8-51. Show in a general orthogonal coordinate system div (grad v×grad w) = 0
8-52. For ˆe 1 ,ˆe 2 ,ˆe 3 independent orthogonal unit vectors (base vectors), one can
express any vector A as
A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 eˆ 3 ,
where A 1 , A 2 , A 3 are the coordinates of A relative to the base vectors chosen.
(a) Show that these components are the projection of A onto the base vectors
and
A= (A·ˆe 1 )ˆe 1 + (A·ˆe 2 )ˆe 2 + (A·eˆ 3 )ˆe 3.
(b) By selecting any three independent orthogonal vectors,E 1 ,E 2 ,E 3 ,not nec-
essarily of unit length, show that one can write
A=
(
A·E 1
E 1 ·E 1
)
E 1 +
(
A·E 2
E 2 ·E 2
)
E 2 +
(
A·E 3
E 3 ·E 3
)
E 3.
Consequently,
A·Ei
Ei·Ei, i = 1,^2 ,or^3
are the components of A relative to the chosen base vectors E 1 ,E 2 ,E 3.
8-53. Two bases E 1 ,E 2 ,E 3 and E^1 ,E^2 ,E^3 are said to be reciprocal if they satisfy
the condition
Ei·Ej=
{ 1 if i=j
0 if i=j
(i.e., A vector from one basis is orthogonal to two of the vectors from the other
basis). Show that if E 1 ,E 2 ,E 3 is a given set of base vectors, then
E^1 =^1
V
E 2 ×E 3 , E^2 =^1
V
E 3 ×E 1 , E^3 =^1
V
E 1 ×E 2
is a reciprocal basis, where V =E 1 ·(E 2 ×E 3 )is a triple scalar product and represents
the volume of the parallelepiped having the basis vectors for its sides. Show also
that E^1 ·(E^2 ×E^3 ) =^1
V
