy + Ezdz) + ∂∂t (Bxdydz + Bydzdx + Bzdxdy)
Splitting into two parts for convenience: d ′E and ∂B∂t
First,
d′E = d ′ (Exdx + Eydy + Ezdz)
= d ′Exdx + d ′Eydy + d ′Ezdz
= ∂Ex
∂y dydx + ∂Ex
∂z dzdx +∂Ey
∂x dxdy
+∂Ey
∂z dzdy + ∂Ez
∂x dxdz + ∂Ez
∂y dydz
=(∂Ey
∂x −∂Ex
∂y
)dxdy +
(∂Ez
∂y −∂Ey
∂z
)dydz +
(∂Ex
∂z −∂Ez
∂x
)dzdx
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Secondly we consider,d ′E + ∂B
∂t =
d ′ (Exdx + Eydy + Ezdz) + ∂∂t (Bxdydz + Bydzdx + Bzdxdy)
Splitting into two parts for convenience: d ′E and ∂B∂t
First,
d′E = d ′ (Exdx + Eydy + Ezdz)= d ′Exdx + d ′Eydy + d ′Ezdz
= ∂Ex
∂y dydx + ∂Ex
∂z dzdx +∂Ey
∂x dxdy
+∂Ey
∂z dzdy + ∂Ez
∂x dxdz + ∂Ez
∂y dydz
=(∂Ey
∂x −∂Ex
∂y
)dxdy +
(∂Ez
∂y −∂Ey
∂z
)dydz +
(∂Ex
∂z −∂Ez
∂x
)dzdx
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Secondly we consider,d ′E + ∂B
∂t =
d ′ (Exdx + Eydy + Ezdz) + ∂∂t (Bxdydz + Bydzdx + Bzdxdy)
Splitting into two parts for convenience: d ′E and ∂B∂t
First,
d′E = d ′ (Exdx + Eydy + Ezdz)= d ′Exdx + d ′Eydy + d ′Ezdz
= ∂Ex
∂y dydx + ∂Ex
∂z dzdx +∂Ey
∂x dxdy
+∂Ey
∂z dzdy + ∂Ez
∂x dxdz + ∂Ez
∂y dydz
=(∂Ey
∂x −∂Ex
∂y
)dxdy +
(∂Ez
∂y −∂Ey
∂z
)dydz +
(∂Ex
∂z −∂Ez
∂x
)dzdx
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
Secondly we consider,d ′E + ∂B
∂t =
d ′ (Exdx + Eydy + Ezdz) + ∂∂t (Bxdydz + Bydzdx + Bzdxdy)
Splitting into two parts for convenience: d ′E and ∂B∂t
First,
d′E = d ′ (Exdx + Eydy + Ezdz)= d ′Exdx + d ′Eydy + d ′Ezdz
= ∂Ex
∂y dydx + ∂Ex
∂z dzdx +∂Ey
∂x dxdy
+∂Ey
∂z dzdy + ∂Ez
∂x dxdz + ∂Ez
∂y dydz
=(∂Ey
∂x −∂Ex
∂y
)dxdy +
(∂Ez
∂y −∂Ey
∂z
)dydz +
(∂Ex
∂z −∂Ez
∂x
)dzdx
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
∂B∂t = ∂
∂t (Bxdydz + Bydzdx + Bzdxdy)
=∂Bx
∂t dydz +∂By
∂t dzdx + ∂Bz
∂t dxdy
Summing yields,
d′E + ∂B∂t =
(∂Ey
∂x −∂Ex
∂y + ∂Bz
∂t
)dxdy +
(∂Ez
∂y −∂Ey
∂z + ∂Bx
∂t
)dydz
+(∂Ex
∂z −∂Ez
∂x +∂By
∂t
)dzdx
We set d ′E + ∂B∂t equal to zero which yields the following three
equations, ∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
∂B∂t = ∂
∂t (Bxdydz + Bydzdx + Bzdxdy)
=∂Bx
∂t dydz +∂By
∂t dzdx + ∂Bz
∂t dxdy
Summing yields,
d′E + ∂B∂t =
(∂Ey
∂x −∂Ex
∂y + ∂Bz
∂t
)dxdy +
(∂Ez
∂y −∂Ey
∂z + ∂Bx
∂t
)dydz
+(∂Ex
∂z −∂Ez
∂x +∂By
∂t
)dzdx
We set d ′E + ∂B∂t equal to zero which yields the following three
equations, ∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
∂B∂t = ∂
∂t (Bxdydz + Bydzdx + Bzdxdy)
=∂Bx
∂t dydz +∂By
∂t dzdx + ∂Bz
∂t dxdy
Summing yields,
d′E + ∂B∂t =
(∂Ey
∂x −∂Ex
∂y + ∂Bz
∂t
)dxdy +
(∂Ez
∂y −∂Ey
∂z + ∂Bx
∂t
)dydz
+(∂Ex
∂z −∂Ez
∂x +∂By
∂t
)dzdx
We set d ′E + ∂B∂t equal to zero which yields the following three
equations, ∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
These are the equations given by ∇× E. Let’s check it.
∇×E =
∣∣∣∣∣∣i j k∂∂x
∂∂y
∂∂z
Ex Ey Ez
∣∣∣∣∣∣ = i
∣∣∣∣ ∂∂y
∂∂z
Ey Ez
∣∣∣∣− j
∣∣∣∣ ∂∂x
∂∂z
Ex Ez
∣∣∣∣+ k
∣∣∣∣ ∂∂x
∂∂y
Ex Ey
∣∣∣∣=(∂Ez
∂y −∂Ey
∂z
)i +(∂Ex
∂z −∂Ez
∂x
)j +(∂Ey
∂x −∂Ex
∂y
)k
Furthermore,
−∂Bdt = −∂Bx
∂t i− ∂By
∂t j− ∂Bz
∂t k
Equating the components of these two equations yields∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
These are exactly the equations we achieved through the exterioralgebra method.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
These are the equations given by ∇× E. Let’s check it.
∇×E =
∣∣∣∣∣∣i j k∂∂x
∂∂y
∂∂z
Ex Ey Ez
∣∣∣∣∣∣ = i
∣∣∣∣ ∂∂y
∂∂z
Ey Ez
∣∣∣∣− j
∣∣∣∣ ∂∂x
∂∂z
Ex Ez
∣∣∣∣+ k
∣∣∣∣ ∂∂x
∂∂y
Ex Ey
∣∣∣∣=(∂Ez
∂y −∂Ey
∂z
)i +(∂Ex
∂z −∂Ez
∂x
)j +(∂Ey
∂x −∂Ex
∂y
)k
Furthermore,
−∂Bdt = −∂Bx
∂t i− ∂By
∂t j− ∂Bz
∂t k
Equating the components of these two equations yields∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
These are exactly the equations we achieved through the exterioralgebra method.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
These are the equations given by ∇× E. Let’s check it.
∇×E =
∣∣∣∣∣∣i j k∂∂x
∂∂y
∂∂z
Ex Ey Ez
∣∣∣∣∣∣ = i
∣∣∣∣ ∂∂y
∂∂z
Ey Ez
∣∣∣∣− j
∣∣∣∣ ∂∂x
∂∂z
Ex Ez
∣∣∣∣+ k
∣∣∣∣ ∂∂x
∂∂y
Ex Ey
∣∣∣∣=(∂Ez
∂y −∂Ey
∂z
)i +(∂Ex
∂z −∂Ez
∂x
)j +(∂Ey
∂x −∂Ex
∂y
)k
Furthermore,
−∂Bdt = −∂Bx
∂t i− ∂By
∂t j− ∂Bz
∂t k
Equating the components of these two equations yields∂Ey
∂x −∂Ex
∂y = −∂Bz
∂t∂Ez
∂y −∂Ey
∂z = −∂Bx
∂t∂Ex
∂z −∂Ez
∂x = −∂By
∂t
These are exactly the equations we achieved through the exterioralgebra method.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
d′B = 0d′E + ∂B
∂t= 0
∂B∂t
Traditional vector check
We conclude that the two Homogeneous Maxwell Equations,
∇ · B = 0
∇× E = −∂B∂t
are both contained in dα = 0 .
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Inhomogeneous ArgumentReferences
Argument for inhomogeneous equations.
∇ · B = 0∇× E = −∂B
∂t∇ · E = ρ
∇× B− ∂E∂t = J
The equation is ?d ? α = 0
dα = 0
?d ? α = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Inhomogeneous ArgumentReferences
Argument for inhomogeneous equations.
∇ · B = 0∇× E = −∂B
∂t∇ · E = ρ
∇× B− ∂E∂t = J
The equation is ?d ? α = 0
dα = 0
?d ? α = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Inhomogeneous ArgumentReferences
Argument for inhomogeneous equations.
∇ · B = 0∇× E = −∂B
∂t∇ · E = ρ
∇× B− ∂E∂t = J
The equation is ?d ? α = 0
dα = 0
?d ? α = 0
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations
IntroductionThe Space of p-VectorsExterior Differentiation
The Maxwell-Boltzmann EquationsThe Homogeneous Equations
Conclusion
Inhomogeneous ArgumentReferences
1 H. Flanders, Differential Forms with Applications to the PhysicalSciences, (1989)
2 J. E. Marsden, and A. J. Tromba, Vector Calculus, Fifth Edition(2003)
3 T. Watson, Introduction to the Hodge Star Operator: DifferentialForms Special Presentation, (2005)
4 S. Owerre, Maxwell’s Equations in Terms of Differential Forms,(2010)
Thank you.
Joseph Ferrara Exterior Algebra and the Maxwell-Boltzmann Equations