8-54. Let E 1 ,E 2 ,E 3 and E^1 ,E^2 ,E^3 be a system of reciprocal basis. (See previous
problem).
(a) If A=A^1 E 1 +A^2 E 2 +A^3 E 3 find the components A^1 , A^2 , A^3 of A relative to the base
vectors E 1 ,E 2 ,E 3.
(b) If A =A 1 E^1 +A 2 E^2 +A 3 E^3 find the components A 1 , A 2 , A 3 relative to the basis
E^1 ,E^2 ,E^3 .The numbers Aiare called the contravariant components of A and the
numbers Aiare called the covariant components of A.
(c) Using the notation
Ei·Ej=gij =gji, and Ei·Ej=gij =gji,
where E 1 ,E 2 ,E 3 and E^1 ,E^2 ,E^3 is a reciprocal system of basis, show that
Ai=
∑^3
k=1
gik Ak and Ai=
∑^3
k=1
gikAk,
where iis called the free index and kis a summation index. Here gij are called
the conjugate metric components of the space and satisfy
∑^3
j=1
gijgjk =δikis the
Kronecker delta.
(d) Show that
g 11 g 12 g 13
g 21 g 22 g 23
g 31 g 32 g 33
g^11 g^12 g^13
g^21 g^22 g^23
g^31 g^32 g^33
=
1 0 0
0 1 0
0 0 1
or
∑^3
j=1
gijgjk =δik
8-55. Show that in an orthogonal curvilinear coordinate system (u, v, w ),the vec-
tors
(E 1 ,E 2 ,E 3 ) =
(
∂r
∂u ,
∂r
∂v ,
∂r
∂w
)
and (E^1 ,E^2 ,E^3 ) = (grad u, grad v, grad w)
are a reciprocal system of basis.
