Deck 17: Vector Calculus

Question 1

Compute the gradient of the function f(x, y) = sin y + cos x.

A) ( cos y - 2y cos x) i + (2x sin y + sin x) j
B) (2x sin y + sin x) i + ( cos y - 2y cos x) j
C) (2x sin y - sin x) i + ( cos y + 2y cos x) j
D) ( cos y + 2y cos x) i + (2x sin y - sin x) j
E) (2x sin y) i + (2y cos x) j

Answer 1

(2x sin y - sin x) i + ( cos y + 2y cos x) j

Question 2

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

A) i
B) j
C) 0
D) k
E) i + j + k

Answer 2

0

Question 3

If f(x, y, z) = z + cos(z ), find f.

A) 2zx sin(z ) i + 2yz j + ( + sin(z )) k
B) 2x sin(z ) i + 2yz j + ( - sin(z )) k
C) -2x sin(z ) i + 2yz j + ( - sin(z )) k
D) -2zx sin(z ) i + 2yz j + ( - sin(z )) k
E) -2zx cos(z ) i + 2yz j + ( - cos(z )) k

Answer 3

-2zx sin(z ) i + 2yz j + ( - sin(z )) k

Question 4

Compute div F for F = (2x + yz) i + ( + ) j + (x sin(z) + ) k.

A) 2 + 2y + + cos(z) + 3
B) 2 + 2y + z - x cos(z) + 3
C) 2 + 2y + z + x cos(z) + 3
D) 2 + 2y + x cos(z)
E) 2 + 2y + - cos(z) + 3

Question 5

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

A) - i + j - k
B) i + j + k
C) - i + j + k
D) i - j
E) 0

Question 6

Define the curl of a vector field F.

A) F ×
B) F
C) 11ee7bab_8c78_b929_ae82_a3f0e4bb6058_TB9661_11 × F
D) 11ee7bab_8c78_b929_ae82_a3f0e4bb6058_TB9661_11 . F
E) 11ee7bab_8c78_b929_ae82_a3f0e4bb6058_TB9661_11 F

Question 7

Let be a scalar field and F be a vector field, both assumed to be sufficiently smooth. Which of the following expressions is meaningless?

A) $\textbf{ curl (grad }$ 11ee7bac_4a3a_9aaa_ae82_759e3f104991_TB9661_11 )
B) $\textbf{ div (curl F) }$
C) $\textbf{ grad (div F) }$
D) $\textbf{ div (grad }$ 11ee7bac_4a3a_9aaa_ae82_759e3f104991_TB9661_11 )
E) $\textbf{ curl (divF) }$

Question 8

Let = arctan(x) - arctan(z) and = . Find a simplified expression for $\textbf{ grad}$ (11ee7bac_4a3a_9aaa_ae82_759e3f104991_TB9661_11 ) × $\textbf{ grad}$ (11ee7bac_77f1_7c2b_ae82_019616e1397c_TB9661_11 ) .

A) j
B) 0 (zero vector field)
C) j
D) 0 (zero scalar field)
E) - j

Question 9

Compute $\textbf{ div F }$ for $\textbf{ F }$ = sin 2x, cos 2y, tan 2z .

A) 2cos 2x + 2sin 2y + z
B) -cos 2x + sin 2y + z
C) 2cos 2x - 2sin 2y + z
D) cos 2x + sin 2y + z
E) 2cos 2x - 2sin 2y + 2sec z

Question 10

Find https://d2lvgg3v3hfg70.cloudfront.net/TB9661/.F if F (x, y, z) = xy , yz, -xyz .

A) 2y + yz - xy
B) y + 2xyz - xy
C) y + z - xy
D) y + z + 2 xy
E) 2y + z + xy

Question 11

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

A) 2y + 2z + 2x
B) 3y + 2z + x
C) y + z + x
D) 2y -2 z + 2x
E) y + 2z + 3x

Question 12

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.

Question 13

Calculate the divergence of the vector field F(x, y, z) = ( - xz) i + (z - ) j - xy k.

A) - z - zx - 2 - y
B) + zx - 2 + 4xy
C) - z + zx - 2 - 4xy
D) - z + zx - 4xy
E) + zx + 2 + 4xy

Question 14

Calculate the curl of the vector field V = x sin y i + cos y j + xy k.

A) x i + y j - x cos y k
B) x i - y j + x cos y k
C) x i - y j - x cos y k
D) x i + y j + x cos y k
E) -x i + y j + y cos y k

Question 15

Calculate the divergence of the vector field F = y i + x j + xyz k.

A) 5xy
B) 4xy + yz
C) 6xy
D) 2xy + 2yz + xz
E) 4xy + xz

Question 16

. F = F . 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 for any sufficiently smooth vector field F.

Question 17

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

A) + +
B) i + j + k
C) i + j + k
D) 0
E) + +

Question 18

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

A) + +
B) - - -
C) + +
D) 0
E) - +

Question 19

The divergence of a vector field F is defined by

A) F
B) . F
C) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 F
D) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . (11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11F)
E) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × F

Question 20

The curl of a vector field F is defined by

A) .(11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11F)
B) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × F
C)
D) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 F
E) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . F

Question 21

Compute the divergence and the curl of the vector field r = x i + y j + z k.

A) . r = 2, 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × r = 0
B)11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . r = 3, 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11× r = 0
C) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . r = 3, 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × r = r
D) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . r = 1, 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × r = 0
E) 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 . r = 2, 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 × r = r

Question 22

If r = x i + y j + z k and f(u) is any differentiable function of one variable, evaluate and simplify .

A) 0
B) r
C) 2r
D) 3r
E) 4r

Question 23

If r = x i + y j + z k and r = |r|, evaluate and simplify div .

A) 0
B)
C)
D)
E)

Question 24

For r = x i + y j + z k, evaluate and simplify . .

A)
B)
C)
D) |r|
E) 0

Question 25

Let B be a constant vector and let G(r) = (B × r) × r be a vector potential of the solenoidal vector field F. Find F.

A) F = B
B) F = r
C) F = r × B
D) F = 3(B × r)
E) F = (B × r)

Question 26

Verify that the vector field F = (2x y²z² - sin(x)sin(y)) i + (2 x²y z²+ cos(x)cos(y)) j + (2x²y² z + ) k is conservative and find a scalar potential f(x, y, z) for it that satisfies f(0, 0, 0) = 1.

A)
B) f(x, y, z) = + cos(x)sin(y) + + 1
C) f(x, y, z) = + sin(x)cos(y) + + 1
D) f(x, y, z) = + cos(x)sin(y) +
E) f(x, y, z) = xyz + cos(x)sin(y) +

Question 27

If the vector field H = f(r) r, r $\neq$ 0 is solenoidal, find an expression for f(r).

A) f(r) = c , where c is an arbitrary constant
B) f(r) = c , where c is an arbitrary constant
C) f(r) = c , where c is an arbitrary constant
D) f(r) = c , where c is an arbitrary constant
E) f(r) = c , where c is an arbitrary constant

Question 28

Show that div ( r) = (n + 3) .You may use the following fact: grad ( ) = n r

Question 29

A vector field F is called $\textbf{ solenoidal }$ in a domain D if

A) F = 0 in D
B) curl(F) = 0 in D
C) F = in D for some scalar field 11ee7bad_3b7e_852d_ae82_0ffea7b87591_TB9661_11
D) div(F) = 0 in D
E) grad(F) = 0 in D

Question 30

Find all values of the nonzero constant real numbers a, b, and c so that the vector field F = a cos( x + 2y )cosh (c z) i + b cos ( x + 2y)cosh (c z) j + c sin( x + 2y)sinh(c z) k is both $\textbf{ irrotational }$ and $\textbf{ solenoidal }$ .

A) a = - , b = -2, c = 3
B) a = , b = 2, c = 2
C) a = - , b = -2, c = ± 2
D) a = , b = 2, c = ± 3
E) a = , b = -2, c = 9

Question 31

Verify that the vector field F = i + (1 - xy) j - xz k is solenoidal, and find a vector potential G for it having the form G(x, y, z) = (x, y, z) i + y k.

A) (xyz + z) i + y k
B) ( y + z) i + y k
C) (xyz - z) i + y k
D) ( z - z) i + y k
E) xyz i + y k

Question 32

For what value of the constant C is the vector field F = i + C(xy + yz) j + k. solenoidal?
If C has that value, find a vector potential G for F having the form G(x, y, z) = (x, y, z) i + y k.

A) C = -2, G = y i + y k
B) C = -2, G = - y i + y k
C) C = -2, G = x i + y k
D) C = -2, G = - x i + y k
E) C = 2, G = y i + y k

Question 33

Show that there does not exist a twice continuously differentiable vector field G such that $\textbf{ curl G }$ = x i + y j + z k.

Question 34

A vector field F satisfying the equation div F = 0 in domain D is called:

A) irrotational in D
B) a scalar potential
C) solenoidal in D
D) conservative in D
E) a vector potential

Question 35

Let and F be sufficiently smooth scalar and vector fields, respectively.Express the well-known identity https://d2lvgg3v3hfg70.cloudfront.net/TB9661/https://d2lvgg3v3hfg70.cloudfront.net/TB9661/. (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F ) = (11ee7bad_7817_372f_ae82_a36163e56c30_TB9661_11 11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 ) . F + 11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 (11ee7bad_7817_372f_ae82_a36163e56c30_TB9661_11. F) using the notations grad , div or curl.

A) curl (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F) = grad ( ) . F + 11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 div (F)
B) div (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F) = curl (11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 ) . F + 11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 grad (F)
C) div (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F) = grad (11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 ) . F + 11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 div (F)
D) grad (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F) = div (11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 ) . F + 11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 curl (F)
E) curl (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F) = div (11ee7bad_e8b4_1a82_ae82_cd578f612ee6_TB9661_11 ) . F + grad (F)

Question 36

Every conservative vector field is irrotational.

Question 37

If r = x i + y j + z k and k is a constant vector field in R³, then

A) div ( k × r) = 0
B) div ( k × r) = 0.
C) grad ( k . r) = 2k
D) curl ( k × r) = 0
E) curl ( k × r) = 0.

Question 38

Use Green's Theorem to evaluate the line integral counterclockwise around the square with vertices (0, 3), (3, 0), (-3, 0), and (0, -3).

A) 18
B) 180
C) -36
D) 0
E) 36

Question 39

Evaluate the integral ( ) - 2y) dx + (3x - ysin( )) dy counterclockwise around the triangle in the xy-plane having vertices (0, 0), (2, 2), and (2, 0).

A) 5
B) 20
C) 0
D) 10
E) 2

Question 40

Use Green's Theorem to compute + xy) dx + ( + xy) dy counterclockwise around the rectangle having vertices (± 1, 1) and (± 1, 2).

A) -9
B) -12
C) 2
D) 0
E) 12

Question 41

Use Green's Theorem to compute the integral clockwise around the circle of radius 3 centred at the origin.

A) 18 $\pi$
B) 9 $\pi$
C) 127 $\pi$
D) 243 $\pi$
E) 0

Question 42

Use Green's Theorem to compute the integral counterclockwise around the square with vertices at (4, 2), (4, 5), (7, 5), and (7, 2).

A) -198
B) -210
C) -126
D) -72
E) -21

Question 43

Use Green's Theorem to compute the integral where C is the triangle formed by the lines y = -x + 1, x = 0 and y = 0, oriented clockwise.

A) 3
B) 2
C) 1
D) 0
E) $\pi$

Question 44

Let C be a non-self-intersecting closed curve in the xy-plane oriented counterclockwise and bounding a region R having area A and centroid . In terms of these quantities, evaluate the line integral .

A) A
B) A
C) A
D) A
E) -A

Question 45

Evaluate the integral - dx counterclockwise around the closed curve formed by y = x³ and y = x, between the points (0, 0) and (1, 1).

A) 1
B)
C)
D)
E) 0

Question 46

Evaluate clockwise around the triangle with vertices (0, 0), (3, 0), and (3, 3).

A) 27
B) 9
C) -9
D) -27
E) 0

Question 47

Let F = - i + j and let C be the boundary of circle + = 9 oriented counterclockwise. Use Green's Theorem to evaluate

A) 9 $\pi$
B) 0
C) -2 $\pi$
D) 2 $\pi$
E) 3 $\pi$

Question 48

Find the flux of F = x i + 2y j out of the circular disk of radius 2 centred at (3, -5).

A) 8 $\pi$
B) 12 $\pi$
C) 16 $\pi$
D) 24 $\pi$
E) 4 $\pi$

Question 49

If C is the positively oriented boundary of a plane region R having area 3 units and centroid at the point (12, 6), evaluate (i) (ii) dx + 3xy dy

A) (i) 36 (ii) 15
B) (i) -36 (ii) 18
C) (i) -18 (ii) 36
D) (i) -4 (ii) 2
E) (i) 432 (ii) 1080

Question 50

Find the flux of F = 2 y i + j out of the rectangle 0 $\le$ x $\le$ ln(3), 0 $\le$ y $\le$ 2.

A) 4
B) 8
C) 16
D) 32
E) 24

Question 51

Find the flux of F = out of (a) the disk + $\le$ , (b) an arbitrary plane region not containing the origin in its interior or on its boundary, and (c) an arbitrary plane region containing the origin in its interior.

A) (a) 0 $~~~~~~~~$ (b) 0 $~~~~~~~~$ (c) 0
B) (a) 2 $\pi$ $~~~~~~~~$ (b) 0 $~~~~~~~~$ (c) 2 $\pi$
C) (a) 2 $\pi$ a $~~~~~~~~$ (b) 0 $~~~~~~~~$ (c) 2 $\pi$
D) (a) 0 $~~~~~~~~$ (b) 2 $\pi$ $~~~~~~~~$ (c) 0
E) None of the above

Question 52

Use Green's theorem in the plane to show that the area A of a regular plane region R enclosed by a positively oriented, piecewise smooth, simple closed curve C is given by A = dx + x dy).

Question 53

Use Green's theorem in the plane to find the x-coordinate of the centroid of a regular plane region R (with areaA) enclosed by a positively oriented, piecewise smooth, simple closed curve C .

A)
B)
C)
D)
E)

Question 54

Use a line integral to find the area enclosed by the x-axis and one arch of the cycloid given parametrically by the equations x(t) = 3(t - sin(t)), y(t) =3(1 - cos(t)), 0 $\le$ t $\le$ 2 $\pi$ .

A) 36 $\pi$
B) 18 $\pi$
C) 27 $\pi$
D) 54 $\pi$
E) 9 $\pi$

Question 55

Find the flux of F = x i + j + k out of the cube bounded by the coordinate planes and the planes and

A) 0
B) 1
C)
D) 3
E)

Question 56

Evaluate F = x y i + xz j + z y k and S is the sphere of radius 3 with centre at the origin and unit outward normal field .

A) 32 $\pi$
B) 34 $\pi$
C) 36 $\pi$
D) 38 $\pi$
E) 72 $\pi$

Question 57

Evaluate the integral where R is the region + + $\le$ 25 and

A) 12500 $\pi$
B) 2500 $\pi$
C) 6250 $\pi$
D) 1250 $\pi$
E) 25000 $\pi$

Question 58

Use the Divergence Theorem to find the outward flux of F = across the boundary of the region

A) 12 $\pi$
B) 16 $\pi$
C) 3 $\pi$
D) $\pi$
E) 60 $\pi$

Question 59

Find the flux of r = x i + y j + z k out of the cone with base + $\le$ 16, z = 0, and vertex at (0, 0, 3).

A) 46 $\pi$
B) 48 $\pi$
C) 50 $\pi$
D) 52 $\pi$
E) 16 $\pi$

Question 60

Calculate the surface integral where G = (x + y) i + (y + z) j + (z + x) k and S is the sphere with outward normal.

A) 32 $\pi$
B) 16 $\pi$
C) 8 $\pi$
D) 64 $\pi$
E) 256 $\pi$

Question 61

Find the flux of i - xy j +3z k out of the solid region bounded by the parabolic cylinder and the planes , and

A) 208
B) 112
C) 64
D) 48
E) 176

Question 62

Evaluate where S is the first-octant part of the sphere of radius a centred at the origin. (Hint: Even though S is not a closed surface, it is still easiest to use the Divergence Theorem because the integrand in the surface integral is zero on the coordinate planes.)

A) $\pi$
B) $\pi$
C) $\pi$
D) 2 $\pi$
E) $\pi$

Question 63

Let C be a cone whose base is an arbitrarily shaped region in the plane z = h > 0 having area A, and whose vertex is at the origin. By calculating the flux of out of C through its entire surface both directly and by using the Divergence Theorem, find the volume of C.

A) Ah
B) Ah
C) Ah
D) Ah
E) 3 Ah

Question 64

Evaluate the surface integral where is the unit inner normal to the surface S of the region lying between the two paraboloids

A)
B) 1
C) 0
D) 2
E) -1

Question 65

Find the flux of G = (x + 2zy) i + (y - ) j + z k outward through the sphere

A)
B)
C)
D)
E)

Question 66

Use the Divergence Theorem to evaluate the surface integral where S is the part of the cone below z = 2, and is the unit normal to S with positive z-component. (An additional surface must be introduced to enclose a volume.)

A)
B)
C)
D)
E)

Question 67

If F = x i + y j, calculate the flux of F upward through the part of the surface z = 4 - x² - y² that lies above the (x, y) plane by applying the Divergence Theorem to the volume bounded by the surface and the disk that it cuts out of the (x, y) plane.

A) 14 $\pi$
B) 16 $\pi$
C) 18 $\pi$
D) 20 $\pi$
E) 8 $\pi$

Question 68

Find the outward flux of F = ln( + ) i - j + z k across the boundary of the region

A) 4 $\pi$ - 3 $\pi$ ln 2 + 2 $\pi$
B) 4 $\pi$ - 3 $\pi$ ln 2 - 2 $\pi$
C) 4 $\pi$ + 3 $\pi$ ln 2 - 2 $\pi$
D) 4 $\pi$ + 3 $\pi$ ln 2 + 2 $\pi$
E) 4 $\pi$ + $\pi$ ln 2 - 2 $\pi$

Question 69

A certain region R in 3-space has volume 5 cubic units and centroid at the point (2, -3, 4). Find the flux of out of R across its boundary.

A) 60
B) 50
C) 20
D) 15
E) 90

Question 70

Find , where F(x, y, z) = x y i + ln x j - z y k, S is the sphere of radius 3 centred at the origin, and is the unit outward normal field on S.

A) 24 $\pi$
B) 12 $\pi$
C) 36 $\pi$
D) 72 $\pi$
E) 54 $\pi$

Question 71

Given F = 4y i + x j + 2z k, find over the hemisphere with outward normal .

A) 2 $\pi$
B) -2 $\pi$
C) -3 $\pi$
D) 3 $\pi$
E) 0

Question 72

Evaluate the integral of over the portion of the ellipse in the first quadrant, traversed in the counterclockwise direction.

A) -31
B) -32
C) -33
D) -34
E) -30

Question 73

Use Stokes's Theorem to evaluate the line integral where C is the triangle with vertices (0, 0, 1), (0, 1, 1) and (1, 0, 0) with counterclockwise orientation as seen from high on the z-axis.

A) 0
B) 1
C) -1
D) 2
E) -2

Question 74

Let F be a smooth vector field in 3-space satisfying the condition Find the flux of curl F upward through the part of the lying above the xy-plane.

A) 81 $\pi$
B) 72 $\pi$
C) 27 $\pi$
D) 18 $\pi$
E) None of the above

Question 75

Evaluate the line integral where C is the circle given by the parametric equations for

A) -
B) - $\pi$
C) $\pi$
D) 2 $\pi$
E)

Question 76

Evaluate the line integral where C is the circle oriented clockwise as seen from high on the z-axis.

A) 40 $\pi$
B) 45 $\pi$
C) 50 $\pi$
D) 55 $\pi$
E) 35 $\pi$

Question 77

Evaluate , where F = y i + zx j + k and C are the positively oriented boundary of the triangle in which the plane with upward normal, intersects the first octant of space.

A)
B) -
C)
D) -
E) -

Question 78

Use Stokes's Theorem to evaluate the line integral + (2x - y) dy + (y + z) dz,where C is the triangle cut from the plane P with equation by the three coordinate planes. C has orientation inherited from the upward normal on P.

A) 18
B) -6
C) 6
D) 24
E) -24

Question 79

Use Stokes's Theorem to evaluate the line integral where C is the triangle with vertices (1, 1, 1), (0, 1, 0) and (0, 0, 0) oriented counterclockwise as seen from high on the z-axis.

A) 0
B) 1
C) 2
D) -2
E) -1

Question 80

Let F = (z - y) i + (x - z) j + (y - x) k. Compute the work done by the force F in moving an object along the curve of intersection of the cylinder with the plane The orientation of the curve is consistent with the upward normal on the plane.

A) 8 $\pi$
B) 6 $\pi$
C) 4 $\pi$
D) 2 $\pi$
E) 0