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)

changeswhen 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

ANSWERAll 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

jxyxFd

jxiyyxFc

rrFb

r

rrFa

),()(

),()(

)()(

)()(

4

3

2

1

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:

A All vectors point toward the origin.B All vectors point away from the origin.C All vectors point in the same direction.D All vectors have the same length.E Vectors get longer away from the origin.F Vectors get shorter away from the origin.G The vector field has rotational symmetry about the

origin.H Vectors that start on the same circle centered at

the origin have the same length.

ConcepTest • Section 17.3 • Question 4

321 ,, FFF

22321 ),( )( )(yxjiyxFrrF

r

rrF

ConcepTest • Section 17.3 • Answer 4

HF,C,:

HG,E,A,:

HG,D,B,:

3

2

1

F

F

FANSWER

COMMENT:

length.constant have circle same theon vectorsand northeast,point vectorsAll fields. vector theof ginterestinmost theis 3F

ConcepTest • Section 17.3 • Question 5

2321 )( )( )(r

rrFrrFr

rrF

Compare and contrast the following vector fields:

ConcepTest • Section 17.3 • Answer 5

are.y origin the thefromaway farther en shorter thget andorigin therdspoint towa vectorsIn origin. thefromaway move they aslonger get

vectorsin and 1,length have vectorsall in that is differenceonly The

origin. thefromaway point vectorssimilar,look very and

3

21

21

F

FF

FF

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

?for choices possible are following theofWhich . field vector theshows 17.3 Figure

f(x,y)gradfF

(a)x2

(b)-x2

(c) -2x(d)-y2

ConcepTest • Section 17.3 • Answer 6

axis.-y the toparallel isgradient its because possible

not is (d) .2ector constant v theisgradient its because

possiblenot is (c) and axis,-y thefromaway pointsgradient its because possiblenot is (a) axis.-y the towardspoints and

axis x the toparallel is which ,2 since ,(b), 22

i-

ix-)grad(x-x 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.