Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VectorCalculus1 due 05/03/2008 at 02:00am EDT.

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1 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus due 05/03/008 at 0:00am EDT.. ( pt) rochesterlibrary/setvectoralculus/ur V.pg onsider the transformation T : x = u 37v, y = 37u+ 37 v A. ompute the Jacobian: (x,y) (u,v) = B. The transformation is linear, which implies that it transforms lines into lines. Thus, it transforms the square S : 37 u 37, 37 v 37 into a square T (S) with vertices: T(37, 37) = (, ) T(-37, 37) = (, ) T(-37, -37) = (, ) T(37, -37) = (, ). Use the transformation T to evaluate the integral RR T (S) x + y da. ( pt) rochesterlibrary/setvectoralculus/ur V.pg ompute the gradient vector fields of the following functions: A. 5x + 4y B. x y 9,. 5x + 4y D. f (x,y,z) = 5x + 4y + z + k E. f (x,y,z) = 5x + 4y + z f (x,y,z) = i+ j+ k 3. ( pt) rochesterlibrary/setvectoralculus/ur V 3.pg D. spheres E. lines F. paraboloids G. hyperbolas H. hyperboloids I. ellipses 4. ( pt) rochesterlibrary/setvectoralculus/ur V 4.pg ompute the total mass of a wire bent in a quarter circle with parametric equations: x = cost, y = sint, 0 t π and density function ρ(x,y) = x + y. 5. ( pt) rochesterlibrary/setvectoralculus/ur V 5.pg Let be the curve which is the union of two line segments, the first going from (0, 0) to (-3, -3) and the second going from (-3, -3) to (-6, 0). Z omputer the line integral 3dy + 3dx. 6. ( pt) rochesterlibrary/setvectoralculus/ur V 6.pg Let F be the radial force field F = xi + yj. Find the work done by this force along the following two curves, both which go from (0, 0) to (8, 64). (ompare your answers!) Z A. If is the parabola: x = t, y = t, 0 t 8, then F dr = B. If Zis the straight line segment: x = 8t, y = 64t, 0 t, then F dr = 7. ( pt) rochesterlibrary/setvectoralculus/ur V 7.pg Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field.. F = i + j + k. F = yi + xj 3. F = xi + yj + zk 4. F = xi yj 5. F = yi + xj 6. F = xi + yj + zk 7. F = i + j 8. F = xi + yj 9. F = xi + yj zk 0. F = xi + yj k. F = xi + yj A. circles B. planes. ellipsoids Let be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine Z for the following vector fields F whether the line integrals F dr are positive, negative, or zero and type P, N, or Z as appropriate. A. F = the radial vector field = xi + yj: B. F = the circulating vector field = yi + xj:. F = the circulating vector field = yi xj: D. F = the constant vector field = i + j: 8. ( pt) rochesterlibrary/setvectoralculus/ur V 8.pg onsider a wire in the shape of a helix r(t) = costi + sintj + 5tk, 0 t π with constant density function ρ(x,y,z) =. A. Determine the mass of the wire: B. Determine the coordinates of the center of mass: (,, )

2 . Determine the moment of inertia about the z-axis: 9. ( pt) rochesterlibrary/setvectoralculus/ur V 9.pg Find the work done by the force field F(x,y,z) = 5xi + 5yj + 4k on a particle that moves along the helix r(t) = 3cos(t)i + 3sin(t)j + 4tk,0 t π. 0. ( pt) rochesterlibrary/setvectoralculus/ur V 0.pg A curve is given by a vector function r(t), t 4, with unit tangent T(t), unit normal N(t), and unit binormal B(t). Indicate whether the following line integrals are positive, negative, or zeroz by typing P, N, or Z as appropriate: A. T dr = Z B. N dr = Z. B dr =. ( pt) rochesterlibrary/setvectoralculus/ur V.pg Z Z Suppose that f (x,y) da = 4 where D is the disk x + D y 6. Now suppose E is thezdisk Z x + y 56 and g(x,y) = 3 f ( 4 x, y 4 ). What is the value of g(x,y) da? E. ( pt) rochesterlibrary/setvectoralculus/ur V.pg A lattice point in the plane is a point (a, b) with both coordinates equal to integers. For example, (-, ) is a lattice point but (/, 3) is not. If D(R) is the disk of radius R and center the origin, count the lattice points inside D(R) and call this number L(R) L(R). What is the limit, lim R R?

3 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus due 05/04/008 at 0:00am EDT.. ( pt) rochesterlibrary/setvectoralculus/ur vc.pg For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, f = F). If it is not conservative, type N. A. F(x,y) = (0x + 3y)i + (3x + y)j B. F(x,y) = 5yi + 6xj. F(x,y,z) = 5xi + 6yj + k f (x,y,z) = D. F(x,y) = (5siny)i + (6y + 5xcosy)j E. F(x,y,z) = 5x i + 3y j + 6z k f (x,y,z) = Note: Your answers should be either expressions of x, y and z (e.g. 3xy + yz ), or the letter N B. { (x,y) x + y < }. { (x,y) x y < } D. { (x,y) x y > } E. { (x,y) < x + y < 4 } 6. ( pt) rochesterlibrary/setvectoralculus/ur vc 6.pg Let be the positively oriented circle x + y =. Use Green s Theorem to evaluate the line integral R 0ydx + 0xdy. 7. ( pt) rochesterlibrary/setvectoralculus/ur vc 7.pg Let be the positively oriented square with vertices (0, 0), (,0), (,), (0,). Use Green s Theorem to evaluate the line integral R 4y xdx + 3x ydy.. ( pt) rochesterlibrary/setvectoralculus/ur vc.pg If is the curve given by r(t) = ( + 3sint)i+ ( + sin t ) j+ ( + 3sin 3 t ) k, 0 t π and F is the radial vector field F(x, y, z) = xi + yj + zk, compute the work done by F on a particle moving along. 3. ( pt) rochesterlibrary/setvectoralculus/ur vc 3.pg Suppose is any curve from (0,0,0) to (,,) and F(x,y,z) = (4z + y)i + (5z + x)j + (5y + 4x)k. ompute the line integral R F dr. 4. ( pt) rochesterlibrary/setvectoralculus/ur vc 4.pg Let F(x,y) = yi+xj x +y and let be the circle r(t) = (cost)i + (sint)j, 0 t π. A. ompute Q x Note: Your answer should be an expression of x and y; e.g. 3xy - y B. ompute P y Note: Your answer should be an expression of x and y; e.g. 3xy - y. ompute R F dr Note: Your answer should be a number D. Is F conservative? Type Y if yes, type N if no. 5. ( pt) rochesterlibrary/setvectoralculus/ur vc 5.pg Determine whether the given set is open, connected, and simply connected. For example, if it is open, connected, but not simply connected, type YYN standing for Yes, Yes, No. A. {(x,y) x >,y < } 8. ( pt) rochesterlibrary/setvectoralculus/ur vc 8.pg Find a parametrization of the curve x /3 + y /3 = and use it to compute the area of the interior. 9. ( pt) rochesterlibrary/setvectoralculus/ur vc 9.pg Let F = 6xi + 8yj + 7zk. ompute the divergence and the curl. A. div F = B. curl F = i+ j+ k 0. ( pt) rochesterlibrary/setvectoralculus/ur vc 0.pg Let F = (yz)i + (8xz)j + (3xy)k. ompute the following: A. div F = B. curl F = i+ j+ k. div curl F = Note: Your answers should be expressions of x, y and/or z; e.g. 3xy or z or 5. ( pt) rochesterlibrary/setvectoralculus/ur vc.pg Let F be any vector field of the form F = f (x)i+g(y)j+h(z)k and let G be any vector field of the form F = f (y,z)i + g(x,z)j + h(x,y)k. Indicate whether the following statements are true or false by placing T or F to the left of the statement.. G is irrotational. F is incompressible 3. F is irrotational 4. G is incompressible. ( pt) rochesterlibrary/setvectoralculus/ur vc.pg Let F = 3yi + 5xj. Use the tangential vector form of Green s Theorem to compute the circulation integral R F dr where is the positively oriented circle x + y = 4.

4 3. ( pt) rochesterlibrary/setvectoralculus/ur vc 3.pg Let F = xi + 5yj and let n be the outward unit normal vector to the positively oriented circle x + y = 4. ompute the flux integral R F nds. 4. ( pt) rochesterlibrary/setvectoralculus/ur vc 4.pg A rock with a mass of 0 kilograms is put aboard an airplane in New York ity and flown to Boston. How much work does the gravitational field of the earth do on the rock? Newton-meters 5. ( pt) rochesterlibrary/setvectoralculus/ur vc 5.pg Suppose F = F(x,y,z) is a gradient field with F = f, S is a level surface of f, and is a curve on S. What is the value of the line integral R F dr? 6. ( pt) rochesterlibrary/setvectoralculus/ur vc F.pg A vector field gives a geographical description of the flow of money in a society. In the neighborhood of a political convention, the divergence of this vector field is: A. positive B. negative. zero

5 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus3 due 05/05/008 at 0:00am EDT.. ( pt) ZrochesterLibrary/setVectoralculus3/ur Z vc 3.pg Evaluate + x + y ds where S is the helicoid: r(u,v) = S ucos(v)i + usin(v)j + vk, with 0 u 4,0 v 3π. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3.pg Find the surface area of the part of the sphere x + y + z = 9 that lies above the cone z = x + y 3. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 3.pg A fluid has density and velocity field v = yi + xj + 3zk. Find the rate of flow outward through the sphere x + y + z = 5 4. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 4.pg Let S be the part of the plane 4x + 3y + z = 3 which lies in the first octant, oriented upward. Find the flux of the vector field F = 4i + j + 4k across the surface S. 5. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 5.pg Use Gauss s law to find the charge enclosed by the cube with vertices (±,±,±) if the electric field is E(x,y,z) = 5xi + 6yj + zk. ε 0 6. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 6.pg The temperature u in a star of conductivity 6 is inversely proportional to the distance from the center: u =. x +y +z 7 If the star is a sphere of radius 4, find the rate of heat flow outward across the surface of the star. 7. ( pt) rochesterlibrary/setvectoralculus3/ur Z Z vc 3 7.pg Use Stoke s theorem to evaluate curlf ds where S F(x,y,z) = 5yzi+5xzj+(x +y )zk and S is the part of the paraboloid z = x + y that lies inside the cylinder x + y =, oriented upward. 8. ( pt) rochesterlibrary/setvectoralculus3/ur Z vc 3 8.pg Use Stoke s Theorem to evaluate F dr where F(x,y,z) = xi + yj + 6(x + y )k and is the boundary of the part of the paraboloid where z = 8 x y which lies above the xy-plane and is oriented counterclockwise when viewed from above. 9. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 9.pg Use the divergence theorem to find the outward flux of the vector field F(x,y,z) = x i+5y j+4z k across the boundary of the rectangular prism: 0 x 4,0 y 3,0 z ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 0.pg If a parametric surface given by r (u,v) = f (u,v)i + g(u,v)j + h(u,v)k and 5 u 5, 3 v 3, has surface area equal to, what is the surface area of the parametric surface given by r (u,v) = r (u,v) with 5 u 5, 3 v 3?. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3.pg Suppose F is a radial force field, S is azsphere Z of radius 5 centered at the origin, and the flux integral F ds = 7. S Let S be a sphere of Zradius Z 30 centered at the origin, and consider the flux integral F ds. S (A) If the magnitude of F is inversely proportional to the square Z Z of the distance from the origin,what is the value of F ds? S (B) If the magnitude of F is inversely proportional Zto Z the cube of the distance from the origin, what is the value of F ds? S. ( pt) rochesterlibrary/setvectoralculus3/ur vc 3 F.pg In springtime, the average value over time of the divergence of the vector field which represents air flow is: A. zero B. positive. negative