Question 1

Use a computer algebra system to compute the flux of F across S.S is the surface of the cube cut from the first octant by the planes    &#10;A) 4&#10;B) 1&#10;C) 3&#10;D)  &#10;E) 0.67

Accepted Answer

First, we need to find the boundaries of the cube. The plane equations we have are:

x = 1
y = 1
z = 1

These planes intersect at the point (1,1,1), which is also the upper right corner of the cube. The cube extends from the origin (0,0,0) to this corner.

Next, we need to find the normal vector to the surface S. The surface S is made up of six faces, each of which is perpendicular to a coordinate axis. Therefore, the normal vector to each face is simply the unit vector in the direction of the corresponding axis. For example, the normal vector to the face with equation x = 1 is (1,0,0).

Using a computer algebra system, we can calculate the surface integrals over each face, and then sum them to get the flux across the entire surface S. The surface integral over the face with equation x = 1 is:

integral(integral(F dot (1,0,0), dy, 0, 1-z), dz, 0, 1) = integral(integral(F_x, dy, 0, 1-z), dz, 0, 1)

where F_x is the x-component of F. Similarly, we have:

integral(integral(F dot (0,1,0), dx, 0, 1-z), dz, 0, 1) = integral(integral(F_y, dx, 0, 1-z), dz, 0, 1)

integral(integral(F dot (0,0,1), dx, 0, 1-y), dy, 0, 1) = integral(integral(F_z, dx, 0, 1-y), dy, 0, 1)

and so on for the other faces.

Since F is given by F = (x^2, y^2, z^2), we have F_x = x^2, F_y = y^2, and F_z = z^2. Therefore, we can evaluate each surface integral as follows:

integral(integral(F_x, dy, 0, 1-z), dz, 0, 1) = integral(integral(x^2, dy, 0, 1-z), dz, 0, 1) = 1/3

integral(integral(F_y, dx, 0, 1-z), dz, 0, 1) = integral(integral(y^2, dx, 0, 1-z), dz, 0, 1) = 1/3

integral(integral(F_z, dx, 0, 1-y), dy, 0, 1) = integral(integral(z^2, dx, 0, 1-y), dy, 0, 1) = 1/3

and so on.

Adding up all six surface integrals, we get:

1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 2

Therefore, the flux of F across the surface S is 2. The closest choice to 2 is D, 11ea8a3b_b74a_3a1a_9fff_59488d4bd1a5_TB6897_11.

Question 2

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives,find , where a is the constant vector.

A) 7
B) 3
C) 6
D) 5
E) 8

3

Accepted Answer

3

Question 3

Use the Divergence Theorem to calculate the surface integral   ; that is,calculate the flux of   across     S is the surface of the box bounded by the coordinate planes and the planes  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11ea8a3b_b749_ebf3_9fff_d9eb90665e5f_TB6897_11

Deck 13: Extension I: Vector Calculus