Can Maxwell's Equations in differential form be viewed as equalities of measures?
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Answered by
1
No, maxwell's equation in differential form be viewed as equality is of measures.
This is because Maxwell has four differential equation in which they all varies
on the basis of the electromagnetism actually this has no mesaurments .
This is the reason why there are different qualities of measures.
This is because Maxwell has four differential equation in which they all varies
on the basis of the electromagnetism actually this has no mesaurments .
This is the reason why there are different qualities of measures.
Answered by
9
Yes this is possible
Suppose we choose to model the universe as a 3 dimensional flat Euclidean space R3R3 equipped with the standard topology and the Borel-sigma algebra. We can then define the charge distribution (ρρ) as a signed measures on (R3,σBorelR3,σBorel) which assigns to each measurable set a charge.
Similarly the divergence of a vector field (∇.A⃗ ∇.A→) which is defined as the flux density of a field may also be interpreted as a signed measure on (R3,σBorelR3,σBorel) which assigns to each measurable set a flux (of the field A⃗ A→ "through it").
Now Maxwell's first equation which reads - ∇.E⃗ =ρϵ0∇.E→=ρϵ0 can be interpreted as equating the flux measure and the charge measure. The Lebesgue integral of both sides reproduces the integral form of Maxwell's Equation.
Is this a valid point of view? I can't find too much literature online relating to this approach (perhaps because it's wrong?). Moreover if this is valid, this also justifies the use of the dirac delta function for point charges since the charge density for a point charge should reproduce the dirac measure.
Suppose we choose to model the universe as a 3 dimensional flat Euclidean space R3R3 equipped with the standard topology and the Borel-sigma algebra. We can then define the charge distribution (ρρ) as a signed measures on (R3,σBorelR3,σBorel) which assigns to each measurable set a charge.
Similarly the divergence of a vector field (∇.A⃗ ∇.A→) which is defined as the flux density of a field may also be interpreted as a signed measure on (R3,σBorelR3,σBorel) which assigns to each measurable set a flux (of the field A⃗ A→ "through it").
Now Maxwell's first equation which reads - ∇.E⃗ =ρϵ0∇.E→=ρϵ0 can be interpreted as equating the flux measure and the charge measure. The Lebesgue integral of both sides reproduces the integral form of Maxwell's Equation.
Is this a valid point of view? I can't find too much literature online relating to this approach (perhaps because it's wrong?). Moreover if this is valid, this also justifies the use of the dirac delta function for point charges since the charge density for a point charge should reproduce the dirac measure.
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