= x i + y j + z k. div F = F = P x + Q. y + R

Transcription

1 Abstract The following 25 problems, though challenging at times, in my opinion are problems that you should know how to solve as a students registered in Math C or any other section offering Math olving them for yourself will give you plenty of confidence in doing well on examinations which contains problems like these. The problems themselves tests your understanding of the Line and urface ntegrals of vector and scalar fields, Green s, toke s and the ivergence Theorems (your course material). The problems are not as difficult as they may appear to be. ome problems require a clever use of ideas from even a precalculus and calculus courses combined with the ideas you are currently learning in vector calculus. Only a few of these problems are solvable by students registered in Math For a continuous vector field F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k defined on a region R of space. We define the concepts divergence and curl of the vector field F. = x i + y j + z k nterpretation of ivergence: The divergence has an important physical interpretation. f we imagine F to be the velocity field of a gas (or a fluid), then div F represents the rate of expansion per unit volume under the flow of the gas (or fluid). f div F < 0, the gas (or fluid) is compressing. For a vector field F on the plane, div F measures the rate of expansion per unit area. div F = F = P x + Q y + R z. PALE WHEEL NTERPRETATON of F: uppose that v = v(x, y, z) is the velocity of a moving fluid whose density at (x, y, z) is δ = δ(x, y, z) and let F = δv. Then F dr C is the circulation of the fluid around the closed curve C. By tokes Theorem, the circulation is equal to the flux of F through a surface spanning C: F dr = F n d. C uppose we fix a point Q in the domain of F (i.e., fix a point Q in the moving fluid with velocity v(q)) and a direction u at Q. Let C be a circle of radius ρ centered at Q, whose plane in normal (or perpendicular) to u. f F is continuous at Q, the average value of the u-component of F over the circular disk enclosed by C approaches the u-component of F at Q as ρ 0: ( F u) Q = lim ρ 0 πρ 2 F n d.

2 f we replace the double integral in this last equation by the circlutation, we get ( F u) Q = lim F dr. ρ 0 πρ 2 The left-hand side of this last equation has its maximum value when u is the direction of F. When the radius ρ of disk bounded by C is small, the limit on the right-hand side of this last equation is approximately πρ 2 C F dr, which is the circulation around C divided by the area of the disk (circulation density). uppose that a small paddle wheel of radius ρ is introduced in the fluid at Q, with its axle directed along u. The circulation of the fluid around C will affect the rate of spin of the paddle wheel. The wheel will spin fastest when the circulation integral is maximized; therefore it will spin fastest when the axle of the paddle wheel points in the direction of F = curl F. C curl F = F = R y Q P i + z z R Q j + x x P k. y GREEN THEOREM: Let P = P (x, y) and Q = Q(x, y) be functions with continuous first partial derivatives, defined in a simplyconnected region, of R 2 and on its boundary. Let γ = be the (positively oriented) boundary curve which encloses. Then Q P (x, y) dx + Q(x, y) dy = x P da y circulation or tangential form: F T ds = (curl F) k da = ( F) k da = Q x P da y where F(x, y) = P (x, y)i + Q(x, y)j is a continuous vector field on representing the velocity field of a fluid flow in the plane. flux or normal form: F n ds = div F da = ( F) da = P x + Q da y where F(x, y) = P (x, y)i + Q(x, y)j is a continuous vector field on representing the velocity field of a fluid flow in the plane and n is the outward-pointing unit normal to the positively oriented boundary curve. o if γ : [a, b] R 2 where γ(t) = (x(t), y(t)) is a parametrization of the positively oriented boundary curve of, then n is given by n = γ (t) (y (t)i x (t)j) Area of the region is A() where A() = y dx + x dy. 2 2

3 TOKE THEOREM: The counterclockwise circulation of a continuous vector field F = F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k around the boundary curve of an oriented smooth surface in the direction counterclockwise with respect to the surface s outward pointing unit normal vector field n = n(x, y, z) is calculated as the integral of the outward flux of curl F ( i.e., curl F = F n) through the surface spanned by the curve Z curl F d = ( F) d = ( F n) d = F T ds VERGENCE THEOREM: The outward flux of a continuous vector field F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k across a closed oriented smooth surface in the direction of the surface s outward unit normal field n is calcaluted as: Z F d = F n d = div F dv E REMEMBER: The key to successfully solving the problems is to apply all the learned concepts and formulas. o not jump to any conclusions based on how a problem might appear (difficult or easy). Afterall, what does looks have to do with Mathematics anyway? However, using the formulas and applying the concepts (from past to present) is everything that is required. Before taking on these problems, you should try a few of the exercises in your mith-minton textbook first, it will be like warming up before the main event. GOO LUCK!! 3

4 Antony Foster November 3, 2009 Practice Problems For Math C and Math T tudents Problem. Write parametric equations of the tangent line to the given parameterize curve γ(t) at the given point P = γ(t 0 ). a) γ(t) = (x(t), y(t), z(t)) = (sin 3t, cos 3t, 2t 5/2 ); t 0 =. b) γ(t) = (x(t), y(t), z(t)) = (cos 2 t, 3t t 3, t); t 0 = 0. Problem 2. uppose that a particle traveling on the given path γ(t) flies off on a tangent at γ(t 0 ). Compute the particle s position along the tangent at time t = t. Provide parameterization of the particle s path along the tangent r(t) from r(t 0 ) to r(t ). Make your parameter interval the unit interval [0, ]. a) γ(t) = (x(t), y(t), z(t)) = (4e t, 6t 4, cos t); t 0 = 0, t =. b) γ(t) = (x(t), y(t), z(t)) = (sin e t, t, 4 t 3 ); t 0 =, t = 2. Problem 3. Consider the function f(x, y, z) = x 2 e yz. Compute the rate of change of f at the point P = (, 0, 0) in the direction of v =,,. n what direction from the point (0, ) does the function g(x, y) = x 2 y 2 increase the fastest? Problem 4. Write the equation of the tangent plane to the surface given as 3xy + z 2 = 4 at the point (,, ). Find a unit normal vector n to the surface at the point P (/3, 3, ). 4

5 Problem 5. Captain Ralph is in trouble near the suny side of Mercury. The temperature of the ship s hull when he is at location (x, y, z) will be given by T (x, y, z) = e x2 2y 2 3z 2, where x, y, and z are measured in meters. He is currently at (,, ). a) n what direction should he proceed in order to decrease the temperature most rapidly? b) f the ship travels at e 8 meter per second, how fast will be the temperature decrease if he proceeds in that direction? c) Unfortunately, the metal of the hull will crack if cooled at a rate greater than 4e 2 degrees per second. escribe the set of possible directions in which he may proceed to bring the temperature down at no more than that rate. Problem 6. Find the exact value for the arc-length of the curve γ(t) = (cos t, sin t, t 2 ); 0 t 2π. Also find the exact value for the arc-length of the curve β(t) = (t, t sin t, t cos t) between the two points (0, 0, 0) and (π, 0, π). You might find the following integral formula useful: x2 + a 2 dx = [ x x a 2 + a 2 log(x + ] x 2 + a 2 ) + C. Problem 7. f F is a vector field, a flow line for F is a curve or path γ(t) such that γ (t) = F(γ(t)). That is, F yields the velocity field of the path γ(t). For the vector field F(x, y) = y, x show that the path γ(t) = (cos t, sin t) is a flow line of F. Furthermore, verify that for constants r and t 0 that β(t) = (r cos(t t 0 ), r sin(t t 0 )) are flow lines for F. Problem 8. Let be a smooth surface given by the equation H(x, y, z) = 0 (called a level surface of H). Let P 0 = (x 0, y 0, z 0 ) be a point of and also suppose that the first partial derivatives of H exists and are continuous. a) how that a tangent plane T at the point P 0 has equation H(P 0 ) (r r 0 ) = 0 where H(P 0 ) 0 and r = xi + yj + zk and r 0 = x 0 i + y 0 j + z 0 k are position vectors for the points Q = (x, y, z) and P 0 = (x 0, y 0, z 0 ) in the Tangent plane. b) how that the equation of a line L normal to and passing through P 0 has equation r(t) = r 0 + t H(P 0 ); t R 5

6 Problem 9. Let l be a line in space containing the points A = (x 0, y 0, z 0 ) and B = (x, y, z ). Consider a particle moving along l from point A at time t 0 to a different point B at time t. a) how that position vector r(t) of any point (x, y, z) on l from A to B is given by the formula r(τ) = ( τ)r 0 + τr ; 0 τ or r(t) = r 0 + t t 0 t t 0 (r r 0 ); t 0 t t ; which we used extensively in class and where r 0 and r are position vectors of the points A and B respectively. b) Calculate γ (t) (use the rules of differentiation) and γ (t). Calculate the arc-length of γ(t). Was the result of your calculation any surprise to you? By this, mean did you have some idea of what the result was before making the actual calculation of the arc-length. c) Use the arc-length function which is the integral s = s(t) = t t 0 γ (τ) dτ to find a formula which expresses the parameter t in terms of the arclength parameter s that measures the distance along the curve γ. Use your found formula express the given path γ(t) in part(a) in terms of the arc-length parameter s. Problem 0. Let F = F(x, y, z) be the vector field given by F(x, y, z) = 2xye z i + x 2 e z j + (x 2 ye z + z 2 )k. a) Compute the divergence and the curl of F. b) s there a function f(x, y, z) such that F = f? f so, find f(x, y, z) and say something about F. c)for a simple closed path γ(t) = (cos t, sin t, 2); 0 t 2π. Calculate P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz γ where P (x, y, z), Q(x, y, z) and R(x, y, z) are the x, y and z component functions respectively of the vector field F. 6

7 Problem. Tom awyer s aunt has asked him to whitewash both sides of an old fence whose base is a curve γ which lies in both the first and second quadrants of the xy-plane and whose height is z = h(x, y). Tom estimates that for each 25 ft 2 of whitewashing he lets someone else do for him, the willing victim will pay 5 cents. How much can Tom hope to earn, assuming his aunt will provide whitewash free of charge? Needed information: The base of the fence in the first quadrant is the path γ(t) = (30 cos 3 t, 30 sin 3 t); 0 t π 2. The base of the other part of the fence in the second quadrant is the mirror reflection of the base in the first quadrant. The height of the fence at (x, y) along it base its given by h(x, y) = + y/3 (in feet). Problem 2. Let f denote the scalar function defined by f(x, y, z) = or f(r) = x2 + y 2 + z 2 r where r = xi + yj + zk ( position vector of a point (x, y, z) in space). Calculate 2 f(x, y, z) (recall that: 2 f = ( f) this is called the Laplacian of f. A function f is called harmonic if its laplacian is 0, i.e., 2 f = 0.) Problem 3. For the vector field F(x, y, z) = 2xz 2 i + j + xy 3 zk and the scalar function f(x, y, z) = x 2 y. Calculate a) f b) F c) F f d) F ( f). Problem 4 (Challenging but solvable). Let R denote the region of the xy-plane defined by R = {(x, y) 0 x, 0 y } the unit rectangle in the xy-plane. Let z = f(x, y) = x 2 + y 2. Calculate the exact value double integrals below; n your own words, what do the two values represent? f(x, y) da and f x (x, y) 2 + f y (x, y) 2 + da R R Problem 5. a) Let f : R 3 R be defined by f(x, y, z) = z + e x2 y 2. n what directions starting from (,, ) is f decreasing at 50% of its maximum rate of change? b) f g is a smooth function of (x, y), γ is the unit circle x 2 + y 2 = and is the unit disk = {(x, y) x 2 + y 2 }, is it true that sin(xy) g g [ dx + sin(xy) x y dy = cos(xy) y g y x g ] dx dy? x γ 7

8 Problem 6. (a) Evaluate the integral 0 y e x2 /2 dx dy. (b) Evaluate x dx dy dz, where is the tetrahedron with vertices (0, 0, 0), (, 0, 0), (0, 2, 0), ( 0, 0, ). 2 Problem 7. etermine whether each of the following statements below is true or false. f true, justify (give a brief explanation or quote a relevant theorem from the course), and if false, give an explanation, or a Counterexample. a)for smooth functions f and g, ( f g) = 0. b)there exist conservative vector fields F and G, such that F G = xi + yj + zk. c) By a symmetry argument, the following holds d)the surface integral of vector field xye ex2 2 (x6 y 4 + x 8 y 7 + 5) π z(y 2 + ) 3 dy dx dz = cos 4 (t) sin(t) dt. π F(x, y, z) = (x + y 2 + z 2 )i + (x 2 + y + z 2 )j + (x 2 + y 2 + z)k over the unit sphere is 4π. e) f G is a smooth vector field on R 3, = {(x, y, 0) x 2 + y 2 }, oriented with the upward normal and the upper hemisphere, 2 = {(x, y, z) x 2 + y 2 + z 2 =, z 0}, also orientied with the upward normal, then ( G) d = ( G) d. 2 8

9 Problem 8. Let E be the solid region of R 3 defined by E = {(x, y, z) x 2 + y 2 + z 2 & y x}. a) Compute E ( x 2 y 2 z 2 ) dx dy dz. b) Find the flux of the vector field out of the region E. F(x, y, z) = (x 3 3x)i + (y 3 + xy)j + (z 3 xz)k Problem 9. Let F and G be vector fields given by F(x, y, z) = 4xzi + y 2 j + yzk any G(x, y, z) = i + j + z(x 2 + y 2 ) 2 k. a) Verify the divergence theorem for the vector field F over the surface of the unit cube in the first octant. b) Evaluate the surface integral G n d, where is the surface of the solid cylinder x 2 + y 2, 0 z (including the sides and both lids). Problem 20. a) Let be the parallelogram in the xy-plane with vertices (0, 0), (, ), (, 3), (0, 2). Evaluate the integral xy dx dy. b) Evaluate G(x, y, z) dv = x2 + y 2 + z 2 e (x2 + y 2 + z 2 ) 2 dx dy dz E E where E is the solid region in space given by E = {(x, y, z) x 2 + y 2 + z 2 4 & z x 2 + y 2 }. 9

10 Problem 2. For each of the questions below, indicate if the statement is true or false. f true, justify (give a brief explanation or quote a relevant theorem from the course), and if false, give an explanation or a counterexample. a) f P (x, y) = Q(x, y), then the vector field F(x, y) = P (x, y)i + Q(x, y)k is a gradient field. b) The flux of any gradient field out of a closed surface is zero. c) There is a vector field F such that F = yj. d) f f is a smooth function of two variables x and y and γ is the unit circle x 2 + y 2 = and is the unit disc x 2 + y 2, then xy f f e dx + exy x y dy = e xy[ y f ] y x f dx dy. x γ e) For any smooth function f(x, y, z), we have x x+y f(x, y, z) dz dy dx = y x+y f(x, y, z) dz dx dy. Problem 22. Let E be the solid region in R 3 defined by E = {(x, y, z) x 2 + y 2, z 0, x 2 + y 2 + z 2 4} a) Find the volume of E. b) Find the flux of the vector field F(x, y, z) = (2x 3xy)i yj + 3yzk out of E. Problem 23. Let f(x, y, z) = xyze xy. a) Compute the gradient vector field F = f. b) Let γ be the curve obtained by intersecting the unit sphere x 2 + y 2 + z 2 = with the vertical plane x = /2 and let be the portion of the sphere with x /2. raw a figure including possible orientations for γ and ; state toke s theorem for this region. c) With F as in (a) and as in (b), let G = F + (z y)i + yk, and evaluate the suface integral ( G) d. 0

11 Problem 24. Let be portion of the sphere x 2 + y 2 + z 2 = 4 inside the cylinder (x ) 2 + y 2 =. Evaluate and state the meaning of d. Problem 25. a) Let a, b > 0 and let a point of mass m move long the space-curve γ(t) = (a cos t, b sin t, t 2 ). escribe the curve and find the tangent line at t = π. b) Find the force acting on the particle at t = π. Hint: F = ma. c) efine the function E(z) by π/2 0 z 2 sin 2 t dt. Find a formula for the circumference of the ellipse x 2 a 2 where 0 < a < b, in terms of E(z). + y2 b 2 =,