Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALSin section 11.3, evaluate the two remaining line integrals and hence find the totalarea common to the two ellipses.11.6 By using parameterisations of the form x = a cos n θ and y = a sin n θ for suitablevalues of n, find the area bounded by the curvesx 2/5 + y 2/5 = a 2/5 and x 2/3 + y 2/3 = a 2/3 .11.7 Evaluate the line integral∮[I = y(4x 2 + y 2 ) dx + x(2x 2 +3y 2 ) dy ]Caround the ellipse x 2 /a 2 + y 2 /b 2 =1.11.8 Criticise the following ‘proof’ that π =0.(a) Apply Green’s theorem in a plane to the functions P (x, y) =tan −1 (y/x) andQ(x, y) =tan −1 (x/y), taking the region R to be the unit circle centred on theorigin.(b) The RHS of the equality so produced is∫ ∫y − xdx dy,R x 2 + y2 which, either from symmetry considerations or by changing to plane polarcoordinates, can be shown to have zero value.(c) In the LHS of the equality, set x =cosθ and y =sinθ, yielding P (θ) =θand Q(θ) =π/2 − θ. The line integral becomes∫ 2π0[( π)]2 − θ cos θ − θ sin θ dθ,which has the value 2π.(d) Thus 2π = 0 and the stated result follows.11.9 A single-turn coil C of arbitrary shape is placed in a magnetic field B and carriesa current I. Show that the couple acting upon the coil can be written as∫∫M = I (B · r) dr − I B(r · dr).CFor a planar rectangular coil of sides 2a and 2b placed with its plane verticaland at an angle φ to a uniform horizontal field B, show that M is, as expected,4abBI cos φ k.11.10 Find the vector area S of the part of the curved surface of the hyperboloid ofrevolutionx 2a − y2 + z 2=12 b 2that lies in the region z ≥ 0anda ≤ x ≤ λa.11.11 An axially symmetric solid body with its axis AB vertical is immersed in anincompressible fluid of density ρ 0 . Use the following method to show that,whatever the shape of the body, for ρ = ρ(z) in cylindrical polars the Archimedeanupthrust is, as expected, ρ 0 gV, whereV is the volume of the body.Express the vertical component of the resultant force on the body, − ∫ pdS,where p is the pressure, in terms of an integral; note that p = −ρ 0 gz and that foran annular surface element of width dl, n · n z dl = −dρ. Integrate by parts anduse the fact that ρ(z A )=ρ(z B )=0.410C