An ideal electric dipole in electrostatics is characterizedby its position in 3-space

In Exercises 14, find the area vector of the oriented flat surface. The triangle with vertices (0, 0, 0), (0, 2, 0), (0, 0, 3) oriented in the negative x direction.

In Exercises 14, find the area vector of the oriented flat surface. . Circular disc of radius 5 in the xy-plane, oriented upward.

In Exercises 14, find the area vector of the oriented flat surface. y = 10, 0 x 5, 0 z 3, oriented away from the xz-plane.

In Exercises 14, find the area vector of the oriented flat surface. y = 10, 0 x 5, 0 z 3, oriented away from the xz-plane.

Find an oriented flat surface with area vector 150j .

Let S be the disk of radius 3 perpendicular the the y-axis, centered at (0, 6, 0) and oriented away from the origin. Is  S (xi + yj ) dA a vector or a scalar?

Compute  S (4i + 5 k ) dA , where S is the square of side length 3 perpendicular to the z-axis, centered at (0, 0, 2) and oriented (a) Toward the origin. (b) Away from the origin.

Compute  S (2i + 3 k ) dA , where S is the disk of radius 4 perpendicular to the x-axis, centered at (5, 0, 0) and oriented (a) Toward the origin. (b) Away from the origin.

Let F (x, y, z) = zi . For each of the surfaces in (a)(e), say whether the flux of F through the surface is positive, negative, or zero. The orientation of the surface is indicated by a normal vector.

Repeat Exercise 9 with F (x, y, z) = zi + x k .

Repeat Exercise 9 with the vector field F (r ) = r .

In Exercises 1215, compute the flux of v = i + 2j 3 k through the rectangular region with the orientation shown. 12

In Exercises 1215, compute the flux of v = i + 2j 3 k through the rectangular region with the orientation shown. 13

In Exercises 1215, compute the flux of v = i + 2j 3 k through the rectangular region with the orientation shown. 14

In Exercises 1215, compute the flux of v = i + 2j 3 k through the rectangular region with the orientation shown. 15

For Exercises 1619 find the flux of the constant vector field v =i j + 3 k through the given surface. A disk of radius 2 in the xy-plane oriented upward.

For Exercises 1619 find the flux of the constant vector field v =i j + 3 k through the given surface. A triangular plate of area 4 in the yz-plane oriented in the positive x-direction.

For Exercises 1619 find the flux of the constant vector field v =i j + 3 k through the given surface. A square plate of area 4 in the yz-plane oriented in the positive x-direction.

For Exercises 1619 find the flux of the constant vector field v =i j + 3 k through the given surface. The triangular plate with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1), oriented away from the origin.

In Exercises 2022, find the flux of H = 2i + 3j + 5 k through the surface S. S is the cylinder x2 + y2 = 1, closed at the ends by the planes z = 0 and z = 1 and oriented outward.

In Exercises 2022, find the flux of H = 2i + 3j + 5 k through the surface S. S is the disk of radius 1 in the plane x = 2 oriented in the positive x-direction.

In Exercises 2022, find the flux of H = 2i + 3j + 5 k through the surface S. S is the disk of radius 1 in the plane x + y + z = 1 oriented in upward.

Find the flux of the vector fields in Exercises 2325 out of the closed box 0 x 1, 0 y 2, 0 z 3. F = 3i + 2j +  k

Find the flux of the vector fields in Exercises 2325 out of the closed box 0 x 1, 0 y 2, 0 z 3. G = xi

Find the flux of the vector fields in Exercises 2325 out of the closed box 0 x 1, 0 y 2, 0 z 3. H = zx k

In Exercises 2629, calculate the flux integral.  S (xi + 4j ) dA where S is the disk of radius 5 perpendicular to the x-axis, centered at(3, 0, 0) and oriented toward the origin

In Exercises 2629, calculate the flux integral. S r dA where S is the sphere of radius 3 centered at the origin.

In Exercises 2629, calculate the flux integral. S(sin xi +(y2+z2)j +y2 k )dA where S is a disk of radius in the plane x = 3/2, oriented in the positive x-direction.

In Exercises 2629, calculate the flux integral. S(5i + 5j + 5 k ) dA where S is a disk of radius 3 in the plane x + y + z = 1, oriented upward.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 2 + 3j through the square of side in the xyplane, oriented upward.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 2 + 3j through the unit disk in the yz-plane, centered at the origin and oriented in the positive xdirection.

In Exercises 3052, calculate the flux of the vector field through the surface. F = xi + yj + z k through the square of side 1.6 centered at (2, 5, 8), parallel to the xz-plane and oriented away from the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = z k through a square of side 14 in a horizontal plane 2 units below the xy-plane and oriented downward.

In Exercises 3052, calculate the flux of the vector field through the surface. F = yi + xj and S is the square plate in the yzplane with corners at (0, 1, 1), (0, 1, 1), (0, 1, 1), and (0, 1, 1), oriented in the positive x-direction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 7i + 6j + 5 k and S is a disk of radius 2 in the yzplane, centered at the origin and oriented in the positive x-direction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = xi + 2yj + 3z k and S is a square of side 2 in the plane y = 3, oriented in the positive y-directiction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 7i +6j +5 k and S is a sphere of radius centered at the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 5r through the sphere of radius 2 centered at the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = xi +yj + (z2 + 3) k and S is the rectangle z = 4, 0 x 2, 0 y 3, oriented in the positive zdirection.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 6i + 7j through triangle of area 10 in the plane x + y = 5, oriented in the positive x-direction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 6i + x2j  k , through the square of side 4 in the plane y = 3, centered on the y-axis, with sides parallel to the x and z axes, and oriented in the positive y-direction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = (x + 3)i + (y + 5)j + (z + 7) k through the rectangle x = 4, 0 y 2, 0 z 3, oriented in the positive x-direction.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 7r through the sphere of radius 3 centered at the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 3r through the sphere of radius 2 centered at the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = 2zi + xj + x k through the rectangle x = 4, 0 y 2, 0 z 3, oriented in the positive xdirection.

In Exercises 3052, calculate the flux of the vector field through the surface. F =i +2j through a square of side 2 lying in the plane x + y + z = 1, oriented away from the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = (x2 + y2) k through the disk of radius 3 in the xy-plane and centered at the origin and oriented upward.

In Exercises 3052, calculate the flux of the vector field through the surface. F = cos(x2 + y2) k through the disk x2 + y2 9 oriented upward in the plane z = 1.

In Exercises 3052, calculate the flux of the vector field through the surface. F = ey2+z2 i through the disk of radius 2 in the yzplane, centered at the origin and oriented in the positive x-direction

In Exercises 3052, calculate the flux of the vector field through the surface. F = yi + xj through the disk in the xy-plane with radius 2, oriented upward and centered at the origin.

In Exercises 3052, calculate the flux of the vector field through the surface. F = r through the disk of radius 2 parallel to the xyplane oriented upward and centered at (0, 0, 2).

In Exercises 3052, calculate the flux of the vector field through the surface. F = (2 x)i through the cube whose vertices include the points (0, 0, 0), (3, 0, 0), (0, 3, 0), (0, 0, 3), and oriented outward.

Let B be the surface of a box centered at the origin, with edges parallel to the axes and in the planes x = 1, y = 1, z = 1, and let S be the sphere of radius 1 centered at origin. (a) Indicate whether the following flux integrals are positive, negative, or zero. No reasons needed. (a) B xi dA (b) B yi dA (c) S |x| i dA (d) S(y x)i dA (b) Explain with reasons how you know which flux integral is greater:  S xi dA or  B xi dA ?

Suppose that E is a uniform electric field on 3-space, so E (x, y, z) = a + bj + c k , for all points (x, y, z), where a, b, c are constants. Show, with the aid of symmetry, that the flux of E through each of the following closed surfaces S is zero: (a) S is the cube bounded by the planes x = 1, y = 1, and z = 1. (b) S is the sphere x2 + y2 + z2 = 1. (c) S is the cylinder bounded by x2 + y2 = 1, z = 0, and z = 2.

Water is flowing down a cylindrical pipe of radius 2 cm; its speed is (3(3/4)r2) cm/sec at a distance r cm from the center of the pipe. Find the flux through the circular cross-section of the pipe, oriented so that the flux is positive.

(a) What do you think will be the electric flux through the cylindrical surface that is placed as shown in the constant electric field in Figure 19.16? Why? (b) What if the cylinder is placed upright, as shown in Figure 19.17? Explain.

Let S be part of a cylinder centered on the y-axis. Explain why the three vectors fields F , G , and H have the same flux through S. Do not compute the flux. F = xi + 2yz k G = xi + y sin xj + 2yz k H = xi + cos(x2 + z)j + 2yz k

Find the flux of F = r / r 3 out of the sphere of radius R centered at the origin

Find the flux of F = r /||r||2 out of the sphere of radius R centered at the origin

Consider the flux of the vector field F = r /||r ||p for p 0 out of the sphere of radius 2 centered at the origin. (a) For what value of p is the flux a maximum? (b) What is that maximum value?

Let S be the cube with side length 2, faces parallel to the coordinate planes, and centered at the origin. (a) Calculate the total flux of the constant vector field v = i + 2j +  k out of S by computing the flux through each face separately. (b) Calculate the flux out of S for any constant vector field v = ai + bj + c k . (c) Explain why the answers to parts (a) and (b) make sense.

Let S be the tetrahedron with vertices at the origin and at (1, 0, 0), (0, 1, 0) and (0, 0, 1). (a) Calculate the total flux of the constant vector field v = i + 2j +  k out of S by computing the flux through each face separately. (b) Calculate the flux out of S in part (a) for any constant vector field v . (c) Explain why the answers to parts (a) and (b) make sense.

Let P(x, y, z) be the pressure at the point (x, y, z) in a fluid. Let F (x, y, z) = P(x, y, z) k . Let S be the surface of a body submerged in the fluid. If S is oriented inward, show that S F dA is the buoyant force on the body, that is, the force upward on the body due to the pressure of the fluid surrounding it. [Hint: F dA = P(x, y, z) k dA = (P(x, y, z) dA )  k .]

A region of 3-space has a temperature which varies from point to point. Let T (x, y, z) be the temperature at a point (x, y, z). Newtons law of cooling says that grad T is proportional to the heat flow vector field, F , where F points in the direction in which heat is flowing and has magnitude equal to the rate of flow of heat. (a) Suppose F = k grad T for some constant k. What is the sign of k? (b) Explain why this form of Newtons law of cooling makes sense. (c) Let W be a region of space bounded by the surface S. Explain why Rate of heat loss from W = k  S (grad T ) dA

The z-axis carries a constant electric charge density of units of charge per unit length, with > 0. The resulting electric field is E . (a) Sketch the electric field, E , in the xy-plane, given E (x, y, z)=2xi + yj x2 + y2 . (b) Compute the flux of E outward through the cylinder x2 + y2 = R2, for 0 z h.

An infinitely long straight wire lying along the z-axis carries an electric current I flowing in the  k direction. Amp`eres Law in magnetostatics says that the current gives rise to a magnetic field B given by B (x, y, z) = I 2 yi + xj x2 + y2 . (a) Sketch the field B in the xy-plane. (b) Suppose S1 is a disk with center at (0, 0, h), radius a, and parallel to the xy-plane, oriented in the  k direction. What is the flux of B through S1? Is your answer reasonable? (c) Suppose S2 is the rectangle given by x = 0, a y b, 0 z h, and oriented in the i direction. What is the flux of B through S2? Does your answer seem reasonable?

An ideal electric dipole in electrostatics is characterized by its position in 3-space and its dipole moment vector p . The electric field D , at the point with position vector r , of an ideal electric dipole located at the origin with dipole moment p is given by D (r )=3 (r p )r r 5 p r 3 . Assume p = p k , so the dipole points in the  k direction and has magnitude p. (a) What is the flux of D through a sphere S with center at the origin and radius a > 0? (b) The field D is a useful approximation to the electric field E produced by two equal and opposite charges, q at r 2 and q at r 1, where the distance r 2 r 1 is small. The dipole moment of this configuration of charges is defined to be q(r 2 r 1). Gausss Law in electrostatics says that the flux of E through S is equal to 4 times the total charge enclosed by S. What is the flux of E through S if the charges at r 1 and r 2 are enclosed by S? How does this compare with your answer for the flux of D through S if p = q(r 2 r 1)?

In Problems 6869, explain what is wrong with the statement. For a certain vector field F and oriented surface S, we have S F dA = 2i 3j +  k .

In Problems 6869, explain what is wrong with the statement. If S is a region in the xy-plane oriented upwards then S F dA >  0.

In Problems 7071, give an example of: A nonzero vector field F such that S F dA = 0, where S is the triangular surface with corners (1, 0, 0), (0, 1, 0), (0, 0, 1), oriented away from the origin.

In Problems 7071, give an example of: A nonconstant vector field F (x, y, z) and an oriented surface S such that S F dA = 1.

Are the statements in Problems 7281 true or false? Give reasons for your answer. The value of a flux integral is a scalar.

Are the statements in Problems 7281 true or false? Give reasons for your answer. The area vector A of a flat, oriented surface is parallel to the surface.

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S is the unit sphere centered at the origin, oriented outward and the flux integral S F dA is zero, then F = 0 .

Are the statements in Problems 7281 true or false? Give reasons for your answer. The flux of the vector field F = i through the plane x = 0, with 0 y 1, 0 z 1, oriented in the i direction is positive.

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S is the unit sphere centered at the origin, oriented outward and F = xi +yj +z k = r , then the flux integral S F dA is positive.

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S is the cube bounded by the six planes x = 1, y = 1, z = 1, oriented outward, and F =  k , then S F dA = 0.

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S is an oriented surface in 3-space, and S is the same surface, but with the opposite orientation, then S F dA = S F dA .

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S1 is a rectangle with area 1 and S2 is a rectangle with area 2, then 2 S1 F dA = S2 F dA .

Are the statements in Problems 7281 true or false? Give reasons for your answer. If F = 2G , then S F dA = 2 S G dA .

Are the statements in Problems 7281 true or false? Give reasons for your answer. If S F dA >  S G dA then ||F || > ||G || at all points on the surface S.

For each of the surfaces in (a)(e), pick the vector field F 1, F 2, F 3, F 4, F 5, with the largest flux through the surface. The surfaces are all squares of the same size. Note that the orientation is shown. F 1 = 2i 3j 4 k F 2 =i 2j + 7 k F 3 = 7i + 5j + 6 k F 4 = 11i + 4j 5 k F 5 = 5i + 3j + 5 k