Tensors for Physics

86 7 Fields, Spatial Differential Operators Fig. 7.3 Uniaxial squeeze-stretch field (iv) Uniaxial Squeeze-stretch Field The vect ...

7.2 Vector Fields, Divergence and Curl or Rotation 87 Fig. 7.4 Planar squeeze-stretch field In this case, the gradient field is ...

88 7 Fields, Spatial Differential Operators Fig. 7.5 Solid-like rotation field Fig. 7.6 Simple shear field 7.2.2 Differential Ch ...

7.2 Vector Fields, Divergence and Curl or Rotation 89 The scalar∇λvλ=∇·v:=divvis called thedivergenceof the vectorv. The cross p ...

90 7 Fields, Spatial Differential Operators For the three-dimensional radial field, one finds ∇·v= 3 , ∇×v= 0 , ∇νvμ = 0. The 2D ...

7.3 Special Types of Vector Fields 91 The arguments just presented here, can be reverted. When∇×v=0 holds true for a vector fiel ...

92 7 Fields, Spatial Differential Operators Clearly, due to∇μvμ=εμνλ∇μ∇νAλ=0, the divergence of a vector field given by (7.39) v ...

7.3 Special Types of Vector Fields 93 Thus the yet undetermined coefficients must obey the relationc 2 − 2 c 1 =1. Clearly, ther ...

94 7 Fields, Spatial Differential Operators Application in Electrostatics For static electric fieldsE, the curl vanishes:∇×E=0. ...

7.3 Special Types of Vector Fields 95 Letvμbe given byvμ=∇μΦ, withΦ=Φ(r). Then vμ= dΦ dr ∇μr= dΦ dr r−^1 rμ=r−^1 Φ′rμ, cf. (7.13 ...

96 7 Fields, Spatial Differential Operators Fig. 7.7 Uniaxial tensor fields for the director in a nematic liquid crystal Fig. 7. ...

7.4 Tensor Fields 97 7.4.3 Local Mass and Momentum Conservation, Pressure Tensor Letρ=ρ(r)andv=v(r)be the mass density and the l ...

98 7 Fields, Spatial Differential Operators A fluid composed of particles with an internal rotational degree of freedom, in gen- ...

7.5 Maxwell Equations in Differential Form 99 as in the Gaussiancgs-system of physical units. Such a description is inconvenient ...

100 7 Fields, Spatial Differential Operators are solutions for ρ(−t,r),−j(−t,r). As a consequence of the combined PT-invariance ...

7.5 Maxwell Equations in Differential Form 101 equations, the first one of which is referred to asOersted law,are εμνλ∇νHλ=jμ, ∇ ...

102 7 Fields, Spatial Differential Operators The plane wave is a special case of (7.63). In complex notation, this solution of t ...

7.5 Maxwell Equations in Differential Form 103 The current density and the charge density are the sources for the vector potenti ...

104 7 Fields, Spatial Differential Operators acting on a particle with chargee, moving with velocityv, in the presence of an ele ...

7.6 Rules for Nabla and Laplace Operators 105 7.6 Rules for Nabla and Laplace Operators 7.6.1 Nabla The application of the nabla ...