Tensors for Physics

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

2

1

-1-0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

3

1, 2

-1-0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

3

1, 2

Fig. 5.2 Cross section of uniaxial (left) and biaxial (right) ellipsoids generated by linear mappings
withS(^1 )=S(^2 )= 0 .8,S(^3 )= 1 .4andS(^1 )= 1 .4,S(^2 )= 0 .6,S(^3 )=1, respectively. Theupper
diagrams show the cross sections in the 1–3- and 2–3-planes, thelowerones are for the 1–2-plane.
Thedashed circlecorresponds to the cross sections with the sphere generated by the isotropic part
of the tensor

V=

4

3

πdet(S), A=

2

3

π

(

SμμSνν−SμνSνμ

)

. (5.40)

The expression forAis equivalent toA =^43 π(S(^1 )S(^2 )+S(^2 )S(^3 )+S(^3 )S(^1 )).
Insertion of the decomposition (3.4)ofSinto isotropic and anisotropic parts, i.e. of

Sμν=S ̄δμν+SμνwithS ̄=^13 Sλλyields

ΔA=−

2

3

πSμν Sμν,ΔV=

4

9

π Sμν Sνλ Sλμ +S ̄ΔA (5.41)

for the differencesΔAandΔVbetween the area and the volume of the ellipsoid
and the pertaining sphere with the radiusS ̄generated by the linear mapping with the
isotropic part ofS.