ConcepTest Section 19.2 Question 1 (a) The rectangle is twice as wide in the x-direction, with new corners at the origin, (2, 0, 0), (2, 1, 3), (0, 1, 3). (b) The rectangle is moved so that its corners are at (1, 0, 0), (2, 0, 0), (2, 1, 3), (1, 1, 3). (c) The orientation is changed to downward. (d) The rectangle is tripled in size, so that its new corners are at the origin, (3, 0, 0), (3, 3, 9), (0, 3, 9).

ConcepTest Section 19.2 Answer 1 ANSWER (a) The surface can be considered as made up of two rectangles, the original one and another “next door” along the x-axis. Because the vector field is independent of x, the flux through both rectangles are the same. Thus, the flux has doubled. (b) Because the vector field does not depend on x, the flux is unchanged. (c) The sign is reversed, but the magnitude of the flux remains the same. (d) Tripling each side of the rectangle multiplies its area by 9. However, the surface now extends further up the z-axis, where the vector field is not given. If the vector field is larger further up the z-direction (as suggested by the diagram), then the flux has multiplied by a factor of more than 9.

ConcepTest Section 19.2 Question 2 To calculate the flux through each of the following surfaces, what coordinate system, rectangular, cylindrical, or spherical, would be a good choice? (a) x 2 + y 2 = 9 (b) x – y + z = 5 (c) 9 – x 2 – y 2 – z 2 = 0 (d) y 2 + z 2 = 5

ConcepTest Section 19.2 Answer 2 ANSWER (a) Cylindrical, because the surface is a cylinder. (b) Rectangular, because the surface is a plane which can be written as the graph of a function of x and y. (c) Spherical, because the surface is a sphere. (d) Cylindrical, because the surface is a cylinder. Since the cylinder is centered on the x-axis, we will have to use x in place of z.

ConcepTest Section 19.2 Question 2 To calculate a flux integral through the surfaces (I)–(IV), which area element (a)–(f) would be a good choice? (There may be more than one solution.) If you choose (f), give the function f you would use.

ConcepTest Section 19.2 Answer 3 ANSWER (I) For part of a sphere, use spherical coordinates, (d), or rectangular, (f) with (II)Use rectangular coordinates, (f), with f (x, y) = c + mx + ny. (III) Use cylindrical coordinates, (e). (IV) Since the surface is in the xz-plane, use (b). (V) For the half cylinder, we could use cylindrical coordinates (centered on the y-axis rather than the z-axis), or rectangular, (f) with