In this section we present some of the widely used identities of vector

In this section we present some of the widely used

This preview shows page 128 - 135 out of 171 pages.

In this section, we present some of the widely used identities of vector calculus using the traditional vector notation as well as its equivalent tensor notation. In the following bullet points, f and h are differentiable scalar fields; A , B , C and D are differentiable vector fields; and r = x i e i is the position vector. ∇ · r = n m (213) i x i = n where n is the space dimension.
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5.4 Common Identities in Vector and Tensor Notation 128 ∇ × r = 0 m (214) ijk j x k = 0 ( a · r ) = a m (215) i ( a j x j ) = a i where a is a constant vector. ∇ · ( f ) = 2 f m (216) i ( i f ) = ii f ∇ · ( ∇ × A ) = 0 m (217) ijk i j A k = 0
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5.4 Common Identities in Vector and Tensor Notation 129 ∇ × ( f ) = 0 m (218) ijk j k f = 0 ( fh ) = f h + h f m (219) i ( fh ) = f∂ i h + h∂ i f ∇ · ( f A ) = f ∇ · A + A · ∇ f m (220) i ( fA i ) = f∂ i A i + A i i f ∇ × ( f A ) = f ∇ × A + f × A m (221) ijk j ( fA k ) = f ijk j A k + ijk ( j f ) A k
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5.4 Common Identities in Vector and Tensor Notation 130 A · ( B × C ) = C · ( A × B ) = B · ( C × A ) m m (222) ijk A i B j C k = kij C k A i B j = jki B j C k A i A × ( B × C ) = B ( A · C ) - C ( A · B ) m (223) ijk A j klm B l C m = B i ( A m C m ) - C i ( A l B l ) A × ( ∇ × B ) = ( B ) · A - A · ∇ B m (224) ijk klm A j l B m = ( i B m ) A m - A l ( l B i ) ∇ × ( ∇ × A ) = ( ∇ · A ) - ∇ 2 A m (225) ijk klm j l A m = i ( m A m ) - ll A i
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5.4 Common Identities in Vector and Tensor Notation 131 ( A · B ) = A × ( ∇ × B ) + B × ( ∇ × A ) + ( A · ∇ ) B + ( B · ∇ ) A m (226) i ( A m B m ) = ijk A j ( klm l B m ) + ijk B j ( klm l A m ) + ( A l l ) B i + ( B l l ) A i ∇ · ( A × B ) = B · ( ∇ × A ) - A · ( ∇ × B ) m (227) i ( ijk A j B k ) = B k ( kij i A j ) - A j ( jik i B k ) ∇ × ( A × B ) = ( B · ∇ ) A + ( ∇ · B ) A - ( ∇ · A ) B - ( A · ∇ ) B m (228) ijk klm j ( A l B m ) = ( B m m ) A i + ( m B m ) A i - ( j A j ) B i - ( A j j ) B i ( A × B ) · ( C × D ) = A · C A · D B · C B · D m (229) ijk A j B k ilm C l D m = ( A l C l ) ( B m D m ) - ( A m D m ) ( B l C l )
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5.5 Integral Theorems in Tensor Notation 132 ( A × B ) × ( C × D ) = [ D · ( A × B )] C - [ C · ( A × B )] D m (230) ijk jmn A m B n kpq C p D q = ( qmn D q A m B n ) C i - ( pmn C p A m B n ) D i In vector and tensor notations, the condition for a vector field A to be solenoidal is: ∇ · A = 0 m (231) i A i = 0 In vector and tensor notations, the condition for a vector field A to be irrotational is: ∇ × A = 0 m (232) ijk j A k = 0 5.5 Integral Theorems in Tensor Notation The divergence theorem for a differentiable vector field A in vector and tensor nota- tions is given by: ˚ V ∇ · A = ¨ S A · n m (233) ˆ V i A i = ˆ S A i n i
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5.5 Integral Theorems in Tensor Notation 133 where V is a bounded region in an n D space enclosed by a generalized surface S , and are generalized volume and surface elements respectively, n and n i are unit vector normal to the surface and its i th component respectively, and the index i ranges over 1 , . . . , n . Similarly, the divergence theorem for a differentiable rank-2 tensor field A in tensor notation for the first index is given by: ˆ V i A il = ˆ S A il n i (234) while the divergence theorem for differentiable tensor fields of higher rank A in tensor notation for the index k is given by: ˆ V k A ij...k...m = ˆ S A ij...k...m n k (235) Stokes theorem for a differentiable vector field A in vector and tensor notations is given by: ¨ S ( ∇ × A ) · n = ˆ C A · d r m (236) ˆ S ijk j A k n i = ˆ C A i dx i where C stands for the perimeter of the surface
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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