From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 43

Tab l e 1 . 3 Various deformation tensors of the first kind

Definitions Author Comments
E 1 =C l =S l R ∗ G r RS l =J l G r J l Cauchy ( 1889 , 1890 ) (“left Cauchy-Green”) if G r =I,then
C l =S l =J l J l
E 2 =C r =S r R ∗ G l RS r =J r G l J r Green ( 1839 ) (“right Cauchy-Green”) if G l =I,then
C r =S r =J r J r

E 3 =C l G − 1
l

Grafarend ( 1995 ) (“left-right Cauchy-Green”) if G l =I,then

E 3 =C l

E 4 =C r G − 1 Grafarend ( 1995 ) (“right-left Cauchy-Green”) if G r =I,

then E 4 =C r

r

E 5 =G l C − 1

Grafarend ( 1995 ) (“inverse left-right Cauchy-Green”) Finger ( 1894a )ifG l =I,
then E 5 =E − 1

l

3

E 6 =G r C − 1

Grafarend ( 1995 ) (“inverse right-left Cauchy-Green”) Piola ( 1836 ), if G r =I,
then E 6 =E − 1

r

4

E 7 =C m/ 2
l

Λ m
1

,Λ m ∼{

2 }

Seth ( 1964a , b )( m ∈ Z,m =0) m =2:E 7 =E 1

E 8 =InC l ∼{ In Λ 1 , In Λ 2 }

−

Hencky ( 1928 )

E 9 =C m/ r ∼{ λ m

1

,λ m

2 }

Seth ( 1964a , b ) m =2:E 9 =E 2

E 10 =InC r ∼{ In λ 1 , In λ 2 } Hencky ( 1928 ) −

E 11 =E l = 2

(C l − G l ) Cauchy ( 1889 , 1890 ) (“left Euler-Lagrange”) if G l =I,then

E l = 2 (C l − I)

E 12 =E r = 2 (G r − C r ) Almansi ( 1911 ) (“right Euler-Lagrange”) if G r =I,then

E r = 2

−

(I C r )

E 13 =E l G − 1
l

= 2 (C l G − 1

l

− I) Grafarend ( 1995 ) (“left-right Euler-Lagrange”) if G l =I,then

E 13 =E l

E 14 =E r G − 1 = 2 C r G − 1

(I − ) Grafarend ( 1995 ) (“right-left Euler-Lagrange”) if G r =I,

then E 14 =E r

r r

E 15 = 2 (C − 1

l

G − l )

−

Karni and Reiner ( 1960 ) if G l =Ithen

E 15 = 2 (C − 1

l

− I)

Definitions Author Comments

E 16 = 2 (G − r − C − 1

r

) Karni and Reiner ( 1960 ) if G r =Ithen

E 16 = 2 C − 1

r

(I − )

E 17 =G l E − 1

Grafarend ( 1995 ) (“inverse left-right Euler-Lagrange”) if G l =I,then
E 17 =E − 1
l

l

E 18 =G r E − 1

Grafarend ( 1995 ) (“inverse right-left Euler-Lagrange”) if G r =I,then
E 18 =E − 1

r

r

E 19 =E m/ 2
l

K m/ 2
1

,K m/ 2
2

∼{ } Seth ( 1964a , b )( m ∈ Z,m =0) m =2:E 19 =E 11

E 20 = 2

In E l Hencky ( 1928 ) −

E 21 =E m/ r ∼{ K m/ 2

1

,K m/ 2
2

} Seth ( 1964a , b ) m =2:E 21 =E 12

E 22 = 2

In E r Hencky ( 1928 ) −

cos Ψ l = U 1 | U 2 =

cos Ψ r = u 1 | u 2 =

versus

(1.133)

U 1 U 2

g μν u 1 u 2 G MN

√ G AB

U 2 √ G ΓΔ U 1

√ g αβ u 1 u 2 √ g γδ u 1 u 2

= = .

U 1 U 2

The second additive measure of deformation is the angular shear or the angle of shear (Σ l is of
type “left” and Σ r is of type “right”, respectively)

Σ l =Σ:= Ψ l − Ψ r versus Σ r = σ := Ψ r − Ψ l . (1.134)