Mathematical identities
(Lukas Jelinek, CTU Prague, October 2019)In the following, the rectangular system is characterized by coordinates (x, y, z) and unit vectors x0 y0 z0
. The cylindrical system is characterized by coordinates (ρ, ϕ, z) and unit vectors
ρ0 ϕ0 z0 . The
spherical system is characterized by coordinates (r, θ, ϕ) and unit vectors
r0 θ0 ϕ0 .
0.1
Point Transformations
Rectangular - Cylindrical x=ρcosϕ y=ρsinϕ z=z (1) Cylindrical - Rectangular ρ=px2+y2 tanϕ=y x z=z (2) Rectangular - Spherical x=rcosϕsinθ y=rsinϕsinθ z=rcosθ (3) Spherical - Rectangular r=px2+y2+z2 tanθ= p x2+y2 z tanϕ=y x (4) Cylindrical - Spherical ρ=rsinθ z=rcosθ ϕ=ϕ (5)Cylindrical - Rectangular ρ0=x0cosϕ+y0sinϕ ϕ0=−x0sinϕ+y0cosϕ z0=z0 (8) Rectangular - Spherical
x0=r0sinθcosϕ+θ0cosθcosϕ−ϕ0sinϕ
y0=r0sinθsinϕ+θ0cosθsinϕ+ϕ0cosϕ
z0=r0cosθ−θ0sinθ
(9)
Spherical - Rectangular
r0=x0sinθcosϕ+y0sinθsinϕ+z0cosθ
θ0=x0cosθcosϕ+y0cosθsinϕ−z0sinθ
ϕ0=−x0sinϕ+y0cosϕ (10) Cylindrical - Spherical ρ0=r0sinθ+θ0cosθ ϕ0=ϕ0 z0=r0cosθ−θ0sinθ (11) Spherical - Cylindrical r0=ρ0sinθ+z0cosθ θ0=ρ0cosθ−z0sinθ ϕ0=ϕ0 (12)
0.3
Differential Operators
Rectangular coordinate system∇f =x0 ∂f ∂x +y0 ∂f ∂y +z0 ∂f ∂z ∇ ·F = ∂Fx ∂x + ∂Fy ∂y + ∂Fz ∂z ∇ ×F =x0 ∂F z ∂y − ∂Fy ∂z +y0 ∂F x ∂z − ∂Fz ∂x +z0 ∂F y ∂x − ∂Fx ∂y ∇2f =∇ ·(∇f) =∂2f ∂x2 + ∂2f ∂y2 + ∂2f ∂z2 ∇2F =∇(∇ ·F)− ∇ × ∇ ×F =x0∇2F x+y0∇2Fy+z0∇2Fz (13)
Cylindrical coordinate system ∇f =ρ0∂f ∂ρ +ϕ0 1 ρ ∂f ∂ϕ+z0 ∂f ∂z ∇ ·F = 1 ρ ∂(ρFρ) ∂ρ + 1 ρ ∂Fϕ ∂ϕ + ∂Fz ∂z ∇ ×F =ρ0 1 ρ ∂Fz ∂ϕ − ∂Fϕ ∂z +ϕ0 ∂F ρ ∂z − ∂Fz ∂ρ +z0 1 ρ ∂(ρFϕ) ∂ρ − 1 ρ ∂Fρ ∂ϕ ∇2f =∇ ·(∇f) = 1 ρ ∂ ∂ρ ρ∂f ∂ρ + 1 ρ2 ∂2f ∂ϕ2+ ∂2f ∂z2 ∇2F =∇(∇ ·F)− ∇ × ∇ ×F = ρ0 ∂2F ρ ∂ρ2 + 1 ρ ∂Fρ ∂ρ − Fρ ρ2 + 1 ρ2 ∂2F ρ ∂ϕ2 − 2 ρ2 ∂Fϕ ∂ϕ + ∂2F ρ ∂z2 + ϕ0 ∂2F ϕ ∂ρ2 + 1 ρ ∂Fϕ ∂ρ − Fϕ ρ2 + 1 ρ2 ∂2F ϕ ∂ϕ2 + 2 ρ2 ∂Fρ ∂ϕ + ∂2F ϕ ∂z2 + z0 ∂2F z ∂ρ2 + 1 ρ ∂Fz ∂ρ + 1 ρ2 ∂2F z ∂ϕ2 + ∂2F z ∂z2 (14)
Spherical coordinate system
∇f =r0 ∂f ∂r +θ0 1 r ∂f ∂θ +ϕ0 1 rsinθ ∂f ∂ϕ ∇ ·F = 1 r2 ∂ r2F r ∂r + 1 rsinθ ∂ ∂θ(Fθsinθ) + 1 rsinθ ∂Fϕ ∂ϕ ∇ ×F = r0 rsinθ ∂ ∂θ(Fϕsinθ)− ∂Fθ ∂ϕ +θ0 r 1 sinθ ∂Fr ∂ϕ − ∂(rFϕ) ∂r + ϕ0 r ∂(rFθ) ∂r − ∂Fr ∂θ ∇2f =∇ ·(∇f) = 1 r2 ∂ ∂r r2∂f ∂r + 1 r2sinθ ∂ ∂θ sinθ∂f ∂θ + 1 r2sin2θ ∂2f ∂ϕ2 ∇2F =∇(∇ ·F)− ∇ × ∇ ×F = r0 ∂2Fr ∂r2 + 2 ρ ∂Fr ∂r − 2 r2Fr+ 1 r2 ∂2Fr ∂θ2 + cotθ r2 ∂Fr ∂θ + 1 r2sin2θ ∂2Fr ∂ϕ2 2 ∂F 2 cotθ 2 ∂F + (15)
0.4
Differential Identities
a·(b×c) =b·(c×a) =c·(a×b) a×(b×c) = (a·c)b−(a·b)c (a×b)·(c×d) = (a·c) (b·d)−(a·d) (b·c) ∇ ×(∇f) = 0 ∇ ·(∇ ×F) = 0 ∇(f·g) =g∇f+f∇g ∇ ·(Ff) =F ·(∇f) +f(∇ ·F) ∇ ×(Ff) = (∇f)×F +f(∇ ×F) ∇ f g =g∇f −f∇g g2 ∇ · F f =f(∇ ·F)−F ·(∇f) f2 ∇ × F f = f(∇ ×F)−(∇f)×F f2 ∇(F ·G) = (F · ∇)G+ (G· ∇)F +F ×(∇ ×G) +G×(∇ ×F) ∇ ·(F ×G) =G·(∇ ×F)−F ·(∇ ×G) ∇ ×(F ×G) =F (∇ ·G)−G(∇ ·F) + (G· ∇)F −(F · ∇)G ∇(f◦g) = (f0◦g)∇g ∇ ·(F ◦g) = F0◦g· ∇g ∇ ×(F ◦g) =− F0◦g× ∇g (16)0.5
Integration Identities
Z V (∇ ·F) dV = I S F ·dS Z V (∇f) dV = I S fdS Z V (∇ ×F) dV =− I S F ×dS Z V f ∇2g dV = I S f(∇g)·dS− Z V [(∇f)·(∇g)] dV Z V [F ·(∇ × ∇ ×G)] dV =− I S [F ×(∇ ×G)]·dS+ Z V [(∇ ×F)·(∇ ×G)] dV Z V f ∇2g −g ∇2f dV = I S [f(∇g)−g(∇f)]·dS Z V [G·(∇ × ∇ ×F)−F ·(∇ × ∇ ×G)] dV = I S [F ×(∇ ×G)−G×(∇ ×F)]·dS Z S (∇ ×F)·dS= I l F ·dl Z S (∇f)×dS=− I l fdl (17)0.6
3D Fourier’s Transformation
Definition FT{f(r)}= ˜f(k) = ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ f(r) e−jk·rdxdydz FT−1nf˜(k)o=f(r) = 1 (2π)3 ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ ˜ f(k) ejk·rdkxdkydkz (18)0.7
Trigonometric identities
sin2(α) =1−cos (2α)2 cos2(α) = 1 + cos (2α)
2
sin (α+β) = sin (α) cos (β) + cos (α) sin (β) sin (α−β) = sin (α) cos (β)−cos (α) sin (β) cos (α+β) = cos (α) cos (β)−sin (α) sin (β) cos (α−β) = cos (α) cos (β) + sin (α) sin (β)
sin (α) + sin (β) = 2 sin α+β 2 cos α−β 2
sin (α)−sin (β) = 2 cos
α+β 2 sin α−β 2
cos (α) + cos (β) = 2 cos
α+β 2 cos α−β 2
cos (α)−cos (β) =−2 sin
α+β 2 sin α−β 2 (21)
0.8
Identities involving separation vector
R=r−r0, R=|R|, R0= R R ∇R=R0 ∇Rn=nRn−1∇R=nRn−1R 0 ∇ ·R= 3 ∇ ×R= 0 ∇ ·R0= 2 R ∇ e−jkR=−jke−jkRR0 ∇2 1 R =−4πδ(R) ∇2+k2e− jkR R =−4πδ(R) ∂2 ∂ri∂rj 1 R = 3RiRj−R2δij R5 , R6= 0 (22)
0.9
Useful integrals
Z dx x2+a2 = 1 aarctan x a Z dx √ 1−x2 = arcsin (x) Z dx √ x2−1 = ln x+px2−1 Z dx √ x2+a2 = ln x+px2+a2 Z dx (x2+a2)3/2 = x a2√x2+a2 Z xnlna x dx= x n+1 (n+ 1)2 + xn+1 n+ 1ln a x Z xeaxdx=ax−1 a2 e ax Z x2eaxdx= a 2x2−2ax+ 2 a3 e ax Zsin3(x) dx=−cos (x) +cos
3(x)
3 Z
cos3(x) dx= sin (x)−sin 3(x) 3 Z dx cos (x) = ln 1 + sin (x) cos (x) Z dx sin (x) = ln 1−cos (x) sin (x) Z cos (x) dx (1−a2cos2(x))3/2 = sin (x) (1−a2)p 1−a2cos2(x) Z sin (x) dx 1−a2sin2 (x)3/2 = −cos (x) (1−a2) q 1−a2sin2(x) (23)
