620-233

THE UNIVERSITY OF MELBOURNE SEMESTER 1, 2006 DEPARTMENT OF MATHEMATICS AND STATISTICS 620-233 VECTOR ANALYSIS (ADVANCED) EXAM Exam duration — 3 hours Reading time — 15 minutes This paper consists of 8 pages. Examination Papers with Common Content: • This paper contains questions in common with those for the subject 620-231 Vector Analysis which is being held at the same time. Instructions to Invigilators: • Initially, students are to receive a 14 page script book. Authorized Materials: • No calculators or computers are permitted. • No written or printed material may be brought into the examination room. Instructions to Students: • There are 10 questions on this examination paper. • The number of marks allocated to a question are shown below the question. • The total number of marks for the whole exam is 100. • There are formulas of vector identities, curvilinear coordinates and geometry of surfaces on pages 6-8 that you may use in this examination. This paper may be reproduced and may be lodged in the Ballieu Library. Page 1 of 8. r\ j 1. (a) Let f(x,y) = (2x2,3y2), g(u,v) = u2 - v2 and h = g o f. Calculate — Q I and — at (x,y) — (1,1) using the chain rule. Verify your answer by direct oy substitution. (b) Find the limit as (x, y) — > (0, 0) or show that no limit exists for the following functions: x + y) (c) Let y if (x, y) / (0,0)x2+ y\ 0 if (1,3,) = (0,0) Show that g(x,y) is not continuous at (0,0). r\ (d) Show that — exists at (0,0) and calculate its value.ox 10 marks 2. Using the method of Lagrange multipliers, find the minimum of the distance squared function d(x, y, z) — x2 + y2 + z2 for the points in common with the cone z2 = 4(x2 + y2) and the plane 2y + 4z = 5. 10 marks 3. Let c be the curve parametrized by c(i) = (sin t — tcost,2, cos t + tsin.t), t > 0. (a) Calculate the arc length of c between t = 0 and t = 1-n. (b) Calculate the curvature of the curve. 10 marks 4. (a) Let f(x,y,z) and g(x,y,z) be any C1 scalar functions. Show that V • (fVg) = (b) Let r = (x, y, z) and r = |r|. Using vector identities or otherwise, calculate the following quantities where they are defined i) V2(3r2) ii) V 10 marks Page 2 of 8. 5. (a) Evaluate the integral / / exp(z + y) dx dy where D is the domain bounded by the lines y = 0, y = x and y = — x + 2. (b) Consider the integral />2 /•v'4—x2 I I exp(x2 + y2} dy dx JoJ-^-x2 i. Sketch the domain of integration, carefully labelling all the boundary curves. ii. Let x — ucosv and y = usinv be a change of variables. Calculate the Jacobian for this change of variables and hence evaluate the integral. 10 marks 6. Let R be the solid region bounded by the plane z = 0 and the cone z = 3 — 3>/rr2 + The z coordinate of the centre of mass is given by z = where p,(x, y, z) is the mass density. If p,(x, y, z) is constant, calculate zc. 10 marks 7. Let 5 be a surface parameterised by x = u, y = v, z = u2 + v2, 0 < w < 1, 0<'W<1 (a) Describe the curves of constant u. (b) Find tangent vectors to the curves of constant u and constant v and hence find a normal vector to the surface S. (c) Find the mass of a thin metal sheet in the shape of the surface S if the mass density per unit area, m(x, y) = \/4x2 + 4t/2 + 1. 10 marks Page 3 of 8. 8. Green's theorem can be written as D (a) Verify Green's theorem for the vector field F = (x2,x2 — y2) and the domain D in common with the two regions y < I and y > x1. (b) If the vector field is changed to F = I — , - 1 then Green's theorem is no\x y-lj longer valid for the domain D. Why? 10 marks 9. Let / : D C M — >• K+ be a positive C2 function. A surface of revolution is obtained by rotating the graph of / around the re-axis. A surface of revolution has the parametrisation <&(w, v) = (v,-Q, 0) + f(v)(Q, cosu, sinu), for v e D and u € [0,2-ir). (a) Compute the coefficients of the first fundamental form of a surface of revolution. (b) Compute the coefficients of the second fundamental form of a surface of revo- lution. Figure 1: Gabriel's horn Gabriel's horn is the surface of revolution obtained by rotating the function / : x H> I/a;, x G [1, oo), around the x-axis. (c) Show that the Gausian curvature of Gabriel's horn is negative everywhere. (d) Compute the volume of Gabriel's horn. (e) Compute the area of Gabriel's horn. Can Gabriel's horn store enough polish to make its divine surface shine everywhere? 10 marks Page 4 of 8. 10. In this question we will use Maxwell's equation for the magnetic field, B = V x A, where A is the magnetic vector potential. We will also use the cylindrical coordinates p, <f> and z, and write the unit vectors in this coordinate system as p, </> and z. Consider the infinitely long cylinder x2 + y2 = a2, -oo < z < oo. Let A = Ain for 0 < p < a and A = Aout for p > a, where Ajn and Aout are C2 vector fields. (a) Let A;n = ^B0p(f>. Compute the magnetic field B;n inside the cylinder from Maxwell's equation using cylindrical coordinates. (b) We define the magnetic field outside the cylinder to be zero, Bout = 0. Compute the magnetic flux (f> = JJD B • dS where D is the disk x2 + y2 < R2, z — 0 and R > a. Take the up-pointing normal. (c) Use Stokes' theorem to show that outside the cylinder Aout ^ 0, even though Bout — 0. (d) Assuming that Aout = g(p) <^, compute g(p) using Maxwell's equation and Stokes' theorem. (e) Show that for any C2 function / the vector potential A' = A + V/ produces the same magnetic field. (f) Compute Aout + V/ with f(p,(f>,z) = — |a25o</>. How do you reconcile your result with (c) and (e)? 10 marks Page 5 of 8. Next page for Identities. BASIC IDENTITIES OF VECTOR ANALYSIS Let f(x, y, z) and g(x, y, z) be scalar functions, F and G be vector fields in R3 and /? be any constant. 2. V(/3/) - /3V/ 3- 4. V ( - ) = provided g £ 0. 2 5. V • (F + G) = V • F + V • G 6. Vx(F + G) = VxF + VxG 7. V • (/F) = /V - F + F • V/ 8. V • (F x G) = G • (V x F) - F • (V x G) 9. V • (V x F) = 0 10. Vx (/F) = /VxF + V/xF 11. Vx (V/) = 0 12. V2(/</) = fV2g + gV2f + 2V/ • Vg 13. V • (V/ x V0) - 0 14. V - (fVg - gVf) = fV2g - gV2f Page 6 of 8. Next page for curvilinear coordinates. ORTHOGONAL CURVILINEAR COORDINATES Let MI, ..., un be orthogonal curvilinear coordinates with scale factors hi,...,hn and unit tangent vectors iii,..., (in. Let / = /(iii,..., un) and F = 52^=i -^(^i) • • • > un)&i- Then dV = hi • • • hn d.u\... dun „grad/= > — — Ui^ Aij auj divF curl F = 2 e =i On?d d d hIndf For polar coordinates (r, 9): hi = 1, hi = r. For cylindrical coordinates (p, <f>,z): hi — 1, /i2 = P, h3 = 1. For spherical coordinates (r, 0, (/>): /ia = 1, hi = r, h3 = r sin5. Page 7 of 8. Next page for geometry of surfaces. GEOMETRY OF SURFACES Let $ : U C R2 —>• S C R3 be a regular parametrization, and let n : U -> K3 be a unit normal vector field on S. Then: • The first fundamental form of 5 is ds2 = Edit? + 2Fdudv + Gdv2 where tii — *PU • ^Pu, " — ™u ' ™vi I-7 = ™u ' ™w The area element is dS = ||<£u x <&„ | = \/EG — F2. The second fundamental form has coefficients e = *„„ • n, / = *„„• n, g = *w • n. The principal curvatures of S are TT I / 7"7"9 IV «i, «2 = r/ ± V-n — K, where // and K are the mean and Gaussian curvature given by Ge + Eg-2Ff eg - f2 2(EG-F2) ' EG-F2' Page 8 of 8. End of exam.