Mathematical Methods for Physics and Engineering - Matematica.NET

10.11 EXERCISES∇Φ =∇ · a =∇×a =∇ 2 Φ =1 ∂Φê 1 + 1 ∂Φê 2 + 1 ∂Φê 3h 1 ∂u 1 h 2 ∂u 2 h 3 ∂u 3[1 ∂(h 2 h 3 a 1 )+h 1 h 2 h 3 ∂u 1∣ ∣ ∣∣∣∣∣∣ h 1 ê 1 h 2 ê 2 h 3 ê 3 ∣∣∣∣∣∣1 ∂ ∂ ∂h 1 h 2 h 3 ∂u 1 ∂u 2 ∂u 3h 1 a 1 h 2 a 2 h 3 a 3∂∂u 2(h 3 h 1 a 2 )+∂ ](h 1 h 2 a 3 )∂u 3[ ( )1 ∂ h2 h 3 ∂Φ+ ∂ ( )h3 h 1 ∂Φ+ ∂ ( )]h1 h 2 ∂Φh 1 h 2 h 3 ∂u 1 h 1 ∂u 1 ∂u 2 h 2 ∂u 2 ∂u 3 h 3 ∂u 3Table 10.4 Vector operators in orthogonal curvilinear coordinates u 1 ,u 2 ,u 3 .Φ is a scalar field and a is a vector field.Letting Φ = a 1 h 1 in (10.60) and substituting into the above equation, we find∇×(a 1 ê 1 )=ê2 ∂(a 1 h 1 ) −ê3 ∂(a 1 h 1 ).h 3 h 1 ∂u 3 h 1 h 2 ∂u 2The corresponding analysis of ∇×(a 2 ê 2 ) produces terms in ê 3 and ê 1 , whilst that of∇×(a 3 ê 3 ) produces terms in ê 1 and ê 2 . When the three results are added together, thecoefficients multiplying ê 1 , ê 2 and ê 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 cylindricaland spherical polar coordinates, given in tables 10.2 and 10.3 respectively, areobtained 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 inchapter 26.10.11 Exercises10.1 Evaluate the integral∫ [a(ḃ · a + b · ȧ)+ȧ(b · a) − 2(ȧ · a)b − ḃ|a|2] dtin which ȧ, ḃ are the derivatives of a, b with respect to t.10.2 At time t = 0, the vectors E and B are given by E = E 0 and B = B 0 ,wheretheunit vectors, E 0 and B 0 are fixed and orthogonal. The equations of motion aredEdt = E 0 + B × E 0 ,dBdt = B 0 + E × B 0 .Find E and B at a general time t, showing that after a long time the directionsof E and B have almost interchanged.369