Unified approach to the impulse response and green function in the circuit and field theory part II : multi–dimensional case

Abstract

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function δ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1-4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. In the previous part [5] the concept of the impulse response of linear systems was approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines). Here the phenomena in more dimensions (static and dynamic electromagnetic fields) are treated. It is shown that many formulas in the field theory, which are often postulated in an inductive way as results of the experiments, and therefore appear as “deux ex machina” effects, can be mathematically deduced from a few starting equations.

A note on ”a note on the magnetic vector potential”

“…As shown in [2] and [3] the inverse Fourier transforms of the last formulae yield the Green functions…”

“…3 Solution of equations for Green functions As shown in [2] and [3] the three-dimensional spatial direct and inverse Fourier transform is introduced by…”

Unified approach to the impulse response and green function in the circuit and field theory, part I: one–dimensional case

“…Reformulating (5) in terms of causal functions f c (t) = f (t)1(t), y c (t) = y(t)1(t) leads, due to (122) to the equation with the embodied initial conditions f (0) and…”

“…Notice the difference between the formulas (17) and (21) as well as between (5) and (20). Both are defined for the whole time axis t ∈ (−∞, ∞), but the latter formulate the solution in terms of causal functions f c (t), ie the solution f c (t) equal to zero for t < 0 , and to the true values f (t) for t > 0 , while the former gives f (t) for any t ∈ (−∞, ∞).…”

The Field Of A Step–Like Accelerated Point Charge: Liénard–Wiechert Potentials And Čerenkov Shock Waves Revisited

“…The solution of (9) can be easily obtained eg [5] in form of the spherical delta-function wave-front spreading in space with velocity c as a function of time,…”