Mathematical Methods for Physics and Engineering - Matematica.NET

11.10 EXERCISESeverywhere except on the axis ρ =0,wherev has a singularity. Therefore ∮ C v · drequals zero for any path C that does not enclose the vortex line on the axis and2π if C does enclose the axis. In order for Stokes’ theorem to be valid for allpaths C, we therefore set∇×v =2πδ(ρ),where δ(ρ) is the Dirac delta function, to be discussed in subsection 13.1.3. Now,since ∇×v = 0, except on the axis ρ = 0, there exists a scalar potential ψ suchthat v = ∇ψ. It may easily be shown that ψ = φ, the polar angle. Therefore, if Cdoes not enclose the axis then ∮ ∮v · dr = dφ =0,Cand if C does enclose the axis,∮v · dr =∆φ =2πn,Cwhere n is the number of times we traverse C. Thus φ is a multivalued potential.Similar analyses are valid for other physical systems – for example, in magnetostaticswe may replace the vortex lines by current-carrying wires and the velocityfield v by the magnetic field B.11.10 Exercises11.1 The vector field F is defined byF =2xzi +2yz 2 j +(x 2 +2y 2 z − 1)k.Calculate ∇×F and deduce that F can be written F = ∇φ. Determine the formof φ.11.2 The vector field Q is defined byQ = [ 3x 2 (y + z)+y 3 + z 3] i + [ 3y 2 (z + x)+z 3 + x 3] j + [ 3z 2 (x + y)+x 3 + y 3] k.Show that Q is a conservative field, construct its potential function and henceevaluate the integral J = ∫ Q · dr along any line connecting the point A at(1, −1, 1) to B at (2, 1, 2).11.3 F is a vector field xy 2 i+2j+xk, andL is a path parameterised by x = ct, y = c/t,z = d for the range 1 ≤ t ≤ 2. Evaluate (a) ∫ F dt, (b)∫ F dy and (c) ∫ F · dr.L L L11.4 By making an appropriate choice for the functions P (x, y)andQ(x, y) that appearin Green’s theorem in a plane, show that the integral of x − y over the upper halfof the unit circle centred on the origin has the value − 2 . Show the same result3by direct integration in Cartesian coordinates.11.5 Determine the point of intersection P , in the first quadrant, of the two ellipsesx 2a + y22 b =1 and x 22 b + y22 a =1. 2Taking b