ConcepTest Section 17.3 Question 1 In Problems 1-2, match the formulas (a)-(d) with the graphs of vector fields (I)-(IV). You do not need to find formulas for f, g, h, and k. Math the formulas (a)-(d) with the graphs of the vector fields (I)-(IV). (a) f(x)i (b) g(x)j (c) h(y)i (d) k(y)j

ConcepTest Section 17.3 Answer 1 ANSWER COMMENT: Students should be encouraged to read formulas for qualitative features. All vectors in the fields f(x)i and h(y)i point in the horizontal directions, since they are all multiples of the horizontal vector i. The graphs (II) and (III) are the ones with horizontal vectors, but which is (a) and which is (c)? Since the lengths of the vectors is (II) changes when y changes (moving upward in the graph) but not when x changes (moving horizontally), graph (II) must correspond to (c), h(y)i. Thus (III) goes with (a), f(x)i. All vectors in the fields g(x)j and k(y)j point in the vertical direction, since they are all multiples of the vertical vector j. The graphs in (I) and (IV) are the ones with vertical vectors. The length of the vectors in (I) changes when x changes but not when y changes, so graph (I) must correspond to (b), g(x)j. Thus (IV) goes with (d), k(y)j. So a-(III), b-(I), c-(II),d-(IV)

ConcepTest Section 17.3 Question 2 In Problems 1-2, match the formulas (a)-(d) with the graphs of vector fields (I)-(IV). You do not need to find formulas for f, g, h, and k. Math the formulas (a)-(d) with the graphs of the vector fields (I)-(IV). (a) f(x)i + f(x)j (b) g(x)i - g(x)j (c) h(y)i + h(y)j (d) k(y)i - k(y)j

ConcepTest Section 17.3 Answer 2 ANSWER All vectors in the fields f(x)i + f(x)j = f(x)(i + j) and h(y)i +h(y)j = h(y)(i + j) point parallel to i + j. The graphs in (II) and (III) are the ones with vectors parallel to i + j, but which is (a) and which is (c)? Since the length of vectors in (II) changes when x changes (moving horizontally in the graph) but not when y changes (moving vertically), graph (II) must correspond to (a), f(x)i + f(x)j. Thus (III) goes with (c), h(y)i + h(y)j. All vectors in the fields g(x)i – g(x)j = g(x)(i – j) and k(y)i – k(y)j = k(y)(i – j) point parallel to i – j. The graphs of (I) and (IV) are the ones in this direction. The length of the vectors in (I) changes when y changes but not when x changes, so graph (I) must correspond to (d), k(y)i – k(y)j. Thus (IV) goes with (b), g(x)i – g(x)j. The match up is: a-(II), b-(IV), c-(III), d-(I).

ConcepTest Section 17.3 Question 3 Match the vector field with the appropriate pictures.

ConcepTest Section 17.3 Answer 3 ANSWER COMMENT: Process of elimination helps to do this problem. (a) And (ii), (b) and (i), (c) and (iv), (d) and (iii)

For each of the vector fields in the plane,, Decide which of the properties the vector field has: AAll vectors point toward the origin. BAll vectors point away from the origin. CAll vectors point in the same direction. DAll vectors have the same length. EVectors get longer away from the origin. FVectors get shorter away from the origin. GThe vector field has rotational symmetry about the origin. HVectors that start on the same circle centered at the origin have the same length. ConcepTest Section 17.3 Question 4

ConcepTest Section 17.3 Answer 4 ANSWER COMMENT:

ConcepTest Section 17.3 Question 5 Compare and contrast the following vector fields:

ConcepTest Section 17.3 Answer 5 ANSWER COMMENT: Vector fields of this form are used frequently for line and flux integrals. It is an advantage to be able to visualize them.

ConcepTest Section 17.3 Question 6 (a)x 2 (b)-x 2 (c)-2x (d)-y 2

ConcepTest Section 17.3 Answer 6 ANSWER COMMENT: Ask students for other possible functions whose gradient is the vector field shown.

Rank the length of the gradient vectors at the points marked on the contour plot in Figure 17.4 ConcepTest Section 17.3 Question 7

ConcepTest Section 17.3 Answer 7 ANSWER COMMENT: While the exact direction and length of the vectors is not that important in this problem, a qualitative comparison of their lengths is. All vectors point radially outward. Vectors on the contour z = 7 are longer than vectors on the contour z = 3, which are longer then vectors on the contour z = 1.