By Orestes N. Stavroudis
During this sequel to his publication, "The Optics of Rays, Wavefronts, and Caustics," Stavroudis not just covers his personal study effects, but in addition comprises more moderen advancements. The ebook is split into 3 elements, beginning with simple mathematical options which are extra utilized within the e-book. floor geometry is taken care of with classical arithmetic, whereas the second one half covers the k-function, discussing and fixing the eikonal equation in addition to Maxwell equations during this context. a last half on purposes comprises conclusions drawn or built within the first elements of the ebook, discussing such subject matters because the Cartesian oval, the fashionable Schiefspiegler, Huygen's precept, and Maxwell's version of Gauss' ideal lens.
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Additional info for The Mathematics Of Geometrical And Physical Optics
Recall that t=dP/ds = (dP/dφ)/(ds/dφ) so that t= 1 (0, cos v − , − sin v). 44) From t we can deduce the unit normal vector n and the curvature 1/ρ using Eqs. 44. First we calculate tv = − 1 − cos v (0, sin v, cos v − ). 8 The Ray Equation 25 From this and Eq. 47) and n= 1 (0, sin v, cos v − ). 48) From Eq. 10 we get b = t × n = (1, 0, 0). 49) The explanation for this is that the ellipse is a plane curve that we have located on the y, z-plane. Its b vector must therefore be a unit vector in the x direction.
This we call the condition for exactness; Eq. 22 is then said to be exact. It can happen that Eq. 22 is not exact but that there exists an integrating factor, µ so that µ(P dx + Qdy + Rdz) = 0 is then exact. Then ∂2V = µPy + µy P = µQx + µx Q, ∂x∂y ∂2V = µPz + µz P = µRx + µx R, ∂x∂z ∂2V = µQz + µz Q = µRy + µy R, ∂y∂z which can be rearranged into µ(Py − Qx ) = Qµx − P µy µ(Pz − Rx ) = Rµx − P µz µ(Qz − Ry ) = Rµy − Qµz . 30) which we will call the condition for integrability. In vector form it is H · (∇ × H) = 0.
6 The Weingarten Equations4 The deﬁnition of the second fundamental quantities given in Eq. 31) N = N · Pww = −Nw · Pw . 4 Dickson 1914, pp. 55-56, Korn and Korn 1968, pp. 55–56. 54 4 The Differential Geometry of Surfaces. Since N is a unit vector the derivatives of N2 must vanish so that N · Nv = N · Nw = 0. 32) Nw = Pv + δPw , where α, β, γ and δ are to be determined. The scalar product of these two equations with Pv and Pw yield αE + βF = −L, γE + δF = −M, αF + βG = −M, γF + δG = −N. 33) By solving these for the Greek letters and substituting the result back into the original equations in Eq.