Let I be the flux of F = ey , 2xex2 , z2 through the upper hemisphere S of the unit

Indicate with an arrow the boundary orientation of the boundary curves of the surfaces in Figure 14, oriented by the outward-pointing normal vectors.

Let F = curl(A). Which of the following are related by Stokes Theorem? (a) The circulation of A and flux of F (b) The circulation of F and flux of A

What is the definition of a vector potential?

Which of the following statements is correct? (a) The flux of curl(A) through every oriented surface is zero. (b) The flux of curl(A) through every closed, oriented surface is zero.

Which condition on F guarantees that the flux through S1 is equal to the flux through S2 for any two oriented surfaces S1 and S2 with the same oriented boundary?

In Exercises 510, calculate curl(F) and then apply Stokes Theorem to compute the flux of curl(F) through the given surface using a line integral. F = x + y,z2 4, x y2 + 1) , surface of the wedge-shaped box in Figure 15 (bottom included, top excluded) with outward-pointing normal

In Exercises 510, calculate curl(F) and then apply Stokes Theorem to compute the flux of curl(F) through the given surface using a line integral. F = 3z, 5x, 2y, that part of the paraboloid z = x2 + y2 that lies below the plane z = 4 with upward-pointing unit normal vector 8

In Exercises 510, calculate curl(F) and then apply Stokes Theorem to compute the flux of curl(F) through the given surface using a line integral. F = yz, xz,z3 , that part of the cone z =  x2 + y2 that lies between the two planes z = 1 and z = 3 with upward-pointing unit normal vector

In Exercises 510, calculate curl(F) and then apply Stokes Theorem to compute the flux of curl(F) through the given surface using a line integral. F = yz,xz,xy, that part of the cylinder x2 + y2 = 1 that lies between the two planes z = 1 and z = 4 with outward-pointing unit normal vector

In Exercises 510, calculate curl(F) and then apply Stokes Theorem to compute the flux of curl(F) through the given surface using a line integral. F =  2y,ez, arctan x  , that part of the paraboloid z = 4 x2 y2 cut off by the xy-plane with upward-pointing unit normal vector

In Exercises 1114, apply Stokes Theorem to evaluate  C F dr by finding the flux of curl(F) across an appropriate surface. 11. F = 3y, 2x, 3y, C is the circle x2 + y2 = 9, z = 2, oriented counterclockwise as viewed from above.

In Exercises 1114, apply Stokes Theorem to evaluate  C F dr by finding the flux of curl(F) across an appropriate surface F = yz,xy,xz, C is the square with vertices(0, 0, 2),(1, 0, 2), (1, 1, 2), and (0, 1, 2), oriented counterclockwise as viewed from above.

In Exercises 1114, apply Stokes Theorem to evaluate  C F dr by finding the flux of curl(F) across an appropriate surface F = y,z,x, C is the triangle with vertices (0, 0, 0), (3, 0, 0), and (0, 3, 3), oriented counterclockwise as viewed from above.

In Exercises 1114, apply Stokes Theorem to evaluate  C F dr by finding the flux of curl(F) across an appropriate surface F = y, 2z, 4x, C is the boundary of that portion of the plane x + 2y + 3z = 1 that is in the first octant of space, oriented counterclockwise as viewed from above.

Let S be the surface of the cylinder (not including the top and bottom) of radius 2 for 1 z 6, oriented with outward-pointing normal (Figure 16). (a) Indicate with an arrow the orientation of S (the top and bottom circles). (b) Verify Stokes Theorem for S and F =  yz2, 0, 0  .

Let S be the portion of the plane z = x contained in the halfcylinder of radius R depicted in Figure 17. Use Stokes Theorem to calculate the circulation of F = z,x,y + 2z around the boundary of S (a half-ellipse) in the counterclockwise direction when viewed from above. Hint: Show that curl(F) is orthogonal to the normal vector to the plane.

Let I be the flux of F =  ey , 2xex2 , z2 through the upper hemisphere S of the unit sphere. (a) Let G =  ey , 2xex2 , 0  . Find a vector field A such that curl(A) = G. (b) Use Stokes Theorem to show that the flux of G through S is zero. Hint: Calculate the circulation of A around S. (c) Calculate I . Hint: Use (b) to show that I is equal to the flux of  0, 0, z2 through S.

Let F = 0, z, 1. Let S be the spherical cap x2 + y2 + z2 1, where z 1 2 . Evaluate S F dS directly as a surface integral. Then verify that F = curl(A), where A = (0,x,xz) and evaluate the surface integral again using Stokes Theorem. 19

Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R in Example 4. Use Stokes Theorem to compute: (a) The flux of B through a circle in the xy-plane of radius r

The magnetic field B due to a small current loop (which we place at the origin) is called a magnetic dipole (Figure 18). Let = (x2 + y2 + z2)1/2. For large, B = curl(A), where A =  y 3 , x 3 , 0  (a) Let C be a horizontal circle of radius R with center (0, 0,c), where c is large. Show that A is tangent to C. (b) Use Stokes Theorem to calculate the flux of B through C.

Auniform magnetic fieldBhas constant strength b in the z-direction [i.e., B = 0, 0, b].(a) Verify that A = 1 2B r is a vector potential for B, where r = x,y, 0. (b) Calculate the flux of B through the rectangle with vertices A, B, C, and D in Figure 19. 2

Let F =  x2y,x, 0  . Referring to Figure 19, let C be the closed path ABCD. Use Stokes Theorem to evaluate C F dr in two ways. First, regard C as the boundary of the rectangle with vertices A, B, C, and D. Then treat C as the boundary of the wedge-shaped box with an open top.

Let F =  y2, 2z + x, 2y2 . Use Stokes Theorem to find a plane with equation ax + by + cz = 0 (where a,b,c are not all zero) such that  C F dr = 0 for every closed C lying in the plane. Hint: Choose a,b,c so that curl(F) lies in the plane.

Let F =  z2, 2zx, 4y x2 , and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate  C F dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).

Let F =  y2, x2, z2 . Show that C1 F dr = C2 F dr for any two closed curves lying on a cylinder whose central axis is the z-axis (Figure 21). F

The curl of a vector field F at the origin is v0 = 3, 1, 4. Estimate the circulation around the small parallelogram spanned by the vectors A =  0, 1 2 , 1 2  and B =  0, 0, 1 3  .

You know two things about a vector field F: (i) F has a vector potential A (but A is unknown). (ii) The circulation of A around the unit circle (oriented counterclockwise) is 25. Determine the flux of F through the surface S in Figure 22, oriented with an upward-pointing normal.

Suppose that F has a vector potential and that F(x, y, 0) = k. Find the flux of F through the surface S in Figure 22, oriented with an upward-pointing normal.

Prove that curl(f a) = f a, where f is a differentiable function and a is a constant vector.

Show that curl(F) = 0 if F is radial, meaning that F = f ()x,y,z for some function f (), where =  x2 + y2 + z2. Hint: It is enough to show that one component of curl(F) is zero, because it will then follow for the other two components by symmetry.

Prove the following Product Rule: curl(f F) = f curl(F) + f F

Assume that f and g have continuous partial derivatives of order 2. Prove that  S f (g) dr = S (f ) (g) dS 3

Verify that B = curl(A) for r>R in the setting of Example 4.

Explain carefully why Greens Theorem is a special case of Stokes Theorem.

In this exercise, we use the notation of the proof of Theorem 1 and prove  C F3(x, y, z)k dr = S curl(F3(x, y, z)k) dS 12 In particular, S is the graph of z = f (x, y) over a domain D, and C is the boundary of S with parametrization (x(t), y(t), f (x(t), y(t))). (a) Use the Chain Rule to show that F3(x, y, z)k dr = F3(x(t), y(t), f (x(t), y(t)) fx (x(t), y(t))x (t) + fy (x(t), y(t))y (t) dt and verify that  C F3(x, y, z)k dr =  C0  F3(x, y, z)fx (x, y), F3(x, y, z)fy (x, y) dr where C0 has parametrization (x(t), y(t)). (b) Apply Greens Theorem to the line integral over C0 and show that the result is equal to the right-hand side of Eq. (12). 36.

Let F be a continuously differentiable vector field in R3, Q a point, and S a plane containing Q with unit normal vector e. Let Cr be a circle of radius r centered at Q in S, and let Sr be the disk enclosed by Cr. Assume Sr is oriented with unit normal vector e. (a) Let m(r) and M(r) be the minimum and maximum values of curl(F(P)) e for P Sr. Prove that m(r) 1 r2 Sr curl(F) dS M(r) (b) Prove that curl(F(Q)) e = lim r0 1 r2 Cr F dr This proves that curl(F(Q)) e is the circulation per unit area in the plane S. 17