Tensors for Physics

5.3 Applications 65 enough, such that terms nonlinear in the applied electric field can be disregarded. In this linear regime, o ...

66 5 Symmetric Second Rank Tensors with some positive numberX. In many applications, all principal values are posi- tive,S(i)> ...

5.4 Geometric Interpretation of Symmetric Tensors 67 Fig. 5.1 Perspective view of uniaxial (left) and biaxial (right) ellipsoids ...

68 5 Symmetric Second Rank Tensors -1-0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 1 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 ...

5.5 Scalar Invariants of a Symmetric Tensor 69 5.5 Scalar Invariants of a Symmetric Tensor 5.5.1 Definitions. The traceSμμ, the ...

70 5 Symmetric Second Rank Tensors I 2 = 2 3 s^2 + 2 q^2 , I 3 =3 det ( S ) = 2 s ( 1 9 s^2 −q^2 ) . (5.45) The special planar b ...

5.6 Hamilton-Cayley Theorem and Consequences 71 5.6 Hamilton-Cayley Theorem and Consequences. 5.6.1 Hamilton-Cayley Theorem Any ...

72 5 Symmetric Second Rank Tensors of products of the tensor. Likewise, the expression for the invariantI 3 of a symmetric trace ...

5.7 Volume Conserving Affine Transformation 73 5.7 Volume Conserving Affine Transformation 5.7.1 Mapping of a Sphere onto an Ell ...

74 5 Symmetric Second Rank Tensors The dependence ofAμνand its inverse onQare Aμν=Q^2 /^3 [ δμν+ ( Q−^2 − 1 ) uμuν ] , A−μν^1 =Q ...

Chapter 6 Summary: Decomposition of Second Rank Tensors Abstract This chapter provides a summary of formulae for the decompositi ...

Chapter 7 Fields, Spatial Differential Operators Abstract This chapter is devoted to the spatial differentiation of fields which ...

78 7 Fields, Spatial Differential Operators 7.1 Scalar Fields, Gradient. In physics, potential functions are important examples ...

7.1 Scalar Fields, Gradient 79 the equipotential surfaces are coaxial cylinders. In this case, the values of the potential are d ...

80 7 Fields, Spatial Differential Operators surface. Then one has dΦ=0, and consequently, in this case∇μΦdrμ=0. This means: the ...

7.1 Scalar Fields, Gradient 81 Clearly,viis the velocity of particle “i”. When the force can be derived from a potential,thisman ...

82 7 Fields, Spatial Differential Operators function. Some examples for the special forms of potential functions discussed in Se ...

7.1 Scalar Fields, Gradient 83 (iii) Spherical Symmetry Now the case is considered where the potential depends onrvia the magnit ...

84 7 Fields, Spatial Differential Operators r=|r 1 −r 2 |only, when these particles are effectively round and do not posses any ...

7.2 Vector Fields, Divergence and Curl or Rotation 85 (ii) Linearly Increasing Field Letv(r)be given by vμ=eμeνrν=xeμ. (7.16) In ...