Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALS11.11 Hints and answers11.1 Show that ∇×F = 0. The potential φ F (r) =x 2 z + y 2 z 2 − z.11.3 (a) c 3 ln 2 i +2j +(3c/2)k; (b)(−3c 4 /8)i − c j − (c 2 ln 2)k; (c)c 4 ln 2 − c.11.5 For P , x = y = ab/(a 2 + b 2 ) 1/2 . The relevant limits are 0 ≤ θ 1 ≤ tan −1 (b/a) andtan −1 (a/b) ≤ θ 2 ≤ π/2. The total common area is 4ab tan −1 (b/a).11.7 Show that, in the notation of section 11.3, ∂Q/∂x − ∂P/∂y =2x 2 ; I = πa 3 b/2.11.9 M = I ∫ r × (dr × B). Show that the horizontal sides in the first term and theCwhole of the second term contribute nothing to the couple.11.11 Note that, if ˆn is the outward normal to the surface, ˆn z · ˆn dl is equal to −dρ.11.13 (b) φ = c + z/r.11.15 (a) Yes, F 0 (x − y)exp(−r 2 /a 2 ); (b) yes, −F 0 [(x 2 + y 2 )/(2a)] exp(−r 2 /a 2 );(c) no, ∇×F ≠ 0.11.17 A spiral of radius c with its axis parallel to the z-direction and passing through(a, b). The pitch of the spiral is 2πc 2 . No, because (i) γ is not a closed loop and(ii) the line integral must be zero for every closed loop, not just for a particularone. In fact ∇×f = −2k ≠ 0 shows that f is not conservative.11.19 (a) dS =(2a 3 cos θ sin 2 θ cos φ i +2a 3 cos θ sin 2 θ sin φ j + a 2 cos θ sin θ k) dθ dφ.(b) ∇ · r = 3; over the plane z =0,r · dS =0.The necessarily common value is 3πa 4 /2.11.21 Write r as ∇( 1 2 r2 ).11.23 The answer is 3 √ 3πα/2 in each case.11.25 Identify the expression for ∇ · (E × B) and use the divergence theorem.11.27 (a) The successive contributions to the integral are:1 − 2e −1 , 0, 2+ 1 2 e, − 7 3 , −1+2e−1 , − 1 2 .(b) ∇×F =2xyz 2 i − y 2 z 2 j + ye x k. Show that the contour is equivalent to thesum of two plane square contours in the planes z =0andx = 1, the latter beingtraversed in the negative sense. Integral = 1 (3e − 5).6414