LINE, SURFACE AND VOLUME INTEGRALS
11.11 Hints and answers11.1 Show that∇×F= 0. The potentialφF(r)=x^2 z+y^2 z^2 −z.
11.3 (a)c^3 ln 2i+2j+(3c/2)k;(b)(− 3 c^4 /8)i−cj−(c^2 ln 2)k;(c)c^4 ln 2−c.
11.5 ForP,x=y=ab/(a^2 +b^2 )^1 /^2. The relevant limits are 0≤θ 1 ≤tan−^1 (b/a)and
tan−^1 (a/b)≤θ 2 ≤π/2. The total common area is 4abtan−^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
∫
Cr×(dr×B). Show that the horizontal sides in the first term and the
whole of the second term contribute nothing to the couple.
11.11 Note that, ifˆnis the outward normal to the surface,nˆz·ˆndlis 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 radiuscwith its axis parallel to thez-direction and passing through
(a, b). The pitch of the spiral is 2πc^2. No, because (i)γis not a closed loop and
(ii) the line integral must be zero foreveryclosed loop, not just for a particular
one. In fact∇×f=− 2 k= 0 shows thatfis not conservative.
11.19 (a)dS=(2a^3 cosθsin^2 θcosφi+2a^3 cosθsin^2 θsinφj+a^2 cosθsinθk)dθ dφ.
(b)∇·r= 3; over the planez=0,r·dS=0.
The necessarily common value is 3πa^4 /2.
11.21 Writeras∇(^12 r^2 ).
11.23 The answer is 3
√
3 πα/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 − 2 e−^1 , 0 ,2+^12 e,−^73 ,−1+2e−^1 ,−^12.
(b)∇×F=2xy z^2 i−y^2 z^2 j+yexk. Show that the contour is equivalent to the
sum of two plane square contours in the planesz=0andx= 1, the latter being
traversed in the negative sense. Integral =^16 (3e−5).