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PHYS 3900 Homework Set #04

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PHYS 3900 Homework Set #04
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
PHYS 3900 Homework Set #04
Due: Mon. Mar. 14, 2016, 4:00pm (All Parts!)
All textbook problems assigned, unless otherwise stated, are from the textbook by M. Boas
Mathematical Methods in the Physical Sciences, 3rd ed. Textbook sections are identified
as ”Ch.cc.ss” for textbook Chapter ”cc”, Section ”ss”. Complete all HWPs assigned: only
two of them will be graded; and you don’t know which ones! Read all ”Hints” before you
proceed! Make use of the latest version of the ”PHYS3900 Homework Toolbox”, posted on
the course web site.
Do not use the calculator, unless so instructed! All arithmetic, to the extent required, is
either elementary or given in the problem statement. State all your answers in terms of
√
real-valued elementary functions (+, −, /, ×, power, root, , exp, ln, sin, cos, tan, cot,
arcsin, arccos, arctan, arcot, ...) of integer numbers, e and π; in terms of i where needed;
and in terms of specificpinput variables, as stated in each problem. So, for example, if the
result is, say, ln(7/2) + (e5 π/3), or 179/2 , or (16−4π)−10 , then just state that as your final
answer: no need to evaluate it as decimal number by calculator! Simplify final results to the
largest extent possible; e.g., reduce fractions of integers to the smallest denominator etc..
1
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
HWP 04.01: Divergence and curl of a vector field.
(a) Calculate the divergence and the curl of the vector field
F~ (~r) ≡ [Fx (~r), Fy (~r), Fz (~r)] = [2z 2 , 3z 2 , (4x + 6y)z] where ~r ≡ [x, y, z] .
~
R(b) Use Stokes’s theorem to show that the line integral of F (~r) over any curve L, given by
F~ (~r) · d~r, depends only on the start- and endpoint of L, but not on the trajectory of L
L
between those two points.
Hint: Consider two different curves, L and M, say, which share a common start- and
endpoint. Construct from them a closed curve C to which Stokes’s theorem can be applied.
HWP 04.02: Line integral and scalar potential of a vector field.
(a) Calculate, for the vector field F~ (~r) defined in HWP 04.01, the line integral
Z
~
~
F~ (~r) · d~r
∆Φ(B, A) :=
~ B)
~
L(A→
~ → B)
~ is the straight line segment, drawn from point A
~ ≡ (Ax , Ay , Az ) to point
where L(A
~ ≡ (Bx , By , Bz ), parameterized, e.g., by ~r(τ ) = p~ +τ ~q with parameter variable τ ∈ [−1, +1];
B
~ + A)/2
~
~ − A)/2.
~
~ and ~r(+1) = B
~ !)
where p~ := (B
and ~q := (B
(Check that ~r(−1) = A
Hints: Convert the line integral into an integral of the parameter variable τ over [−1, +1].
Write the resulting τ -integrand in the form aτ 2 + bτ + c with coefficients a, b and c expressed
in terms of px , py , pz , qx , qy , and qz . Integrate! You only need a and c; b drops out. Collect
qz2 -, p2z - and pz qz -terms and express them in terms of Az and Bz by using qz2 +p2z = (Bz2 +A2z )/2
and 2qz pz = (Bz2 − A2z )/2. Then use qα + pα = Bα and qα − pα = −Aα for α ≡ x, y.
(b) Use the result from part (a) to find a scalar potential function Φ(~r) such that line integral
~ A)
~ can be written in terms of Φ(~r) as:
∆Φ(B,
~ A)
~ = Φ(B)
~ − Φ(A)
~
∆Φ(B,
Hint: Try, for example, Φ(~r) = (2x + 3y)z 2 − 427.
(c) By evaluating the partial derivatives of Φ(~r) show explicitly that
~ r) = F~ (~r) .
∇Φ(~
HWP 04.03: Prove the following ”product rules” of vector calculus, using the conventional
product rule, (uv)0 = uv 0 + u0 v, applied to the relevant partial derivatives, as needed:
(a) for two scalar fields Φ(~r) and Ψ(~r)
~
~ + Ψ∇Φ
~ ;
∇(ΦΨ)
= Φ∇Ψ
~ r) and B(~
~ r)
(b) for two vector fields A(~
~ · (A
~ × B)
~ =B
~ · (∇
~ × A)
~ −A
~ · (∇
~ × B)
~ ;
∇
2
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
~ r) and a scalar field Φ(~r)
(c) for a vector field A(~
~ × (ΦA)
~ = Φ(∇
~ × A)
~ −A
~ × (∇Φ)
~
∇
.
HWP 04.04: Volume and surface integration by parts.
~ · (ΦA)
~ = Φ∇
~ ·A
~ +A
~ · ∇Φ
~
(a) Use Gauss’s theorem and (without proof) the product rule ∇
to prove the following ”integration-by-parts” rule for a scalar field Φ(~r) and a vector field
~ r)
A(~
Z
Z
I
~ ·A
~ dv = − A
~ · ∇Φ
~ dv + ΦA
~ · d~a
Φ∇
V
V
S
where V is a 3D volume with volume elements dv; and S is the oriented closed surface that
encloses V, with outward directed surface area element vectors d~a.
(b) Use Gauss’s theorem and the product rule HWP 04.03 Part(b) to prove the following
~ r) and B(~
~ r)
”integration-by-parts” rule for vector fields A(~
I
Z
Z
~ × B)
~ · d~a
~
~
~
~
~
~
A · (∇ × B) dv + (A
B · (∇ × A) dv =
S
V
V
where V is a 3D volume with volume elements dv; and S is the oriented closed surface that
encloses V, with outward directed surface area element vectors d~a.
(c) Use Stokes’s theorem and the product rule from HWP 04.03 Part(c), to prove the fol~ r)
lowing ”integration-by-parts” rule for a scalar field Φ(~r) and a vector field A(~
Z
Z
I
~ × ∇Φ)
~
~ · d~r
~ × A)
~ · d~a = (A
· d~a + ΦA
Φ(∇
S
S
C
where S is an oriented open surface with surface area element vectors d~a; and C is the
closed curve that encloses S with line element vectors d~r, encircling the d~a-vectors with
right-handed orientation.
HWP 04.05: Gauss for curls, Stokes for gradients.
(a) Use the integration-by-parts rule from HWP 04.04 Part (b), to prove the ”curl version”
~ r):
of Gauss’s theorem for any vector field B(~
Z
I
~
~
~ × d~a
∇ × B dv = − B
V
S
where V is a 3D volume with volume elements dv; and S is the oriented closed surface that
encloses V, with outward directed surface area element vectors d~a.
~ r) in HWP 04.04 Part (b) is simply some arbitrary constant
Hint: Assume the vector field A(~
~ independent of ~r. Then, apply the triple product rule [i.e., for any three vectors ~u,
vector A
~v , w,
~ the triple product
obeys: (~u ×H~v ) · w
~ = ~u · (~v × w)]
~ to write the surface integral in HWP
H
~
~
~
~
~ · ... from
04.04 Part (b) as S (A × B) · d~a = S A · (B × d~a) Then factor out the constant A
R
H
~ ×A
~ = 0 to obtain 0 = A
~ ·[ ∇
~ ×B
~ dv + B
~ × d~a]. Then set, e.g.,
the integrals and use ∇
V
S
3
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
~ = [1, 0, 0] ≡ x̂ or A
~ = [0, 1, 0] ≡ ŷ or A
~ = [0, 0, 1] ≡ ẑ, to prove the ”curl-Gauss” theorem
A
separately for the x-, y- and z-components of the volume and surface integrals involved.
(b) Use the integration-by-parts rule from HWP 04.04 Part (c), to prove the ”gradient
version” of Stokes’s theorem for any scalar field Φ(~r):
Z
I
~
∇Φ × d~a = − Φ d~r
S
C
where S is an oriented open surface with surface area element vectors d~a; and C is the
closed curve that encloses S with line element vectors d~r, encircling the d~a-vectors with
right-handed orientation.
~ is a constant vector, apply the triple
Hints: Use the same tricks as in Part (a): assume A
~ · ... from the two non-zero integrals on
product rule to the surface integral, and factor out A
the right-hand side of HWP 04.04 Part (c).
HWP 04.06: Proving Cauchy’s integral theorem by Stokes’ theorem.
Consider the plane of complex numbers
ζ = x + iy ,
with x = Re(ζ) , y = Im(ζ) ,
as the x-y-plane, embedded in a 3D space with position vectors ~r = [x, y, z]. Note that the
symbol ”ζ” (not ”z”!) is used here to denote a complex variable. This is done in order to
clearly distinguish this complex variable from the real-valued variable ”z” which is used here
to denote the 3rd coordinate of the 3D coordinate vector ~r.
Also consider a complex-valued function f (ζ) defined in terms of real-valued functions u(x, y)
and v(x, y) by
f (ζ) = u(x, y) + iv(x, y) ,
with u(x, y) = Re[f (ζ)] , v(x, y) = Im[f (ζ)] .
on some open set S in the complex ζ-plane.
~ r) and H(~
~ r) by
Given f (ζ) = u(x, y) + iv(x, y), we can define two 3D vector fields G(~
~ r) ≡ [Gx (~r), Gy (~r), Gz (~r)] := [ +u(x, y), −v(x, y), 0 ]
G(~
and
~ r) ≡ [Hx (~r), Hy (~r), Hz (~r)] := [ +v(x, y), +u(x, y), 0 ] .
H(~
~ r) and H(~
~ r) will be referred to as the ”associated 3D vector fields of f (ζ)”.
The fields G(~
They are both defined on a 3D open prismatic/cylindrical volume V which has the open set
S as its cross-sectional area and a mantle parallel to the z-axis, as shown in Fig. 04.06:
V := {~r | ~r = [x, y, z] with x + iy ∈ S and − ∞ < z < ∞} .
Note that V is singly connected in 3D space if S is singly connected in the 2D complex
~ r) and H(~
~ r) have the special property that their z-components
ζ-plane. Note also that G(~
(Gz , Hz ) are zero and their x- and y-components (Gx , Gy , Hx , Hy ) are independent of z.
4
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
Consider further a curve C inside the complex open set S, defined by some parameterization
ζ(τ ) = x(τ ) + iy(τ ) on a real τ -interval [a, b]. C can then also be regarded as a curve in the
x-y-plane in the 3D open set V, parameterized by
~r(τ ) = [ x(τ ), y(τ ), 0 ] .
(a) Show that
Z
Z
~ r) · d~r + i
G(~
f (ζ)dζ =
C
C
Z
~ r) · d~r .
H(~
C
~ r) and H(~
~ r), have vanishing curls,
(b) Show that the associated 3D vector fields, of f (ζ), G(~
~ × G(~
~ r) = 0
∇
~ × H(~
~ r) = 0
∇
and
for all ~r ∈ V ,
if f (ζ) is complex differentiable for all ζ ∈ S.
Hint: If f (ζ) is complex differentiable then u(x, y) and v(x, y) obey the Cauchy-Riemann
differential equations. You may use these equations here without proof.
(c) Assume now that f (ζ) is complex differentiable for all ζ ∈ S; that S is singly connected
in the 2D complex ζ-plane; and that C is a simple closed curve in S, enclosing a singly
~ r) and H(~
~ r),
connected interior I = Interior(C) ⊂ S. Use the associated 3D vector fields G(~
the results from Parts (a) and (b), and Stokes’ theorem to prove Cauchy’s integral theorem:
I
f (ζ)dζ = 0 .
C
5
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
Fig. 04.06 6
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
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