Mathematical Methods for Physics and Engineering - Matematica.NET

10.12 HINTS AND ANSWERS(a) Express z and the perpendicular distance ρ from P to the z-axis in terms ofu 1 ,u 2 ,u 3 .(b) Evaluate ∂x/∂u i , ∂y/∂u i , ∂z/∂u i ,fori =1, 2, 3.(c)Find the Cartesian components of û j and hence show that the new coordinatesare mutually orthogonal. Evaluate the scale factors and the infinitesimalvolume element in the new coordinate system.(d) Determine and sketch the forms of the surfaces u i =constant.(e) Find the most general function f of u 1 only that satisfies ∇ 2 f =0.10.12 Hints and answers10.1 Group the term so that they form the total derivatives of compound vectorexpressions. The integral has the value a × (a × b)+h.10.3 For crossed uniform fields, ẍ+(Bq/m) 2 x = q(E −Bv 0 )/m, ÿ =0,mż = qBx+mv 0 ;(b) ξ = Bqt/m; the path is a cycloid in the plane y =0;ds =[(dx/dt) 2 +(dz/dt) 2 ] 1/2 dt.10.5 g = ¨r ′ − ω × (ω × r), where ¨r ′ is the shell’s acceleration measured by an observerfixed in space. To first order in ω, the direction of g is radial, i.e. parallel to ¨r ′ .(a) Note that s is orthogonal to g.(b) If the actual time of flight is T ,use(s +∆)· g =0toshowthatT ≈ τ(1 + 2g −2 (g × ω) · v + ···).In the Coriolis terms, it is sufficient to put T ≈ τ.(c) For this situation (g × ω) · v =0andω × v = 0; τ ≈ 43 s and ∆ = 10–15 mto the East.10.7 (a) Evaluate (dr/du) · (dr/du).(b) Integrate the previous result between u =0andu =1.(c) ˆt =[ √ 2(1 + u 2 )] −1 [(1 − u 2 )i +2uj +(1+u 2 )k]. Use dˆt/ds =(dˆt/du)/(ds/du);ρ −1 = |dˆt/ds|.(d) ˆn =(1+u 2 ) −1 [−2ui +(1− u 2 )j]. ˆb =[ √ 2(1 + u 2 )] −1 [(u 2 − 1)i − 2uj +(1+u 2 )k].Use dˆb/ds =(dˆb/du)/(ds/du) and show that this equals −[3a(1 + u 2 ) 2 ] −1 ˆn.(e) Show that dˆn/ds = τ(ˆb − ˆt) =−2[3 √ 2a(1 + u 2 ) 3 ] −1 [(1 − u 2 )i +2uj].10.9 Note that dB =(dr · ∇)B and that B = Bˆt, withˆt = dr/ds. Obtain(B · ∇)B/B =ˆt(dB/ds)+ˆn(B/ρ) and then take the vector product of ˆt with this equation.10.11 To integrate sec 2 φ(sec 2 φ +tan 2 φ) 1/2 dφ put tan φ =2 −1/2 sinh ψ.10.13 Work in Cartesian coordinates, regrouping the terms obtained by evaluating thedivergence on the LHS.10.15 (a) 2z(x 2 +y 2 +z 2 ) −3 [(y 2 +z 2 )(y 2 +z 2 −3x 2 )−4x 4 ]; (b) 2r −1 cos θ (1−5sin 2 θ cos 2 φ);both are equal to 2zr −4 (r 2 − 5x 2 ).10.17 Use the formulae given in table 10.2.(a) C = −B 0 /(µ 0 a); B(ρ) =B 0 ρ/a.(b) B 0 ρ 2 /(3a) forρa.(c) [B0 2/(2µ 0)][1 − (ρ/a) 2 ].10.19 Recall that ∇×∇φ = 0 for any scalar φ and that ∂/∂t and ∇ act on differentvariables.10.21 Two sets of paraboloids of revolution about the z-axis and the sheaf of planescontaining the z-axis. For constant u, −∞