By Edward J Rothwell; Michael J Cloud

Content material: Introductory options -- Notation, conventions, and symbology -- the sector idea of electromagnetics -- historic point of view -- Formalization of box idea -- The resources of the electromagnetic box -- Macroscopic electromagnetics -- inspired vs. secondary resources -- floor and line resource densities -- cost conservation -- Magnetic cost -- Maxwell's conception of electromagnetism -- the concept -- The Maxwell-Minkowski equations -- Connection to mechanics -- The well-posed nature of the concept -- strong point of options to Maxwell's equations -- Constitutive kinfolk -- Maxwell's equations in relocating frames -- box conversions below Galilean transformation -- box conversions below Lorentz transformation -- The Maxwell-Boffi equations -- Large-scale type of Maxwell's equations -- floor relocating with consistent pace -- relocating, deforming surfaces -- Large-scale kind of the Boffi equations -- the character of the 4 box amounts -- Maxwell's equations with magnetic assets -- Boundary (jump) stipulations -- Boundary stipulations throughout a desk bound, skinny resource layer -- Boundary stipulations throughout a desk bound layer of box discontinuity -- Boundary stipulations on the floor of an ideal conductor -- Boundary stipulations throughout a desk bound layer of box discontinuity utilizing similar resources -- Boundary stipulations throughout a relocating layer of box discontinuity -- basic theorems -- Linearity -- Duality -- Reciprocity -- Similitude -- Conservation theorems -- The wave nature of the electromagnetic box

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2 2 A simple example of a symmetric dyadic is the unit dyadic I¯ deﬁned by I¯ = ˆi1ˆi1 + ˆi2ˆi2 + ˆi3ˆi3 . This quantity often arises in the manipulation of dyadic equations, and satisﬁes A · I¯ = I¯ · A = A for any vector A. In matrix form I¯ is the identity matrix: 100 ¯ = 0 1 0. [I] 001 The components of a dyadic may be complex. 72) holds for any B. This requires that a¯ = a¯ . Taking the transpose we can write T a¯ = (¯a∗ )T = a¯ † where “†” stands for the conjugate-transpose operation.

This leads to three possibilities. If k x = k y = 0, we have the product solution ψ(x, y) = (ax x + bx )(a y y + b y ). 106) If k y is real and nonzero, then ψ(x, y) = (A x e−k y x + Bx ek y x )(A y e jk y y + B y e− jk y y ). 107) as ψ(x, y) = (A x sinh k y x + Bx cosh k y x)(A y sin k y y + B y cos k y y). ) If k x is real and nonzero we have ψ(x, y) = (A x sin k x x + Bx cos k x x)(A y sinh k x y + B y cosh k x y). 111) holding in the region 0 < x < L 1 , 0 < y < L 2 , −∞ < z < ∞, together with the boundary conditions V (0, y) = V1 , V (L 1 , y) = V2 , V (x, 0) = V3 , V (x, L 2 ) = V4 .

It is found that ψ is bounded at x = ±1 only if λ = n(n + 1) where n ≥ m is an integer. These λ are the eigenvalues of the Sturm–Liouville problem, and the corresponding ψn (x) are the eigenfunctions. 92) has two solutions known as associated Legendre functions. The solution bounded at both x = ±1 is the associated Legendre function of the ﬁrst kind, denoted Pnm (x). The second solution, unbounded at x = ±1, is the associated Legendre function of the second kind Q m n (x). 2 tabulates some properties of these functions.