Formulation

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Formulation

  1. 1. Formulas/transformations of vectors in three coordinates system Cartesian Coordinates System(X,Y,Z): ∧ ∧ ∧ DIFFERENTIAL LENGTH VECTOR : dl = dx a x + dy a y + dz a z DIFFERENTIAL VOLUME ELEMENT : dV = dx dy dz −∞ < X < ∞, − ∞ < Y < ∞, − ∞ < Z < ∞ ∧ DIFFERENTIAL SURFACE ELEMENTS: dS x = dy dz a x − ∞ < Y < ∞, − ∞ < Z < ∞ ∧ dS y = dx dz a y − ∞ < X < ∞, − ∞ < Z < ∞ ∧ dS z = dy dz a x − ∞ < X < ∞, − ∞ < Y < ∞ DISTANCE BETWEEN TWO POINTS: [ d = ( x1 − x 2 ) 2 + ( y1 − y 2 ) 2 + ( z1 − z 2 ) 2 ] 1/ 2 ∧ ∂V ∧ ∂V ∧ ∂V GRADIENT OF SCALAR V : ∇ = ax V +a y +az ∂x ∂y ∂z ∂Ax ∂A y ∂Az DIVERGENCE OF VECTOR A : ∇• A = + + ∂x ∂y ∂Z ∧ ∧ ∧ ax ay az ∇ A × ∂ ∂ ∂ CURL OF VECTOR A: = ∂x ∂y ∂z Ax A y Az ∂2V ∂2V ∂2V LAPLACIAN OF A SCALAR V: ∇2V = 2 + 2 + ∂x ∂y ∂Z 2A VECTOR A IS SAID TO BE SOLENOIDAL (OR DIVERGENCELESS ) if ∇ A = • 0A VECTOR A IS SAID TO BE IRROTATIONAL( OR POTENTIAL) IF ∇ A= × 0 (BOTH STATEMENT ARE TRUE IN ALL THE COORDINATE SYSTEMS)DIVERGENCE THEORM(GREENS THEORM) : ∫A •ds =∫∇•A S V dVSTOCKS THEORM: ∫ A • dl L = ∫(∇×A ) •dS SCOMPUTATION FORMULAS ON GRADIENT: (a ) ∇ V +U ) =∇ +∇ ( V U (b) ∇ UV ) =V∇ +U∇ ( U V  V  U∇ −V∇ V U (c ) ∇ = U  U2 ( d ) ∇ n = nV n −1∇ V V where U and V are scalars and n is int eger
  2. 2. Cylindrical coordinates system ( ρ ,φ , z)RELATIONSHIP BETWEEN (X,Y,Z) AND ( ( ρ , φ , z ) : X = ρ cos φ y φ = tan −1  .Y = ρ sin φ x 0 ≤ ρ < ∞ , 0 ≤ φ < 2π , − ∞ < z < ∞ Z =Z ρ= 2 x +y 2DIFFERENTIAL LENGTH VECTOR : dl = dρ aρ + ρ dφ aφ + dza z ˆ ˆ ˆDIFFERENTIAL VOLUME ELEMENT : dv = ρ dρ dφ dz 0 ≤ ρ < ∞ , 0 ≤ φ < 2π , − ∞ < z < ∞DIFFERENTIAL SURFACE ELEMENTS : ds ρ = ρ dφ dz a ρ ˆ 0 ≤ φ < 2π , − ∞ < z < ∞ dsφ = dρ dz aφ ˆ 0 ≤ ρ < ∞, −∞ < z < ∞ ds z = ρ dφ dρ a z ˆ 0 ≤ ρ < ∞ , 0 ≤ φ < 2πDISTANCE BETWEEN TWO POINTS : d 2 = ρ1 2 + ρ2 2 − 2 ρ1 ρ2 cos(φ1 −φ2 ) + ( z 2 − z1 ) 2Transformation of A from cylindrical to cartesian coordinates system Aρ   cos φ − sin φ 0   Ax      Aφ  =  sin φ cos φ 0   A y A   0 0 1   Az  z   Transformation of A from cartesian to cylinderical coordinates system Ax   cos φ sin φ 0   Aρ      A y  =  − sin φ cos φ 0   Aφ A   0 0 1   Az  z    ∂ ∧ V 1 ∂ ∧ V ∂V ∧GRADIENT OF A SCALAR V: ∇ = V aρ+ aφ + az ∂ρ ρ ∂φ ∂Z 1 ∂AφDIVERGENCE OF A VECTOR A: ∇• A = 1 ∂ ρ ∂ρ ( ρ Aρ + ρ ∂φ )+ ∂Az ∂Z aρ ρ Aφ Az 1∂ ∂ ∂CURL OF A VECTOR A: ∇ × A= ρ ∂ρ ∂φ ∂Z Aρ ρ Aφ Az
  3. 3. 1 ∂  ∂V  1 ∂2V  ∂2V LAPLACIAN OF A SCALAR V: ∇2V = ρ + 2 +  ρ ∂ρ  ∂ρ  ρ ∂φ 2  ∂z 2     
  4. 4. Spherical Coordinate System (r ,θ, φ) y φ = tan −1   X = r sin θ cos φ x .Y = r sin θ sin φ r = x2 + y2 +z2 0 ≤ r < ∞ , - π ≤ θ < π , 0 < φ < 2π Z =r cos θ x2 +y 2 θ =tan −1 z Differenial length vector : dl = dr a r + r dθ aθ + r sin θ dφ aφ ˆ ˆ ˆ DIFFERENTIAL VOLUME ELEMENT : dV = r 2 sin θ dr dθ dφ 0 ≤ r < ∞, - π ≤ θ < π , 0 < φ < 2 dIFFERENTIAL SURFACE ELEMENT : ds r = r 2 sin θ dθ dφ a r ˆ - π ≤θ < π, 0 < φ < 2π dsθ = r sin θ dr dθ aφ ˆ 0 ≤ r < ∞, 0 < φ < 2π dsφ = r dr dθ aφ ˆ 0 ≤ r < ∞, - π ≤ θ < π DISTANCE BETWEEN TWO POINTS : d 2 = r12 + r2 + 2r1 r2 cos θ1 cos θ2 − 2 r1 r2 sin θ1 sin θ2 cos(θ1 −θ 2 Transformation of A from Cartesian to spherical coordinate system  Ar   sin θ cos φ sin θ sin φ cos θ   Ax        Aθ  =  cos θ cos φ cos θ sin φ − sin θ   A y    Aφ   − sin φ     cos φ 0   Az    Transformation of A from spherical to cartesian coordinates system  Ax   sin θ cos φ cos θ cos φ − sin φ   Ar        A y  =  sin θ sin φ cos θ sin φ cos φ   Aθ   A   sin θ    z  − sin θ 0   Aφ    Transformation of A from spherical to cylindrical coordinates system  Aρ   sin θ cos θ 0   Ar        Aφ  = 0 0 1   Aθ  A   cos θ    z   − sin θ 0   Aφ    Transformation of A from cylindrical to spherical coordinates system  Ar   sin θ 0 cos θ   Aρ        Aθ  =  cos θ 0 − sin θ   Aφ     Aφ   0  A     1 0  z  ∂V ∧ 1 ∂ ∧ V 1 ∂ ∧ V GRADIENT OF A SCALAR V: ∇ = V ar + aθ + aφ ∂r r ∂θ r sin θ ∂φDIVERGENCE OF A VECTOR A: ∇• A = r 1 ∂ 2 2 ∂r ( r Ar + 1 ) ∂ r sin ϑ ∂ϑ ( Aϑ Sin ϑ) + 1 ∂Aφ r Sin ϑ ∂φ ∧ ∧ ∧ a r r aϑ r Sinϑ aφ 1 ∂ ∂ ∂ CURL OF A VECTOR A: ∇ × A= r 2 Sinϑ ∂ r ∂ ϑ ∂ φ Ar rAφ r Sinϑ Aφ
  5. 5. LAPLACIAN OF A SCALER FIELD, V: 1 ∂  ∂V  1 ∂  ∂V  1  ∂2V ∇2V = ρ + sin θ +   ρ ∂ρ  ∂ρ  r 2 sin θ ∂θ    ∂θ  r 2 sin 2 θ  ∂φ 2   
  6. 6. LAPLACIAN OF A SCALER FIELD, V: 1 ∂  ∂V  1 ∂  ∂V  1  ∂2V ∇2V = ρ + sin θ +   ρ ∂ρ  ∂ρ  r 2 sin θ ∂θ    ∂θ  r 2 sin 2 θ  ∂φ 2   

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