10.11 EXERCISES
∇Φ=
1
h 1∂Φ
∂u 1ˆe 1 +1
h 2∂Φ
∂u 2ˆe 2 +1
h 3∂Φ
∂u 3ˆe 3∇·a =1
h 1 h 2 h 3[
∂
∂u 1(h 2 h 3 a 1 )+∂
∂u 2(h 3 h 1 a 2 )+∂
∂u 3(h 1 h 2 a 3 )]
∇×a =1
h 1 h 2 h 3∣
∣
∣∣
∣
∣∣
h 1 ˆe 1 h 2 eˆ 2 h 3 ˆe 3
∂
∂u 1∂
∂u 2∂
∂u 3
h 1 a 1 h 2 a 2 h 3 a 3∣
∣
∣∣
∣
∣∣
∇^2 Φ=
1
h 1 h 2 h 3[
∂
∂u 1(
h 2 h 3
h 1∂Φ
∂u 1)
+
∂
∂u 2(
h 3 h 1
h 2∂Φ
∂u 2)
+
∂
∂u 3(
h 1 h 2
h 3∂Φ
∂u 3)]
Table 10.4 Vector operators in orthogonal curvilinear coordinatesu 1 ,u 2 ,u 3.
Φ is a scalar field andais a vector field.Letting Φ =a 1 h 1 in (10.60) and substituting into the above equation, we find
∇×(a 1 eˆ 1 )=ˆe 2
h 3 h 1∂
∂u 3(a 1 h 1 )−eˆ 3
h 1 h 2∂
∂u 2(a 1 h 1 ).The corresponding analysis of∇×(a 2 ˆe 2 ) produces terms in ˆe 3 andeˆ 1 , whilst that of
∇×(a 3 ˆe 3 ) produces terms ineˆ 1 andˆe 2. When the three results are added together, the
coefficients multiplyingeˆ 1 ,ˆe 2 andˆe 3 are the same as those obtained by writing out (10.62)
explicitly, thus proving the stated result.
The general expressions for the vector operators in orthogonal curvilinearcoordinates are shown for reference in table 10.4. The explicit results for cylindrical
and spherical polar coordinates, given in tables 10.2 and 10.3 respectively, are
obtained by substituting the appropriate set of scale factors in each case.
A discussion of the expressions for vector operators in tensor form, whichare valid even for non-orthogonal curvilinear coordinate systems, is given in
chapter 26.
10.11 Exercises10.1 Evaluate the integral
∫[
a( ̇b·a+b· ̇a)+ ̇a(b·a)−2( ̇a·a)b− ̇b|a|^2
]
dtin which ̇a, ̇bare the derivatives ofa,bwith respect tot.
10.2 At timet= 0, the vectorsEandBare given byE=E 0 andB=B 0 ,wherethe
unit vectors,E 0 andB 0 are fixed and orthogonal. The equations of motion are
dE
dt
=E 0 +B×E 0 ,
dB
dt=B 0 +E×B 0.
FindEandBat a general timet, showing that after a long time the directions
ofEandBhave almost interchanged.