Analisis Vectorial

Appendix A

Vector algebra and Analysis formulas used in this book
Vector identities

A × (B × C) = (A · C)B − (A · B)C (A × B) × C

(A.1) (A.2)

= (C · A)B − (C · B)A

Vector differentialoperators in rectangular, cylindrical and spherical coordinate systems Rectangular coordinates

∂Ψ ∂Ψ ∂Ψ +y ˆ +z ˆ (A.3) ∂x ∂y ∂z ∂Ax ∂Ay ∂Az ∇·A = + + (A.4) ∂x ∂y ∂z ¶ µ ¶ µ ¶ µ ∂Ay ∂Ax ∂Az ∂Ay ∂Ax∂Az − +y ˆ − +z ˆ − (A.5) ∇×A = x ˆ ∂y ∂z ∂z ∂x ∂x ∂y ∂2Ψ ∂2Ψ ∂2Ψ ∇2 Ψ = + + (A.6) ∂x2 ∂y 2 ∂z 2 ∇Ψ = x ˆ ∇2 A = x∇2 Ax + y ∇2 Ay + z ∇2 Az ˆ ˆ ˆ 133 (A.7)

134APPENDIX A. VECTOR ALGEBRA ANDANALYSIS FORMULAS USED IN THIS BOOK Cylindrical coordinates

∇Ψ = ρ ˆ ∇·A = ∇×A = ∇2 Ψ = ∇2 A =

∂Ψ ˆ 1 ∂Ψ ∂Ψ +φ +z ˆ (A.8) ∂ρ ρ ∂φ ∂z 1 ∂Aφ 1 ∂ ∂Az (ρAρ ) + + (A.9) ρ ∂ρ ρ ∂φ ∂z ¶ µ ¶ µ ¶ µ ∂Aφ 1 ∂Azˆ ∂Aρ − ∂Az + z 1 ∂ (ρAφ ) − ∂Aρ − +φ ˆ (A.10) ρ ˆ ρ ∂φ ∂z ∂z ∂ρ ρ ∂ρ ∂φ µ ¶ 2 2 1 ∂ ∂Ψ 1 ∂ Ψ ∂ Ψ + (A.11) ρ + 2 ρ ∂ρ ∂ρ ρ ∂φ2 ∂z 2 ³ ´ ∇ ∇·A −∇×∇×A (A.12)

Spherical coordinates

∇Ψ = r ˆ ∇·A =∇×A = ∇2 Ψ = = ∇2 A =

ˆ ∂Ψ ˆ 1 ∂Ψ φ ∂Ψ +θ + (A.13a) ∂r r ∂θ r sin θ ∂φ ∂ 1 ∂Aφ 1 ∂ ¡ 2 ¢ 1 r Ar + (Aθ sin θ) + (A.13b) r2 ∂r r sin θ ∂θ r sin θ ∂φ ¶ ˆµ µ ¶ ˆµ ¶ ∂Aθ φ ∂ ∂Ar r ˆ ∂ θ 1 ∂Ar ∂ (Aφ sin θ) −+ − (rAφ ) + (rAθ ) −(A.13c) r sin θ ∂θ ∂φ r sin θ ∂φ ∂r r ∂r ∂θ µ ¶ µ ¶ 2 1 ∂ 1 ∂ ∂Ψ 1 ∂Ψ ∂ Ψ (A.13d) r2 + 2 sin θ + 2 2 r2 ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 µ ¶ ∂ ∂2Ψ 1 1 ∂ 2 (rΨ) ∂Ψ 1 + 2 (A.13e) sinθ + 2 2 r ∂r2 r sin θ ∂θ ∂θ r sin θ ∂φ2 ³ ´ ∇ ∇·A −∇×∇×A (A.13f)

135 Relations between differential operators ∇ × (∇Ψ) = 0 (A.14a) (A.14b) (A.14c) (A.14d) (A.14e) (A.14f) (A.14g) (A.14h) (A.14j)∇ × (∇ × A) = ∇(∇ · A) − ∇2 A ∇ (ΦΨ) = Φ∇Ψ + Ψ∇Φ

∇ · (∇ × A) = 0 ∇ · (∇Ψ) = ∇2 Ψ

∇ · (A × B) = B · ∇ × A − A · ∇ × B

∇ × (ΨA) = Ψ∇ × A − A × ∇Ψ

∇ · (ΨA) = Ψ∇ · A + A · (∇Ψ)

∇ × (A ×B) = A∇ · B − (A · ∇)B − B∇ · A + (B · ∇)A dA dΨ dA ∇A (Ψ) = ∇Ψ dΨ dA ∇ × A (Ψ) = ∇Ψ × dΨ ∇(ΨA) = (∇Ψ) A − Ψ(∇A) ∇ · A (Ψ) = ∇Ψ ·

(A.14i) ∇(A · B) = A × ∇ × B + (A · ∇)B + B × ∇ × A + (B · ∇)A...