Physics 3900 University of Georgia Spring 2019 Instructor: HBSch¨uttler

PHYS 3900 Homework Set #04 Due: Mon. Mar. 18, 2019, 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 specific input variables, as stated in each problem. So, for example, if the result is, say, ln(7/2) + p(e5π/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 University of Georgia Spring 2019 Instructor: HBSch¨uttler

HWP 04.01: Divergence and curl of a vector field. (a) Calculate the divergence and the curl of the vector field

~ 2 2 F (~r) ≡ [Fx(~r),Fy(~r),Fz(~r)] = [2z , 3z , (4x + 6y)z] where ~r ≡ [x, y, z] .

(b) Use Stokes’s Theorem to show that the line integral of F~ (~r) over any curve L, given by Z F~ (~r) · d~r, L depends only on the start- and endpoint of L, but not on the trajectory of L between those two points. Hint: (1) Consider two different curves, L and M, say, which share a common start- and endpoint. (2) For your orientation and visualization, make a drawing to sketch L and M. (3) By reversal and concatenation, i.e., by anti-concatenation, construct from L and M a closed H ~ curve C to which Stokes’s Theorem can be applied. (3) Then use the result for C F (~r) · d~r, R ~ R ~ from Stokes’s Theorem. to prove that L F (~r) · d~r and M F (~r) · d~r are the same.

HWP 04.02: Line integral and scalar potential of a vector field. (a) For F~ (~r) defined in HWP 04.01, calculate by parameterization the line integral Z ~ IL := F (~r) · d~r L where L is a straight line segment, drawn from some point ~rA ≡ [Ax,Ay,Az] to some other point ~rB ≡ [Bx,By,Bz]. Parameterize L by ~r(τ) = ~p + τ~q, with parameter variable τ ∈ [−1, +1], where ~p := (~rB + ~rA)/2 and ~q := (~rB − ~rA)/2. Check that indeed ~r(τ) = ~rA for τ = −1 and ~r(τ) = ~rB for τ = +1 . Express IL in terms of Ax,Ay,Az,Bx,By,Bz. 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: 2 2 why? Collect qz -, pz- and pzqz-terms and express them in terms of Az and Bz by using 2 2 2 2 2 2 qz + pz = (Bz + Az)/2 and 2qzpz = (Bz − Az)/2. Then use qα + pα = Bα and qα − pα = −Aα for α ≡ x, y. (b) Use the result from part (a) to find a so-called ”scalar potential” function, Φ(~r), such that line integral IL can be written in terms of Φ(~r) as:

IL = Φ(~rB) − Φ(~rA) . Hint: Try, for example, a function Φ(~r) of the general form

Φ(~r) = g xn zm + h yn zm + C.

Here, g and h are constant coefficients and n and m are integer exponents whose values (g, h, n, m) you need to find. Explain why the constant C can be chosen arbitrarily.

2 Physics 3900 University of Georgia Spring 2019 Instructor: HBSch¨uttler

(c) By evaluating the partial derivatives of the Φ(~r) found in part (b) show explicitly that

∇~ Φ(~r) = F~ (~r) .

Using this result, explain the result you found in part (b) in light of the Gradient Theorem.

HWP 04.03: Prove the following ”product rules” of vector calculus, using the conventional product rule, (uv)0 = uv0 + u0v, applied to the relevant partial derivatives, as needed: (a) For two scalar fields Φ(~r) and Ψ(~r)

∇~ (ΦΨ) = Φ ∇~ Ψ + Ψ ∇~ Φ .

(b) For two vector fields A~(~r) and B~ (~r)

∇~ · (A~ × B~ ) = B~ · (∇~ × A~) − A~ · (∇~ × B~ ) .

(c) For a vector field A~(~r) and a scalar field Φ(~r)

∇~ × (ΦA~) = Φ(∇~ × A~) − A~ × (∇~ Φ) .

(d) For a vector field A~(~r) and a scalar field Φ(~r)

∇~ · (ΦA~) = Φ∇~ · A~ + A~ · ∇~ Φ .

HWP 04.04: Volume and surface integration by parts. (a) Use Gauss’s Theorem and the product rule from HWP 04.03 part (d), to prove the following ”integration-by-parts” rule for a scalar field Φ(~r) and a vector field A~(~r) Z Z I Φ(∇~ · A~) dv = − A~ · (∇~ Φ) dv + Φ A~ · d~a 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 from HWP 04.03 part (b) to prove the following ”integration-by-parts” rule for vector fields A~(~r) and B~ (~r) Z Z I B~ · (∇~ × A~) dv = A~ · (∇~ × B~ ) dv + (A~ × B~ ) · d~a 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.

3 Physics 3900 University of Georgia Spring 2019 Instructor: HBSch¨uttler

(c) Use Stokes’s Theorem and the product rule from HWP 04.03 part (c), to prove the following ”integration-by-parts” rule for a scalar field Φ(~r) and a vector field A~(~r) Z Z I Φ(∇~ × A~) · d~a = (A~ × ∇~ Φ) · d~a + ΦA~ · d~r 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. (d) Use the integration-by-parts equation derived in part (a) to prove the following two Green’s Identities, GI1 and GI2, for any two scalar fields Φ(~r) and Ψ(~r): Z I [GI1] Φ(∇2Ψ) + (∇~ Ψ) · (∇~ Φ) dv = Φ(∇~ Ψ) · d~a ; V S Z I [GI2] Ψ(∇2Φ) − Φ(∇2Ψ) dv = Ψ(∇~ Φ) − Φ(∇~ Ψ) · d~a. V S Here, ∇2 denotes the Lapace operator,

2 ~ ~ 2 2 2 ∇ Ψ = ∇ · (∇Ψ) = ∂xΨ + ∂y Ψ + ∂z Ψ . The V is again 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. Hints: (1) To prove GI1, note that ∇~ Ψ is a vector field, and so is A~ in part (a). (2) To prove GI2, write down another GI1 equation, just like the one stated above, but with Φ and Ψ interchanged, i.e., replace Φ by Ψ and Ψ by Φ. Then combine these two GI1 equations to get GI2.

HWP 04.05: Gauss for curls, Stokes for gradients, Gauss for gradients. (a) Use the integration-by-parts rule from HWP 04.04 Part (b), to prove the ”curl version” of Gauss’s Theorem for any vector field B~ (~r): Z I ∇~ × B~ dv = − B~ × d~a 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. Hint: Assume the vector field A~(~r) in HWP 04.04 Part (b) is simply some arbitrary constant vector A~ independent of ~r. Then, apply the triple product rule [i.e., for any three vectors ~u, ~v, ~w, the triple product obeys: (~u ×~v) · ~w = ~u · (~v × ~w)] to write the surface integral in HWP H ~ ~ H ~ ~ ~ 04.04 Part (b) as S (A × B) · d~a = S A · (B × d~a) Then factor out the constant A · ... from ~ ~ ~ R ~ ~ H ~ the integrals and use ∇ × A = 0 to obtain 0 = A · [ V ∇ × B dv + S B × d~a]. Then set, e.g., A~ = [1, 0, 0] ≡ xˆ or A~ = [0, 1, 0] ≡ yˆ or A~ = [0, 0, 1] ≡ zˆ, to prove the ”curl-Gauss” Theorem separately for the x-, y- and z-components of the volume and surface integrals involved.

4 Physics 3900 University of Georgia Spring 2019 Instructor: HBSch¨uttler

(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. Hints: Use the same tricks as in Part (a): assume A~ is a constant vector, apply the triple product rule to the surface integral, and factor out A~ · ... from the two non-zero integrals on the right-hand side of HWP 04.04 Part (c). (c) Use the integration-by-parts rule from HWP 04.04 Part (a), to prove the ”gradient version” of Gauss’s Theorem for any scalar field Φ(~r): Z I ∇~ Φ dv = Φ d~a 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. Hints: Use similar tricks as in Parts (a) and (b): assume A~ is a constant vector, and factor out A~ · ... from the two non-zero integrals on the right-hand side of HWP 04.04 Part (a). Etc., as above.

HWP 04.06: To be announced.

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