Appendix A Vector Analysis

Transcription

1 Appendix A Vector Analysis A.1 Orthogonal Coordinate Systems A.1.1 Cartesian (Rectangular Coordinate System The unit vectors are denoted by x, ŷ, ẑ in the Cartesian system. By convention, ( x, ŷ, ẑ triplet form a right-handed system and obey the following cyclic relations Differential length is defined as x ŷ = ẑ, ẑ x = ŷ, ŷ ẑ = x (A.1 dl = dx x + dy ŷ + dz ẑ (A. Differential surface areas with normal vectors x, ŷ, and ẑ are respectively defined as ds x = dydz x ds y = dx dz ŷ ds z = dx dy ẑ (A.3 The differential volume element is dv = dx dydz (A.4 A.1. Cylindrical Coordinate System The unit vectors are denoted by ρ, φ, ẑ in the cylindrical system. By convention, ( ρ, φ, ẑ triplet form a right-handed system and obey the following cyclic relations Springer International Publishing Switzerland 015 K. Barkeshli, Advanced Electromagnetics and Scattering Theory, DOI /

2 346 Appendix A: Vector Analysis Differential length is defined as ρ φ = ẑ, ẑ ρ = φ, φ ẑ = ρ (A.5 dl = dρ ρ + ρ dφ φ + dz ẑ (A.6 Differential surface areas with normal vectors ρ, φ, and ẑ are respectively defined as ds ρ = ρ dφ dz ρ ds φ = dρ dz φ ds z = ρ dρ dφ ẑ (A.7 The differential volume element is dv = ρ dρ dφ dz (A.8 A.1.3 Spherical Coordinate System The unit vectors are denoted by r, θ, φ in the spherical system. By convention, ( r, θ, φ triplet form a right-handed system and obey the following cyclic relations Differential length is defined as r θ = φ, φ r = θ, θ φ = r (A.9 dl = dr r + rdθ θ + r sin θ dφ φ (A.10 Differential surface areas with normal vectors r, θ, and φ are respectively defined as ds r = r sin θ dθ dφ r ds θ = r sin θ dr dφ θ ds φ = rdrdθ φ (A.11 The differential volume element is dv = r sin θ dr dθ dφ (A.1

3 Appendix A: Vector Analysis 347 A. Coordinate Transformations Consider a point P whose coordinates in Cartesian, cylindrical, and spherical coordinate systems, respectively are (x, y, z, (ρ, φ, z, and (r,θ,φ. The Cartesian and cylindrical coordinates are related by x = ρ cos φ y = ρ sin φ (A.13 The Cartesian and spherical coordinates are related by x = r sin θ cos φ y = r sin θ sin φ z = r cos θ (A.14 The spherical and cylindrical coordinates are related by ρ = r sin θ z = r cos θ (A.15 A.3 Vector Transformations Consider a three-dimensional vector F. We can express F in terms of the unit vectors in Cartesian, cylindrical, and spherical systems respectively as: F = F x x + F y ŷ + F z ẑ F = F ρ ρ + F φ φ + F z ẑ F = F r r + F θ θ + F φ φ (A.16 A.3.1 Cartesian-Cylindrical Vector Transformations Cartesian to cylindrical components are related by F ρ = F x cos φ + F y sin φ F φ = F x sin φ + F y cos φ (A.17 F z = F z

4 348 Appendix A: Vector Analysis Cylindrical to Cartesian components are related by F x = F ρ cos φ F φ sin φ F y = F ρ sin φ + F φ cos φ (A.18 F z = F z A.3. Cartesian-Spherical Vector Transformations Cartesian to spherical components are related by F r = F x sin θ cos φ + F y sin θ sin φ + F z cos θ F θ = F x cos θ cos φ + F y cos θ sin φ F z sin θ F φ = F x sin φ + F y cos φ (A.19 Spherical to Cartesian components are related by F x = F r sin θ cos φ + F θ cos θ cos φ F φ sin φ F y = F r sin θ sin φ + F θ cos θ sin φ F φ cos φ F z = F r cos θ F θ sin θ (A.0 A.3.3 Cylindrical-Spherical Vector Transformations Cylindrical to spherical components are related by F r = F ρ sin θ + F z cos θ F θ = F ρ cos θ F z sin θ (A.1 F φ = F φ Spherical to cylindrical components are related by F ρ = F r sin θ + F θ cos θ F φ = F φ F z = F r cos θ F θ sin θ (A.

5 Appendix B Vector Calculus In the following, scalar functions are denoted by lower-case letters and vector functions are denoted by upper-case bold letters. The notations for vector components are same as shown in (A.17. B.1 Differential Operators B.1.1 Cartesian System f = f x x + f y ŷ + f z ẑ (B.1 F = F x x + F y y + F z z (B. F = ( Fz y F ( y Fx x + z z F ( z Fy ŷ + x x F x ẑ y (B.3 f = f x + f y + f z (B.4 Springer International Publishing Switzerland 015 K. Barkeshli, Advanced Electromagnetics and Scattering Theory, DOI /

6 350 Appendix B: Vector Calculus B.1. Cylindrical System f = f ρ ρ + 1 f + ρ φ φ f z ẑ (B.5 F = 1 ρ ( 1 F φ ρ Fρ + ρ ρ φ + F z z (B.6 F = ( 1 F z ρ + 1 ρ φ F φ z ( ρ ( Fρ ρ + ( ρ Fφ F ρ φ z F z ρ ẑ φ (B.7 f = 1 ρ ( ρ f + 1 f ρ ρ ρ φ + f z (B.8 B.1.3 Spherical System f = f r r + 1 f + r θ θ 1 f r sin θ φ φ (B.9 F = 1 (r r F r + 1 r r sin θ θ (F θ sin θ + 1 F φ r sin θ φ (B.10 F = 1 [ ( Fφ sin θ F ] θ r r sin θ θ φ [ 1 F r + r sin θ φ 1 ( ] rfφ θ + 1 r r r [ r (rf θ F ] r φ (B.11 θ

7 Appendix B: Vector Calculus 351 f = 1 ( r r f + r r 1 r sin θ ( sin θ f + θ θ 1 r sin θ f φ (B.1 B. Differentiation ( f F = f F + F f (B.13 (E F = F E E F (B.14 ( f F = f F + f F (B.15 (E F = E F F E + (F E (E F (B.16 (E F = (E F + (F E + E ( F + F ( E (B.17 F = ( F F (B.18 ( F = 0 (B.19 f = 0 (B.0 B.3 Integration Theorems In the following V is a three-dimensional volume with differential volume element dv, S is a closed two-dimensional surface enclosing V with differential surface

8 35 Appendix B: Vector Calculus element ds and outward unit normal vector n. An open surface is denoted by and its bounding contour by C. V Fdv = S F nds (Divergence Theorem (B.1 fdv = f nds (B. V S Fdv = n F ds (B.3 V S V ( f g g f dv = S ( f g g f nds (B.4 ( F nds = C F dl (Stoke s Theorem (B.5 n fds= f dl (B.6 C

9 Appendix C Bessel Functions C.1 Gamma Function In the 18th century, Swiss mathematician Leonhard Euler concerned himself with the problem of interpolating the factorial function (n! =n(n 1(n between non-integer values. This problem eventually led him to gamma function defined as Ɣ(z = e t t z 1 dt 0 (C.1 The integral in (C.1 is convergent for all complex numbers except negative integers and zero. It can be shown that for positive integers Ɣ(n = (n 1! (C. Two useful properties of the gamma function are Ɣ(zƔ(1 z = π sin(πz (Reflection Formula (C.3 ( z 1 Ɣ(zƔ z + 1 = πɣ(z (Duplication Formula (C.4 It is important to note that the reflection formula in (C.3 is only valid for non-integer values of z. Springer International Publishing Switzerland 015 K. Barkeshli, Advanced Electromagnetics and Scattering Theory, DOI /

10 354 Appendix C: Bessel Functions C. Bessel Functions Bessel functions are the solutions of Bessel s equation d y dz + 1 dy z dz + (1 ν z y = 0 (C.5 where ν which may be complex is called the order of the equation and the solution. C..1 Bessel Functions of the First Kind J ν (z The series representation for the Bessel function of the first kind of order ν is J ν (z = k=0 ( 1 k ( z k+ν (C.6 k!ɣ(k + ν + 1 It can be easily verified that J ν (z satisfies the differential equation in (C.5. Bessel functions of integer order are of particular interest in electromagnetic problems. For integer values of ν the expression in (C.6 is simplified to J n (z = k=0 Furthermore it can be shown that for n = 0, 1,,... ( 1 k ( z k+n, n = 0, 1,,... (C.7 k!(k + n! J n (z = ( 1 n J n (z (C.8 Figure C.1 shows the plot for Bessel functions of the first kind of orders 0, 1,, and 3. As it can be seen from (C.6 the analytical expression for Bessel functions is rather complicated. In many electromagnetic boundary value problems, depending on the application, only Bessel functions of small arguments (for near-field analysis and large arguments (for far-field analysis are of interest. In these cases it is much more convenient to use concise asymptotic expressions which have been derived for Bessel functions. It can be shown that 1 ifn = 0 lim J n(z z 0 + ( z n! if n = 1,, 3,... (C.9

11 Appendix C: Bessel Functions 355 Fig. C.1 Bessel function of the first kind Fig. C. Large argument asymptotic expressions for J 1 (z and J (z and lim J n(z z πz cos (z π 4 nπ (C.10 Figure C. compares the values for J 1 (z and J (z with their large argument asymptotic expression from (C.10 which have been denoted by J 1 (z and J (z respectively.

12 356 Appendix C: Bessel Functions Generating Function The expression w(z, t = e z (t 1/t, 0 < t < (C.11 is called the generating function for J n (z. It can be shown that w(z, t = e z (t 1/t = J n (zt n n= (C.1 The Bessel generating function has many useful applications in electromagnetic boundary value problems. For example setting t = je jφ in (C.1 gives us the expansion of a plane wave in terms of an infinite sum of cylindrical waves e jzcos φ = j n J n (ze jnφ n= (C.13 C.. Bessel Functions of the Second Kind (Neumann Functions N ν (z Bessel functions of the second kind, also known as the Neumann functions are defined by the formula N ν (z = cos(νπj ν(z J ν (z sin(νπ (C.14 By inspection it can be verified that N ν (z satisfies the differential equation in (C.5. For integer values of ν the expression in (C.14 is an indeterminate form and has to be evaluated at the limit N n (z = lim ν n N ν (z, n = 1,, 3,... (C.15 The limit in (C.15 can be evaluated using l Hospital s rule, leading to N n (z = ( z π J n(z ln 1 n 1 (n k 1! ( z k n π k! k=0 1 ( 1 k ( z k+n [ψ(k + n ψ(k + 1] π k!(n + k! k=0 (C.16

13 Appendix C: Bessel Functions 357 Fig. C.3 Neumann functions of orders 0, 1,, and 3 where ψ(z is the digamma function defined as ψ(z = d dz ln Ɣ(z = Ɣ (z Ɣ(z (C.17 Similar to Bessel functions of the first kind, it can be shown that for n = 0, 1,,... N n (z = ( 1 n N n (z (C.18 Figure C.3 shows the plot for Neumann functions of orders 0, 1,, and 3. As it can be seen from the plot, N n contains a singularity at zero due to the logarithm term in (C.16. The asymptotic expressions for N n (z are lim N n(z z 0 + π ln ( γ z (n 1! π where γ = is Euler s constant and if n = 0 ( z n if n = 1,, 3,... (C.19 lim z N n(z πz sin (z π 4 nπ (C.0 Figure C.4 compares the values for N 1 (z and N (z with their large argument asymptotic expression from (C.0 which have been denoted by Ñ 1 (z and Ñ (z respectively.

14 358 Appendix C: Bessel Functions Fig. C.4 Large argument asymptotic expressions for N 1 (z and N (z C..3 Bessel Functions of the Third Kind (Hankel Functions Bessel functions of the first kind and Neumann functions are the orthogonal eigenfunctions of the Bessel equation, hence any linear combination of them is also a solution to (C.5. Two particularly important such combinations are the Hankel function of the first kind of order n defined as H (1 n (z = J n(z + jn n (z (C.1 and the Hankel function of the second kind of order n defined as H ( n (z = J n(z jn n (z (C. Hankel functions are often encountered in electromagnetic radiation problems. For large arguments, the asymptotic expressions for Hankel functions are [ ( lim H (1 z n (z πz exp j z π 4 nπ ] (C.3 [ ( lim H ( z n (z πz exp j z π 4 nπ ] (C.4

15 Appendix C: Bessel Functions 359 C..4 Formulas for Bessel Functions In the following Z n (z represents J n (z, N n (z, H n (1 (z, H n ( (z or any linear combinations of these functions. Recurrence and Differentiation Formulas Z n (z = Z n 1(z Z n+1 (z (C.5 n z Z n(z = Z n 1 (z + Z n+1 (z (C.6 J n (zn n (z N n(zj n (z = zπ (C.7 J n (zj n (z J n(zj n (z = zπ sin(nπ (C.8 Integrals z n+1 Z n (z dz = z n+1 Z n+1 (z (C.9 z 1 n Z n (z dz = z 1 n Z n 1 (z (C.30 [ ] zzn z (kz dz = Zn (kz Z n 1(kzZ n+1 (kz (C.31 π J n (z = 1 cos(nφ z sin φdφ π 0 (C.3