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Hints and answers

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Hints and answers
10.12 HINTS AND ANSWERS
(a) Express z and the perpendicular distance ρ from P to the z-axis in terms of
u1 , u2 , u3 .
(b) Evaluate ∂x/∂ui , ∂y/∂ui , ∂z/∂ui , for i = 1, 2, 3.
(c) Find the Cartesian components of ûj and hence show that the new coordinates are mutually orthogonal. Evaluate the scale factors and the infinitesimal
volume element in the new coordinate system.
(d) Determine and sketch the forms of the surfaces ui = constant.
(e) Find the most general function f of u1 only that satisfies ∇2 f = 0.
10.12 Hints and answers
10.1
10.3
10.5
Group the term so that they form the total derivatives of compound vector
expressions. The integral has the value a × (a × b) + h.
For crossed uniform fields, ẍ+(Bq/m)2 x = q(E −Bv0 )/m, ÿ = 0, mż = qBx+mv0 ;
(b) ξ = Bqt/m; the path is a cycloid in the plane y = 0; ds = [(dx/dt)2 +
(dz/dt)2 ]1/2 dt.
g = r̈ − ω × (ω × r), where r̈ is the shell’s acceleration measured by an observer
fixed in space. To first order in ω, the direction of g is radial, i.e. parallel to r̈ .
(a) Note that s is orthogonal to g.
(b) If the actual time of flight is T , use (s + ∆) · g = 0 to show that
T ≈ τ(1 + 2g −2 (g × ω) · v + · · · ).
In the Coriolis terms, it is sufficient to put T ≈ τ.
(c) For this situation (g × ω) · v = 0 and ω × v = 0; τ ≈ 43 s and ∆ = 10–15 m
to the East.
10.7
10.9
10.11
10.13
10.15
10.17
(a) Evaluate (dr/du) · (dr/du).
(b) Integrate
√ the previous result between u = 0 and u = 1.
(c) t̂ = [ 2(1 + u2 )]−1 [(1 − u2 )i + 2uj + (1 + u2 )k]. Use dt̂/ds = (dt̂/du)/(ds/du);
ρ−1 = |dt̂/ds|.
√
(d) n̂ = (1 + u2 )−1 [−2ui + (1 − u2 )j]. b̂ = [ 2(1 + u2 )]−1 [(u2 − 1)i − 2uj + (1 + u2 )k].
Use db̂/ds = (db̂/du)/(ds/du) and show that this equals −[3a(1 + u2 )2 ]−1 n̂.
√
(e) Show that dn̂/ds = τ(b̂ − t̂) = −2[3 2a(1 + u2 )3 ]−1 [(1 − u2 )i + 2uj].
Note that dB = (dr · ∇)B and that B = B t̂, with t̂ = dr/ds. Obtain (B · ∇)B/B =
t̂(dB/ds) + n̂(B/ρ) and then take the vector product of t̂ with this equation.
To integrate sec2 φ(sec2 φ + tan2 φ)1/2 dφ put tan φ = 2−1/2 sinh ψ.
Work in Cartesian coordinates, regrouping the terms obtained by evaluating the
divergence on the LHS.
(a) 2z(x2 +y 2 +z 2 )−3 [(y 2 +z 2 )(y 2 +z 2 −3x2 )−4x4 ]; (b) 2r−1 cos θ (1−5 sin2 θ cos2 φ);
both are equal to 2zr −4 (r2 − 5x2 ).
Use the formulae given in table 10.2.
(a) C = −B0 /(µ0 a); B(ρ) = B0 ρ/a.
(b) B0 ρ2 /(3a) for ρ < a, and B0 [ρ/2 − a2 /(6ρ)] for ρ > a.
(c) [B02 /(2µ0 )][1 − (ρ/a)2 ].
10.19
10.21
Recall that ∇ × ∇φ = 0 for any scalar φ and that ∂/∂t and ∇ act on different
variables.
Two sets of paraboloids of revolution about the z-axis and the sheaf of planes
containing the z-axis. For constant u, −∞ < z < u2 /2; for constant v, −v 2 /2 <
z < ∞. The scale factors are hu = hv = (u2 + v 2 )1/2 , hφ = uv.
375
VECTOR CALCULUS
10.23
The tangent vectors are as follows: for u = 0, the line joining (1, 0, 0) and
(−1, 0, 0); for v = 0, the line joining (1, 0, 0) and (∞, 0, 0); for v = π/2, the line
(0, 0, z); for v = π, the line joining (−1, 0, 0) and (−∞, 0, 0).
ψ(u) = 2 tan−1 eu + c, derived from ∂[cosh u(∂ψ/∂u)]/∂u = 0.
376
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