10.12 HINTS AND ANSWERS
(a) Expresszand the perpendicular distanceρfromPto thez-axis in terms of
u 1 ,u 2 ,u 3.
(b) Evaluate∂x/∂ui,∂y/∂ui,∂z/∂ui,fori=1, 2 ,3.
(c) Find the Cartesian components ofuˆjand hence show that the new coordi-
nates are mutually orthogonal. Evaluate the scale factors and the infinitesimal
volume element in the new coordinate system.
(d) Determine and sketch the forms of the surfacesui=constant.
(e) Find the most general functionfofu 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 vector
expressions. The integral has the valuea×(a×b)+h.
10.3 For crossed uniform fields,x ̈+(Bq/m)^2 x=q(E−Bv 0 )/m,y ̈=0,m ̇z=qBx+mv 0 ;
(b)ξ=Bqt/m; the path is a cycloid in the planey=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 observer
fixed in space. To first order inω, the direction ofgis radial, i.e. parallel to ̈r′.
(a) Note thatsis orthogonal tog.
(b) If the actual time of flight isT,use(s+∆)·g=0toshowthatT≈τ(1 + 2g−^2 (g×ω)·v+···).In the Coriolis terms, it is sufficient to putT≈τ.
(c) For this situation (g×ω)·v=0andω×v= 0 ;τ≈43 s and ∆ = 10–15 m
to the East.10.7 (a) Evaluate (dr/du)·(dr/du).
(b) Integrate the previous result betweenu=0andu=1.
(c) ˆt=[
√
2(1 +u^2 )]−^1 [(1−u^2 )i+2uj+(1+u^2 )k]. Usedˆt/ds=(dˆt/du)/(ds/du);
ρ−^1 =|dˆt/ds|.
(d)ˆn=(1+u^2 )−^1 [− 2 ui+(1−u^2 )j].ˆb=[√
2(1 +u^2 )]−^1 [(u^2 −1)i− 2 uj+(1+u^2 )k].
Usedˆb/ds=(dbˆ/du)/(ds/du) and show that this equals−[3a(1 +u^2 )^2 ]−^1 ˆn.
(e) Show thatdˆn/ds=τ(ˆb−ˆt)=−2[3√
2 a(1 +u^2 )^3 ]−^1 [(1−u^2 )i+2uj].
10.9 Note thatdB=(dr·∇)Band thatB=Bˆt,withˆt=dr/ds.Obtain(B·∇)B/B=
ˆt(dB/ds)+ˆn(B/ρ) and then take the vector product ofˆtwith this equation.
10.11 To integrate sec^2 φ(sec^2 φ+tan^2 φ)^1 /^2 dφput tanφ=2−^1 /^2 sinhψ.
10.13 Work in Cartesian coordinates, regrouping the terms obtained by evaluating the
divergence on the LHS.
10.15 (a) 2z(x^2 +y^2 +z^2 )−^3 [(y^2 +z^2 )(y^2 +z^2 − 3 x^2 )− 4 x^4 ]; (b) 2r−^1 cosθ(1−5sin^2 θcos^2 φ);
both are equal to 2zr−^4 (r^2 − 5 x^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)forρ<a,andB 0 [ρ/ 2 −a^2 /(6ρ)] forρ>a.
(c) [B^20 /(2μ 0 )][1−(ρ/a)^2 ].10.19 Recall that∇×∇φ= 0 for any scalarφand that∂/∂tand∇act on different
variables.
10.21 Two sets of paraboloids of revolution about thez-axis and the sheaf of planes
containing thez-axis. For constantu,−∞<z<u^2 /2; for constantv,−v^2 / 2 <
z<∞. The scale factors arehu=hv=(u^2 +v^2 )^1 /^2 ,hφ=uv.