Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUSUse this formula to show that the area of the curved surface x 2 + y 2 − z 2 = a 2between the planes z =0andz =2a is(πa 2 6+ √ 1 sinh −1 2 √ )2 .210.12 For the functionz(x, y) =(x 2 − y 2 )e −x2 −y 2 ,find the location(s) at which the steepest gradient occurs. What are the magnitudeand direction of that gradient? The algebra involved is easier if plane polarcoordinates are used.10.13 Verify by direct calculation that∇ · (a × b) =b · (∇×a) − a · (∇×b).10.14 In the following exercises, a, b and c are vector fields.(a) Simplify∇×a(∇ · a) + a × [∇×(∇×a)] + a ×∇ 2 a.(b) By explicitly writing out the terms in Cartesian coordinates, prove that[c · (b · ∇) − b · (c · ∇)] a =(∇×a) · (b × c).(c) Prove that a × (∇×a) =∇( 1 2 a2 ) − (a · ∇)a.10.15 Evaluate the Laplacian of the functionzx 2ψ(x, y, z) =x 2 + y 2 + z 2(a) directly in Cartesian coordinates, and (b) after changing to a spherical polarcoordinate system. Verify that, as they must, the two methods give the sameresult.10.16 Verify that (10.42) is valid for each component separately when a is the Cartesianvector x 2 y i + xyz j + z 2 y k, by showing that each side of the equation is equal toz i +(2x +2z) j + x k.10.17 The (Maxwell) relationship between a time-independent magnetic field B and thecurrent density J (measured in SI units in A m −2 ) producing it,∇×B = µ 0 J,can be applied to a long cylinder of conducting ionised gas which, in cylindricalpolar coordinates, occupies the region ρaand B = B(ρ) forρ