Vector calculus

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Vector calculus

  1. 1. Vector Calculus
  2. 2. Vector Calculus COORDINATE SYSTEMS• RECTANGULAR or Cartesian • CYLINDRICAL Choice is based on • SPHERICAL symmetry of problem Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL
  3. 3. Cylindrical Symmetry Spherical Symmetry
  4. 4. Cartesian Coordinates Or Rectangular Coordinates P (x, y, z z) −∞ < x < ∞ P(x,y,z) −∞ < y < ∞ y −∞ < z < ∞ x A vector A in Cartesian coordinates can be written as ( Ax , Ay , Az ) or Ax a x + Ay a y + Az a zwhere ax,ay and az are unit vectors along x, y and z-directions.
  5. 5. Cylindrical Coordinates z P (ρ, Φ, z) 0≤ρ <∞ z P(ρ, Φ, z) 0 ≤ φ < 2π −∞ < z < ∞ x Φ ρ y A vector A in Cylindrical coordinates can be written as ( Aρ , Aφ , Az ) or Aρ aρ + Aφ aφ + Az a zwhere aρ,aΦ and az are unit vectors along ρ, Φ and z-directions. x= ρ cos Φ, y=ρ sin Φ, z=z y ρ = x + y , φ = tan 2 2 −1 ,z = z x
  6. 6. The relationships between (ax,ay, az) and (aρ,aΦ, az)are a x = cos φaρ − sin φaφ a y = sin φaρ − cos φaφ az = az or aρ = cos φa x + sin φa y aφ = − sin φa x + cos φa y az = azThen the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)areA = ( Ax cos φ + Ay sin φ )aρ + (− Ax sin φ + Ay cos φ )aφ + Az a z
  7. 7. Aρ = Ax cos φ + Ay sin φ Aφ = − Ax sin φ + Ay cos φ Az = AzIn matrix form we can write  Aρ   cos φ sin φ 0  Ax   A  = − sin φ cos φ 0  Ay   φ     Az   0    0 1  Az   
  8. 8. Spherical Coordinates z P (r, θ, Φ) 0≤r <∞ P(r, θ, Φ) θ r 0 ≤θ ≤π 0 ≤ φ < 2π x Φ yA vector A in Spherical coordinates can be written as ( Ar , Aθ , Aφ ) or Ar ar + Aθ aθ + Aφ aφ where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions. x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ x2 + y2 −1 y r = x 2 + y 2 + z 2 , θ = tan −1 , φ = tan z x
  9. 9. The relationships between (ax,ay, az) and (ar,aθ,aΦ)are a x = sin θ cos φar + cos θ cos φaθ − sin φaφ a y = sin θ sin φar + cos θ sin φaθ + cos φaφ a z = cos θar − sin θaθ or ar = sin θ cos φa x + sin θ sin φa y + cos θa z aθ = cos θ cos φa x + cos θ sin φa y − sin θa z aφ = − sin φa x + cos φa yThen the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are A = ( Ax sin θ cos φ + Ay sin θ sin φ + Az cos θ )ar + ( Ax cos θ cos φ + Ay cos θ sin φ − Az sin θ )aθ + (− Ax sin φ + Ay cos φ )aφ
  10. 10. Ar = Ax sin θ cos φ + Ay sin θ sin φ + Az cos θ Aθ = Ax cos θ cos φ + Ay cos θ sin φ − Az sin θ Aφ = − Ax sin φ + Ay cos φIn matrix form we can write  Ar   sin θ cos φ sin θ sin φ cos θ   Ax      Aθ  = cos θ cos φ cos θ sin φ − sin θ   Ay     Aφ   − sin φ cos φ 0   Az      
  11. 11. z z P(r, θ, Φ) Cartesian Coordinates P(x,y,z) θ r P(x, y, z) y y xx Φ Spherical Coordinates Cylindrical Coordinates P(r, θ, Φ) z P(ρ, Φ, z) z P(ρ, Φ, z) r y x Φ
  12. 12. Differential Length, Area and Volume Cartesian CoordinatesDifferential displacementdl = dxa x + dya y + dza zDifferential area dS = dydza x = dxdza y = dxdya z Differential Volume dV = dxdydz
  13. 13. Cylindrical Coordinates ρ ρ ρ ρ ρ ρ ρρ ρ ρ ρ
  14. 14. Differential Length, Area and Volume Cylindrical CoordinatesDifferential displacement dl = dρa ρ + ρdφaφ + dza zDifferential area dS = ρdφdza ρ = dρdzaφ = ρdρdφa z Differential Volume dV = ρdρdφdz
  15. 15. Spherical Coordinates
  16. 16. Differential Length, Area and Volume Spherical CoordinatesDifferential displacement dl = drar + rdθaθ + r sin θdφaφDifferential area dS = r sin θdθdφar = r sin θdrdφaθ = rdrdθaφ 2 Differential Volume dV = r sin θdrdθdφ 2
  17. 17. Line, Surface and Volume IntegralsLine Integral ∫ A.dl LSurface Integral ψ = ∫ A.dS S Volume Integral ∫ p dv V v
  18. 18. Gradient, Divergence and Curl The Del Operator• Gradient of a scalar function is a vector quantity. ∇f Vector• Divergence of a vector is a scalar ∇. A quantity.• Curl of a vector is a vector ∇× A quantity.• The Laplacian of a scalar A ∇ A 2
  19. 19. Del OperatorCartesian Coordinates ∂ ∂ ∂ ∇ = ax + a y + az ∂x ∂y ∂zCylindrical Coordinates ∂ 1 ∂ ∂ ∇= aρ + aφ + a z ∂ρ ρ ∂φ ∂zSpherical Coordinates ∂ 1 ∂ 1 ∂ ∇ = ar + aθ + aφ ∂r r ∂θ r sin θ ∂φ
  20. 20. Gradient of a ScalarThe gradient of a scalar field V is a vector that representsboth the magnitude and the direction of the maximum spacerate of increase of V. ∂V ∂V ∂V ∇V = ax + ay + az ∂x ∂y ∂z ∂V 1 ∂V ∂V ∇V = aρ + aφ + az ∂ρ ρ ∂φ ∂z ∂V 1 ∂V 1 ∂V ∇V = ar + aθ + aφ ∂r r ∂θ r sin θ ∂φ
  21. 21. Divergence of a VectorThe divergence of A at a given point P is the outward flux perunit volume as the volume shrinks about P. ∫ A.dS divA = ∇. A = lim S ∆v →0 ∆v ∂A ∂A ∂A ∇. A = + + ∂x ∂y ∂z 1 ∂ 1 ∂Aφ ∂Az ∇. A = ( ρAρ ) + + ρ ∂ρ ρ ∂φ ∂z
  22. 22. Curl of a VectorThe curl of A is an axial vector whose magnitude is themaximum circulation of A per unit area tends to zero andwhose direction is the normal direction of the area when thearea is oriented to make the circulation maximum.  A.dl   ∫  curlA = ∇ × A =  lim L  an  ∆s →0 ∆S      maxWhere ΔS is the area bounded by the curve L and an is the unitvector normal to the surface ΔS
  23. 23.  ax ay az   aρ ρaφ az  ∂ ∂ ∂ 1∂ ∂ ∂∇× A =   ∇× A =    ∂x ∂y ∂z  ρ  ∂ρ ∂φ ∂z   Ax  Ay Az    Aρ  ρAφ Az   Cartesian Coordinates Cylindrical Coordinates  ar raθ r sin θaφ  1 ∂ ∂ ∂  ∇× A = 2   r sin θ  ∂r ∂θ ∂φ   Ar  rAθ r sin θAφ   Spherical Coordinates
  24. 24. Divergence or Gauss’ TheoremThe divergence theorem states that the total outward flux ofa vector field A through the closed surface S is the same as thevolume integral of the divergence of A. ∫ A.dS = ∫ ∇. Adv V
  25. 25. Stokes’ TheoremStokes’s theorem states that the circulation of a vector field A around aclosed path L is equal to the surface integral of the curl of A over the opensurface S bounded by L, provided A and × A ∇ are continuous on S ∫ A.dl = ∫ (∇ × A).dS L S

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