G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv
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1 1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N to the D. g x, g y, 1 dx dy surface G(x, y, z) = c 5. one version of ds E. for a surface G(x, y, z) = c G G 6. surface of revolution F. ds G. x = u cos v, y = f(u), z = u sin v H. x = u + v, y = v, z = u v I. 1 + g 2 x + g 2 y du dv
2 2 2. Match equivalent expressions. 1. grad φ A. F 2. div F B. F 3. curl F C. 2 φ 4. div grad φ, D. φ same as Laplacian of φ 3. Let φ = e (x2 +y 2 +z 2 )/2 = e r2 /2, with, as usual, r = x, y, z, and (in this context) r 2 = r 2. a. Find grad φ and show that grad φ = φ r. b. Find 2 φ and show that 2 φ = (r 2 3)φ. c. Find curl grad φ =
3 3 4. Let E = x 2 + y 2, x 2 + y 2, 1 1 z 2. Let F = sin y + cos y, x(cos y sin y) + y, 2z. Let B = x, y, 0. And let u = 1, 1, 0. a. Find and simplify u (E B) = b. Find div F = c. Find curl F = d. Find div curl F =
4 N.B. 1. The abstract Stokes Theorem in our context has one of four concrete forms: Fundamental Theorem for Line Integrals, Green s Theorem, Divergence (Gauss ) Theorem, Stokes Theorem (classical version). In the rest of this exam, any problem labelled with E&U means to explain and use the appropriate theorem in solving the problem. Explain and Use means to state the (name of the) appropriate theorem and use it to solve the problem. 2. As usual, is used to indicate the (one-less-dimensional) boundary of the curve/surface/volume written as C, S, V respectively (E&U) Write an equivalent integral using the appropriate theorem. State which theorem you are using. Indicate clearly the cases where the result always gives zero. a. S grad φ dr = b. curl F ds = V c. grad φ ds = V d. C grad φ dr, where C joins point P to point Q.
5 6. In the plane, let C be the curve consisting of three pieces: C 1 the line segment from (0, 4) to (1, 4), C 2 the arc of the curve xy = 4 from (1, 4) to (4, 1), and C 3 the line segment from (4, 1) to (4, 0). a. Calculate the line integral ( x 2 ) dy = C 5 b. (E&U) Deduce an equivalent double integral over the region enclosed by adding to C the line segments C 4 from (4, 0) to (0, 0) and C 5 from (0, 0) to (0, 4). Take C as the boundary to get the positive orientation.
6 6 7. Parametrize the ellipsoid x2 a 2 + y2 b 2 + z2 c 2 = 1 as with 0 u π, 0 v 2π. a. Find r u, r v, and ds. x = a sin u cos v, y = b sin u sin v, z = c cos u b. Let F = 0, 0, z. Calculate the flux integral S F ds. c. (E&U) Use your answer for part #b to find the volume of the ellipsoid. Check that your result agrees for the case of the sphere, with a = b = c = R.
7 8. Consider the surface of revolution formed by rotating z = e x,0 x ln 8, about the z-axis. Parametrize this surface as with 0 u ln 8, 0 v 2π. a. Find r u, r v, and ds. x = u cos v, y = u sin v, z = e u 7 b. Write (do not evaluate) an integral for the area of the surface. c. The boundary of S, S, is the circle x 2 + y 2 = 2 in the plane z =. (Fill in.) d. (E&U) Let F = y, x, 0. Find the line integral F dr by calculating the appropriate flux integral. S e. Enclose the volume V by including (with the surface) the top disk 0 r, z = (fill in cylindrical coordinates). On the top disk, the outward unit normal N =. f. (E&U) Find the value of F ds with F = y, x, 0. V
8 8 Study on Normals, Surface Area, and Volumes. These problems use the main theorems to relate normals, surface area, and volumes. Recall our notation ds = N ds. In these problems, it will generally be useful to use the explicit form N ds. 9. (Fill in.) Let S be given by G(x, y, z) = c. Then, G is to S and N = G G is a unit normal to S. So, N N =. And N T = for any vector T tangent to S. 10. (E&U) Let S be a surface with unit normal N and boundary (the curve) C. Let F satisfy curl F = N on S. What can you deduce about the area of the surface given the value of C F dr?
9 11. Let V be a volume with boundary (the surface) S. Let F satisfy div F = a, a non-zero constant, on V. a. (E&U) Express the volume of V in terms of S F ds. 9 b. What if div F = 0 on V? 12. (E&U) Let V be a volume with boundary (the surface) S. Let F = N (the outward unit normal) on S. Show that div F dv =the surface area of S. V
10 Consider the sphere G(x, y, z) = x 2 + y 2 + z 2 = R 2. a. Calculate G and N = G/ G. b. Show that div N = 3 R. c. (E&U) Use F = N on the sphere and the Divergence Theorem to find the surface area of the sphere of radius R. (This is the technique indicated in the previous problem.)
11 1. Matching Let v, w be two vectors in R 3 such that the (oriented) angle between them is less than π. Let v = v and w = w denote their respective lengths. 1. v is parallel to w a. (v/v) (w/w) 2. v is perpendicular to w b. v w = 1 3. v and w have the same length c. v w = 0 4. unit vector in the direction of v d. v v = 1 5. v is a unit vector e. v/w 6. cosine of the angle f. v/v between v and w 7. area of parallelogram g. v w spanned by v and w 8. normal to the plane h. v w spanned by v and w i. v w = 0, 0, 0 j. v v = w w
12 2 2. Matching 1. volume of sphere of radius R a. 6a 2 2. surface area of sphere of radius R b. a 3 3. volume of cylinder of height h c. abc and radius R 4. surface area of cylinder d. (a + b + c) 2 a 2 b 2 c 2 of height h and radius R 5. volume of a cube of side a e. πa 2 6. surface area of a cube of side a f. 4πR 3 /3 7. volume of a rectangular g. πr 2 h parallelepiped of sides a, b, c 8. surface area of a rectangular h. 2πRh parallelepiped of sides a, b, c i. 4πR 2 j. 4πR 2 /3
13 3. Let V 1 be a solid ball (sphere) of radius R with boundary S 1 = V 1, sphere of radius R. Let V 2 be the circular cylinder of height h and radius R with boundary S 2 = V 2 = B T L, centered at the origin, sitting on the xy-plane. I.e., L = {(x, y, z): x 2 + y 2 = R 2, 0 z h}, B = {(x, y, z): x 2 + y 2 R 2, z = 0}, T = {(x, y, z): x 2 + y 2 R 2, z = h}. In parts a.-e., use the Stokes family of theorems to evaluate these integrals. In each case, state which theorem you are using (Fund. line int ls, Green, Gauss, Stokes) and show clearly how you use it. In every case, we let F = x, y, z, φ satisfies div grad φ = 8, and G = curl H for some vector field H. a. F ds S 1 3 b. G ds S 2 c. B F dr d. T grad φ dr e. grad φ ds S 2 f. Calculate directly: T F ds
14 4. Let F = c ˆn for some positive constant c, where ˆn is the unit normal to a surface S with area SA. a. Explain/Show that c = F. b. Explain/Show that S F ds = c times SA. 4 c. Let F = 2x, 2y, 2z. Find φ such that F = grad φ by inspection. Use the fundamental theorem for line integrals to calculate C F dr where C is the arc of the curve x = 2/(1 + t 4 ), y = t 2 t, z = 3, 0 t Let x = 3 cos u, y = 4 sin u, z = v parametrize S, the (surface of) an elliptic cylinder, where 0 u 2π, 0 v h. Let F = x, y, xy. Calculate the flux integral S F ds.
15 6. In the plane, let C = C 1 C 2, where C 1 is the line segment from (0, 5) to (0, 5), C 2 is the arc of the circle of radius 5 centered at the origin from (0, 5) to (0, 5) passing through (5, 0). Sketch. Let F = x 2 + y 2, x + y. a. Use Green s Theorem to find C F dr by calculating the appropriate integral over the semicircular region enclosed by C. 5 b. Calculate C 2 F dr directly. c. Use the results of a. and b. to find C 1 F dr without further ado.
16 7. Let S denote the rectangle with vertices (in order for positive orientation) (4, 0, 0) to (0, 4, 0) to (0, 4, 8) to (4, 0, 8) back to (4, 0, 0). a. Taking (4, 0, 0) as origin in the rectangle, find two vectors in the (plane of the) rectangle. Calculate the area of the rectangle. 6 b. Find an equation of the plane containing the rectangle. c. Find ˆn, the unit normal to the rectangle. d. Use Stokes Theorem to calculate S F dr where F = z, 0, y. (Hint: express curl F as a multiple of ˆn and use Problem 4a,b with curl F replacing F.) e. Let C 3 denote the line segment from (0, 4, 8) to (4, 0, 8). Calculate C 3 z dx + y dz directly.
17 7 8. Consider the cone of height 2, radius 1 as a surface of revolution z = 2(1 x 2 + y 2 ), i.e., z = 2 2r in cylindrical coordinates. Let F = x, y, z. a. Calculate the flux integral F ds over the cone. b. Enclose a volume by adjoining the disk B in the xy-plane of radius R centered at the origin. Calculate B F ds directly. c. Explain/Use one of the Stokes family to find the volume enclosed by the cone. d. Find the surface area of the cone.
18 1. Matching v and w are non-collinear vectors in R 3. Let v = v and w = w denote their respective lengths. 1. unit vector a. (v/v) (w/w) 2. v is perpendicular to w b. v w = vw 3. v and w have the same length c. v w = 0 4. unit normal to the plane d. v w = 1 spanned by v and w 5. cosine of the angle e. v/v, v 0 between v and w 6. area of parallelogram f. v w spanned by v and w 7. normal to the plane g. (v w)/ v w spanned by v and w h. v w i. v v = w w
19 2 2. a. Write an equation of the plane spanned by v = 2,1,1 and w = 1,1,2 passing through the point (1,1,1). b. Write an equation of the plane containing the points (0,1,0), (2,1,0), and (0, 1, 1). c. Write an equation of the line passing through the points (0,2,0) and (2,1,1). d. Write parametric equations for the directed line segment from (0, 2, 0) to (2,1,1). e. Find a unit normal to the plane tangent to the surface xy + yz = 5 at the point (3,1,2).
20 3 3. a. Let φ = e x2 + e yz. Find: φ = div gradφ = 2 φ = curl gradφ = b. Let F 1 = x 2 + y 2 + z 2,zy 2 /2,xz 2 /2 and F 2 = xz,y 2 /2,xy. Find: divf 1 = curlf 2 = div curlf 2 =
21 4 4. From the given data, for c through i, use the Stokes family of theorems as appropriate to evaluate these integrals. In each case, state which theorem you are using (Fund. Line Integrals, Stokes, Gauss), show the conversion, and give an exact evaluation. (Note that S and V are completely separate.) Surface S, with ds = 8, dive = 9, curle ds = 3. S And B is such that curlb ds = 5 ds. Volume V, with dv = 10, φ such that S φ ds = 5. Let F = x,z,y. V V a. area of the surface S equals b. volume of V equals c. E dr = d. S φ dr = e. S B dr = f. S E ds = g. h. V V curle ds = F ds = i. V 2 φdv = V
22 5 5. Let x = 4 cos u, y = sin u, z = v parametrize S, the (surface of) an elliptic cylinder, where 0 u 2π, 0 v h. Let F = xy,y,z. *a. Calculate the flux integral S F ds. *b. Use the fact that F ds = curl yz,0,0 ds on the top (elliptical disk) of the cylinder, z = h. Calculate yz,0,0 dr around the boundary of C the top of the cylinder and use Stokes Theorem to get the flux F ds on the top. Then explain why the integral on the bottom equals S zero and use Gauss Theorem to deduce divfdv. V
23 6 6. In the yz-plane, let C be the curve consisting of three pieces: C 1 the line segment from (0,0,0) to (0,10,0), C 2 the arc of the parabola z = 100 y 2 from (0,10,0) to (0,0,100), and C 3 the line segment from (0,0,100) back to (0,0,0). *a. Calculate the line integral C F dr where F = x,y,1. b. From your answer, it it possible that there is a function φ such that F = φ? Explain in the context of either the fundamental theorem for line integrals or Stokes theorem. 7. * Let F = x,y,z. Calculate the flux integral F ds on the surface formed by the plane 3x+5y +2z = 15 cut off by the coordinate axes. First write an integral over the region that is the projection of the surface onto the xy-plane. You may use geometry to evaluate it. S
24 7 8. Consider the paraboloid cut off by the xy-plane: x 2 + y 2 + z = 16, z 0. a. Write an integral in cylindrical coordinates for the area of the surface of the paraboloid. DO NOT EVALUATE. *b. Write an integral for the volume enclosed by the paraboloid (cut off by the xy-plane). Use cylindrical coordinates and evaluate it. *c. Verify Gauss Theorem for F = x,y,z on the enclosed volume with boundary surfaces consisting of the top paraboloid and the bottom disk cut off in the xy-plane. Calculate the flux integral F ds explicitly for each boundary surface. Then compare with your result of part b. (common value is 384π) S
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