PHYS 3900 Homework Set #04 Due: Wed. Mar. 7, 2018, 4:00pm (All Parts!)

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1 PHY 3900 Homework et #04 Due: Wed. Mar. 7, 2018, 4:00pm (All Parts!) All textbook problems assigned, unless otherwise stated, are from the textbook by M. Boas Mathematical Methods in the Physical ciences, 3rd ed. Textbook sections are identified as h.cc.ss for textbook hapter cc, ection ss. omplete 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 PHY3900 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. tate 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. o, for example, if the result is, say, ln(7/2) + (e 5 π/3), or 17 9/2, or (16 4π) 10, then just state that as your final answer: no need to evaluate it as decimal number by calculator! implify final results to the largest extent possible; e.g., reduce fractions of integers to the smallest denominator etc.. 1

2 HWP 04.01: Divergence and curl of a vector field. (a) alculate the divergence and the curl of the vector field F ( r) [F x ( r), F y ( r), F z ( r)] = [2z 2, 3z 2, (4x + 6y)z] where r [x, y, z]. (b) Use tokes s theorem to show that the line integral of F ( r) over any curve L, given by 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: onsider two different curves, L and M, say, which share a common start- and endpoint. onstruct from them a closed curve to which tokes s theorem can be applied. 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 I L := F ( r) d r L where L is a straight line segment, drawn from some point r A [A x, A y, A z ] to some other point r B [B x, B y, B z ]. Parameterize L by r(τ) = p + τ q, with parameter variable τ [ 1, +1], where p := ( r B + r A )/2 and q := ( r B r A )/2. heck that indeed r(τ) = r A for τ = 1 and r(τ) = r B for τ = +1. Express I L in terms of A x, A y, A z, B x, B y, B z. Hints: onvert 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 p x, p y, p z, q x, q y, and q z. Integrate! You only need a and c; b drops out: why? ollect q 2 z-, p 2 z- and p z q z -terms and express them in terms of A z and B z by using q 2 z + p 2 z = (B 2 z + A 2 z)/2 and 2q z p z = (B 2 z A 2 z)/2. Then use q α + p α = B α and q α p α = A α 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 I L can be written in terms of Φ( r) as: I L = Φ( r B ) Φ( r A ). Hint: Try, for example, a function Φ( r) of the general form Φ( r) = g x n z m + h y n z m +. 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 can be chosen arbitrarily. (c) By evaluating the partial derivatives of the Φ( r) found in part (b) show explicitly that Φ( r) = F ( r). 2

3 HWP 04.03: Prove the following product rules of vector calculus, using the conventional product rule, (uv) = uv + u v, 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) (Φ A) = Φ( A) A ( Φ). HWP 04.04: olume and surface integration by parts. (a) Use Gauss s theorem and (without proof) the product rule (ΦA) = Φ A + A Φ to prove the following integration-by-parts rule for a scalar field Φ( r) and a vector field A( r) Φ A dv = A Φ dv + ΦA d a where is a 3D volume with volume elements dv; and is the oriented closed surface that encloses, with outward directed surface area element vectors d a. (b) Use Gauss s theorem and the product rule HWP Part(b) to prove the following integration-by-parts rule for vector fields A( r) and B( r) B ( A) dv = A ( B) dv + ( A B) d a where is a 3D volume with volume elements dv; and is the oriented closed surface that encloses, with outward directed surface area element vectors d a. (c) Use tokes s theorem and the product rule from HWP Part(c), to prove the following integration-by-parts rule for a scalar field Φ( r) and a vector field A( r) Φ( A) d a = ( A Φ) d a + ΦA d r where is an oriented open surface with surface area element vectors d a; and is the closed curve that encloses with line element vectors d r, encircling the d a-vectors with right-handed orientation. HWP 04.05: Gauss for curls, tokes for gradients. (a) Use the integration-by-parts rule from HWP Part (b), to prove the curl version of Gauss s theorem for any vector field B( r): B dv = B d a 3

4 where is a 3D volume with volume elements dv; and is the oriented closed surface that encloses, with outward directed surface area element vectors d a. Hint: Assume the vector field A( r) in HWP 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 Part (b) as ( A B) d a = A ( B d a) Then factor out the constant A... from the integrals and use A = 0 to obtain 0 = A [ B dv + B d a]. Then set, e.g., A = [1, 0, 0] ˆx or A = [0, 1, 0] ŷ or A = [0, 0, 1] ẑ, to prove the curl-gauss theorem separately for the x-, y- and z-components of the volume and surface integrals involved. (b) Use the integration-by-parts rule from HWP Part (c), to prove the gradient version of tokes s theorem for any scalar field Φ( r): Φ d a = Φ d r where is an oriented open surface with surface area element vectors d a; and is the closed curve that encloses 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 Part (c). HWP 04.06: Proving auchy s integral theorem by tokes theorem. onsider 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 in the complex ζ-plane. Given f(ζ) = u(x, y) + iv(x, y), we can define two 3D vector fields G( r) and H( r) by G( r) [G x ( r), G y ( r), G z ( r)] := [ +u(x, y), v(x, y), 0 ] and H( r) [H x ( r), H y ( r), H z ( r)] := [ +v(x, y), +u(x, y), 0 ]. 4

5 The fields G( r) and H( r) will be referred to as the associated 3D vector fields of f(ζ). They are both defined on a 3D open prismatic/cylindrical volume which has the open set as its cross-sectional area and a mantle parallel to the z-axis, as shown in Fig : := { r r = [x, y, z] with x + iy and < z < }. Note that is singly connected in 3D space if is singly connected in the 2D complex ζ-plane. Note also that G( r) and H( r) have the special property that their z-components (G z, H z ) are zero and their x- and y-components (G x, G y, H x, H y ) are independent of z. onsider further a curve inside the complex open set, defined by some parameterization ζ(τ) = x(τ) + iy(τ) on a real τ-interval [a, b]. can then also be regarded as a curve in the x-y-plane in the 3D open set, parameterized by r(τ) = [ x(τ), y(τ), 0 ]. (a) how that f(ζ)dζ = G( r) d r + i H( r) d r. (b) how that the associated 3D vector fields, of f(ζ), G( r) and H( r), have vanishing curls, G( r) = 0 and H( r) = 0 for all r, if f(ζ) is complex differentiable for all ζ. Hint: If f(ζ) is complex differentiable then u(x, y) and v(x, y) obey the auchy-riemann differential equations. You may use these equations here without proof. (c) Assume now that f(ζ) is complex differentiable for all ζ ; that is singly connected in the 2D complex ζ-plane; and that is a simple closed curve in, enclosing a singly connected interior I = Interior(). Use the associated 3D vector fields G( r) and H( r), the results from Parts (a) and (b), and tokes theorem to prove auchy s integral theorem: f(ζ)dζ = 0. 5

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