How do the three fundamental theorems of calculus look like?

general:

What does the Helmholz Theorem tells us?

And what general rules apply for potentials?

With the boundary condition “fields go to zero at infinity” the field is uniquely determined by its divergence and curl.

Curl-less (or “irrotational”) fields -> Theorem 1:

What do you know about field lines, flux and the Gauss law?

What is the curl of E?

Zero due to superposition principle one can apply Stockes’ theorem for all individual point charges and realise that the integral around a closed path is zero.

Electric potential, poisson and laplace equation?

Units of Potential. In our units, force is measured in newtons and charge in coulombs, so electric fields are in newtons per coulomb. Accordingly, potential is newton-meters per coulomb, or joules per coulomb. A joule per coulomb is a volt.

Laplace Equation in 1 and 3 dimenstions?

its solutions are the harmonic funcitons

Gauss's law in the presence of dielectrics?

it makes reference only to free charges, and free charge is the stuff we control. Bound charge comes along for the ride: when we put the free charge in place, a certain polarization automatically ensues, by the mechanisms of Sect. 4.1, and this polarization produces the bound charge. In a typical problem, therefore, we know ρ f , but we do not (initially) know ρb; Eq. 4.23 lets us go right to work with the information at hand. In particular, whenever the requisite symmetry is present, we can immediately calculate D by the standard Gauss’s law methods.

Definition of current and its link to the continuity equation?

Divergence and curl of B?

Magnetic vector potential?

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