Problem 1RCC: What is a vector field? Give three examples that have physical meaning. Problem 2RCC Problem 3RCC Problem 4RCC: (a) Define the line integral of a vector field F along a smooth curve C given by a vector function... Problem 5RCC Problem 6RCC Problem 7RCC Problem 8RCC: Write expressions for the area enclosed by a curve C in terms of line integrals around C. Problem 9RCC Problem 10RCC Problem 11RCC Problem 12RCC Problem 13RCC Problem 14RCC Problem 15RCC Problem 16RCC Problem 1RQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 2RQ Problem 3RQ Problem 4RQ Problem 5RQ Problem 6RQ Problem 7RQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 8RQ Problem 9RQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 10RQ Problem 11RQ Problem 12RQ Problem 13RQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 1RE: A vector field F, a curve C, and a point P are shown. (a) Is C F dr positive, negative, or zero?... Problem 2RE: Evaluate the line integral. 2. C x ds, C is the arc of the parabola y = x2 from (0, 0) to (1, 1) Problem 3RE: Evaluate the line integral. 3. C yz cos x ds, C: x = t, y = 3 cos t, z = 3 sin t, 0 t Problem 4RE: Evaluate the line integral. 4. C y dx + (x + y2) dy, C is the ellipse 4x2 + 9y2 = 36 with... Problem 5RE: Evaluate the line integral. 5. C y3 dx + x2 dy, C is the arc of the parabola x = 1 y2 from (0, 1)... Problem 6RE: Evaluate the line integral. 6. C xy dx + ey dy + xz dz, C is given by r(t) = t4 i + t2 j + t3 k, 0 ... Problem 7RE: Evaluate the line integral. 7. C xy dx + y2 dy + yz dz, C is line segment from (1,0, 1), to (3,4, 2) Problem 8RE: Evaluate the line integral. 8. C F dr, where F(x, y) = xy i + x2 j and C is given by r(t) = sin t i... Problem 9RE: Evaluate the line integral. 9. C F dr, where F(x,y,z) = ez i + xz j + (x + y) k and C is given by... Problem 10RE: Find the work done by the force field F(x, y, z) = z i + x j+y k in moving a particle from the point... Problem 11RE: Show that F is a conservative vector field. Then find a function f such that F = f. 11. F(x, y) = (1... Problem 12RE: Show that F is a conservative vector field. Then find a function f such that F = f. 12. F(x,y,z) =... Problem 13RE: Show that F is a conservative and use this fact to evaluate C F dr along the given curve. 13. F(x,... Problem 14RE: Show that F is a conservative and use this fact to evaluate C F dr along the given curve. 14. F(x,... Problem 15RE: Verify that Greens Theorem is true for the line integral C xy2 dx x2y dy, where C consists of the... Problem 16RE: Use Greens Theorem to evaluate C 1+x3dx + 2xydy where C is the triangle with vertices (0, 0), (1,... Problem 17RE Problem 18RE: Find curl F and div F if F(x, y, z) = e-x sin y i + e-y sin z j + e-z sin x k Problem 19RE: Show that there is no vector field G such that curl G = 2x i + 3yz j xz2 k Problem 20RE: If F and G are vector fields whose component functions have continuous first partial derivatives,... Problem 21RE: If C is any piecewise-smooth simple closed plane curve and f and g are differentiable functions,... Problem 22RE: If f and g are twice differentiable functions, show that 2(fg) = f 2 g + g2 f + 2f g Problem 23RE: If f is a harmonic function, that is, 2f = 0, show that the line integral fy dx fx dy is... Problem 24RE Problem 25RE: Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices (0,... Problem 27RE: Evaluate the surface integral. 27. S z dS, where S is the part of the paraboloid z = x2 + y2 that... Problem 28RE: Evaluate the surface integral. 28. s (x2z + y2z)dS, where S is the part of the plane z = 4 + x + y... Problem 29RE: Evaluate the surface integral. 29. S F dS, where F(x, y, z) = xz i 2y j + 3x k and S is the sphere... Problem 30RE: Evaluate the surface integral. 30. S F dS, where F(x, y, z) = x2 i xy j + z k and S is the part of... Problem 31RE: Verify that Stokes Theorem is true for the vector field F(x, y, z) = x2 i + y2 j + z2 k, where S is... Problem 32RE: Use Stokes Theorem to evaluate s curl F dS, where F(x, y, z) = x2yz i + yz2 j + z3exy k, S is the... Problem 33RE: Use Stokes Theorem to evaluate C F dr, where F(x, y, z) = xy i + yz j + zx k, and C is the triangle... Problem 34RE: Use the Divergence Theorem to calculate the surface integral S F dS, where F(x, y, z) = x3 i + y3 j... Problem 35RE: Verify that the Divergence Theorem is true for the vector field F(x, y, z) = x i + y j + z k, where... Problem 36RE: Compute the outward flux of F(x, y, z) = xi+yj+zk(x2+y2+z2)32 through the ellipsoid 4x2 + 9y2 + 6z2... Problem 37RE: Let F(x, y, z) = (3x2 yz 3y) i + (x3z 3x) j + (x3y + 2z) k Evaluate C F dr, where C is the curve... Problem 38RE Problem 39RE Problem 40RE: If the components of F have continuous second partial derivatives and S is the boundary surface of a... Problem 41RE Problem 1P: 1. Let S be a smooth parametric surface and let P be a point such that each line that starts at P... Problem 2P: Find the positively oriented simple closed curve C for which the value of the line integral C (y3 ... Problem 3P: Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector n... Problem 5P: Prove the following identity: (F G) = (F )G + (G )F + F curl G + G curl F Problem 6P format_list_bulleted