Deck 13: Vector Calculus

Question 1

Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y \sin z \right) \mathbf { i } + \left( y ^ { 3 } + z ^ { 2 } \sin z \right) \mathbf { j } + \left( z ^ { 3 } + x \right) \mathbf { k }$ and let S be the boundary surface of the solid E bounded by $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } , z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ , and $z = 0$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $\frac { 62 \pi } { 5 }$

B) $\frac { \pi } { 2 }$
C) $\pi$
D) $\frac { 4 \pi } { 3 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 192 \pi } { 5 }$

G) $\frac { 186 \pi } { 5 }$

H) $4 \pi$

Answer 1

$\frac { 186 \pi } { 5 }$

Question 2

Find the flux of across the surface of the solid bounded by , and the planes .

Answer 2

Question 3

Let $\mathbf { F } ( x , y , z ) = \sin \left( y ^ { 2 } + z ^ { 2 } \right) \mathbf { i } + \cos \left( x ^ { 2 } + z ^ { 2 } \right) \mathbf { j } + e ^ { z ^ { 2 } + y ^ { 2 } } \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { \pi } { 4 }$
C) $\frac { \pi } { 3 }$
D) $\frac { \pi } { 2 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 3 \pi } { 4 }$

G) $\pi$

H) $\frac { 3 \pi } { 2 }$

Answer 3

$0$

Question 4

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { \pi } { 2 }$
C) $\pi$
D) $\frac { 4 \pi } { 3 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 8 \pi } { 3 }$

G) $2 \pi$

H) $4 \pi$

Question 5

Let $\mathbf { F } ( x , y , z ) = x y i$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D)
$\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F)
$\frac { 3 } { 4 }$


G)
$1$

H) $\frac { 3 } { 2 }$

Question 6

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H) $\frac { 3 } { 2 }$

Question 7

Let and let S be the surface of the rectangular box bounded by the planes , and . Evaluate the surface integral .

Question 8

Let and let S be the boundary surface of the solid . Evaluate the surface integral .

Question 9

Let and let S be the surface of the solid bounded by the spheres and . Evaluate the surface integral .

Question 10

Evaluate , where S is the cube bounded by the planes and , and n is the outward normal.

Question 11

Let and let S be the surface with equation . Evaluate the surface integral .

Question 12

Let $\mathbf { F } ( x , y , z ) = x y ^ { 2 } \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F)
$\frac { 3 } { 4 }$

G) $1$

H) $\frac { 3 } { 2 }$

Question 13

Use the Divergence Theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = x ( y - 1 ) \mathbf { i } + 2 y z \mathbf { j } - \left( z ^ { 2 } + y z \right) \mathbf { k }$ and S is the surface of the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ , bounded by the planes $z = 0$ and $z = 3$ .

A) $12 \pi$

B) $4 \pi$
C) $- 3 \pi$
D) $6 \pi$
E) $- 12 \pi$
F) $- 4 \pi$

G) $3 \pi$

H) $- 6 \pi$

Question 14

Find the flux of across the surface of the solid bounded by the paraboloid and the plane.

Question 15

Let and let S be the surface of the tetrahedron with vertices , and . Evaluate the surface integral .

Question 16

Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H)
$\frac { 3 } { 2 }$

Question 17

Use the Divergence Theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = x ( y + 1 ) \mathbf { i } + 2 y z \mathbf { j } - \left( z ^ { 2 } + y z \right) \mathbf { k }$ and S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .

A) $- 4 \pi$

B) $\frac { 4 \pi } { 3 }$
C) $- \frac { 3 \pi } { 2 }$
D) $1$
E) $4 \pi$
F) $- \frac { 4 \pi } { 3 }$

G) $\frac { 3 \pi } { 2 }$

H) $\frac { 2 \pi } { 3 }$

Question 18

Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D)
$\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H) $\frac { 3 } { 2 }$

Question 19

Let and let S be the boundary surface of the solid . Evaluate the surface integral .

Question 20

Evaluate the flux integral over the boundary of the ball .

Question 21

Evaluate , where S is the boundary surface of the solid sphere and

Question 22

Evaluate , where S is the boundary surface of the region outside the sphere and inside the ball and .

Question 23

Let $\mathbf { F } ( x , y , z ) = 2 x y ^ { 3 } z ^ { 4 } \mathbf { i } + 3 x ^ { 2 } y ^ { 2 } z ^ { 4 } \mathbf { j } + 4 x ^ { 2 } y ^ { 3 } z ^ { 3 } \mathbf { k }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , \quad 0 \leq t \leq 2 \pi$ .

A) $\pi$

B) $\pi \sqrt { 2 }$
C) $\pi \sqrt { 3 }$
D) $2 \pi$
E) $\frac { \sqrt { 2 } } { 2 } \pi$
F) $\frac { \sqrt { 3 } } { 2 } \pi$

G) $\frac { \pi } { 2 }$

H) $0$

Question 24

Let S be the outwardly-oriented surface of a solid region E where the volume of E is . If and , evaluate the surface integral .

Question 25

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the circle $x = 3 \cos t , y = 3 \sin t , z = 2,0 \leq t \leq 2 \pi$ .

A) $27 \pi$

B) $99 \pi$
C) $18 \pi$
D) $54 \pi$
E) $- 27 \pi$
F) $- 99 \pi$

G) $- 18 \pi$

H) $- 54 \pi$

Question 26

Evaluate , where and S is the sphere .

Question 27

Use Stokes' Theorem to evaluate where and S is the part of the paraboloid that lies inside the cylinder , oriented upward.

Question 28

Let $\mathbf { F } ( x , y , z ) = x \mathbf { j }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , 0 \leq \mathrm { t } \leq 2 \pi$ .

A) $\pi$

B) $\pi \sqrt { 2 }$
C) $\pi \sqrt { 3 }$
D) $2 \pi$
E) $\frac { \sqrt { 2 } } { 2 } \pi$
F) $\frac { \sqrt { 3 } } { 2 } \pi$

G) $\frac { \pi } { 2 }$

H) $\frac { \pi } { 4 }$

Question 29

Let $\mathbf { F } ( x , y , z ) = x \mathbf { j }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the rectangular path from $( 0,0,0 )$ to $( 1,0,1 )$ to $( 1,1,1 )$ to $( 0,1,0 )$ to $( 0,0,0 )$ .

A) $1$

B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $2$
E) $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { \sqrt { 3 } } { 2 }$

G) $\frac { 1 } { 2 }$

H) $\frac { 1 } { 4 }$

Question 30

Let $\mathbf { F } ( x , y , z ) = - y \mathbf { j } + x \mathbf { j } + e ^ { z ^ { 2 } } \mathbf { k }$ . Evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot \mathrm { d } \mathbf { S }$ over the surface S given by $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ , with downward orientation.

A) $2 \pi$

B) $\pi$
C) $\frac { \pi } { 2 }$
D) $3 \pi$
E) $- 2 \pi$
F) $- \pi$

G) $- \frac { \pi } { 2 }$

H) $- 3 \pi$

Question 31

Find the flux of across the surface of the solid bounded by the paraboloid and the plane.

Question 32

Evaluate , where and S is the sphere .

Question 33

Find the flux of across the surface of the solid bounded by the paraboloid and the plane.

Question 34

Let . Let C be the rectangular path from to to to to . Use Stokes' Theorem to evaluate the line integral , where T is the unit tangent vector to C.

Question 35

Find the flux of across the surface of the solid .

Question 36

Evaluate , where the path C is the curve of intersection of the paraboloid with the plane .

Question 37

Let $\mathbf { F } ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 2 x$ .

A) $1$

B) $- 1$
C) $\sqrt { 2 }$
D) $- \sqrt { 2 }$
E) $\sqrt { 3 }$
F) $- \sqrt { 3 }$

G) $2$

H) $0$

Question 38

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the circle $x = 3 \sin t , y = 3 \cos t , z = 2,0 \leq t \leq 2 \pi$ .

A) $27 \pi$

B) $99 \pi$
C) $18 \pi$
D) $54 \pi$
E) $- 27 \pi$
F) $- 99 \pi$

G) $- 18 \pi$

H) $- 54 \pi$

Question 39

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the triangle with vertices $( 1,0,0 ) , ( 0.1 .0 )$ , and $( 0.0 .1 )$ , oriented counter clockwise as viewed from above.

A) $- 11$

B) $- \frac { 11 } { 2 }$
C) $- \frac { 7 } { 2 }$
D) $7$
E) $11$
F) $\frac { 11 } { 2 }$

G) $\frac { 7 } { 2 }$

H) $- 7$

Question 40

Let and let S be the boundary surface of the solid . Evaluate the surface integral .

Question 41

A surface has the shape of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between $z = 0$ and $z = 1$ with the density function $\rho ( x , y , z ) = 1 - z$ . Find the mass of the surface.

A) $\sqrt { 2 } \pi$
B) $\frac { \sqrt { 2 } \pi } { 3 }$
C) $6$

D) $1$
E) $2 \sqrt { 2 } \pi$
F) $2 \pi$
G) $\frac { 2 \sqrt { 2 } \pi } { 3 }$
H) $0$

Question 42

Consider the surfaces : , and : , and let F be a vector field with continuous partial derivatives everywhere. Why do we know that ?

Question 43

Use Stokes' Theorem to evaluate , where C is the triangle with vertices , and , oriented counterclockwise as viewed from above.

Question 44

Use Stokes' Theorem to evaluate where and S is the part of the hemisphere that lies inside the cylinder , oriented in the direction of the positive x-axis.

Question 45

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = x \mathbf { i } + y \mathbf { j } + x \mathbf { k }$ and S is the part of the plane $z = 1 - x - y$ in the first octant with downward orientation.

A) $\frac { 1 } { 3 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 4 }$
D) $1$
E) $- \frac { 1 } { 3 }$
F) $- \frac { 1 } { 2 }$
G) $- \frac { 1 } { 4 }$
H) $- 1$

Question 46

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = ( x , y , z ) = y \mathbf { j } + z \mathbf { k }$ and S is the cube bounded by $x = \pm 1 , \quad y = \pm 1 , \quad z = \pm 1$ .

A) $16$
B) $8$
C) $- 8$
D) $1$
E) $32$
F) $4$
G) $- 4$
H) $0$

Question 47

Evaluate the surface integral $\iint _ { S } ( x + y + z ) d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

A) $1$
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $\frac { 3 } { 2 }$
E) $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { \sqrt { 3 } } { 2 }$
G) $\frac { \sqrt { 2 } } { 3 }$
H) $\frac { 3 \sqrt { 2 } } { 2 }$

Question 48

Verify that Stokes' Theorem is true for the vector field and the cone , oriented upward.

Question 49

Let $\mathbf { F } ( x , y , z ) = \mathbf { k }$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ and has upward orientation.

A) $1$
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $2$
E) $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { \sqrt { 3 } } { 2 }$
G) $\frac { 1 } { 2 }$
H) $\frac { 1 } { 4 }$

Question 50

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } = ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is the part of the surface $z = 1 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y -$ plane and has upward orientation.

A) $- 3 \pi$
B) $- \frac { 3 \pi } { 4 }$

C) $- \frac { 3 \pi } { 2 }$
D) $1$
E) $3 \pi$
F) $\frac { 3 \pi } { 4 }$
G) $\frac { 3 \pi } { 2 }$
H) $\frac { 4 \pi } { 3 }$

Question 51

Let S be the parametric surface . Use Stokes' Theorem to evaluate , where .

Question 52

Use Stokes' Theorem to evaluate , where C is the curve of intersection of the paraboloid and the cylinder , oriented counterclockwise as viewed from above.

Question 53

An upper hemisphere is given by $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ with the density function $\rho ( x , y , z ) = z$ . Find the mass of the sphere.

A) $32 \pi$
B) $8 \pi$
C) $64 \pi$
D) $1$
E) $16 \pi$
F) $4 \pi$
G) $\frac { 32 \pi } { 3 }$
H) $\frac { 64 \pi } { 3 }$

Question 54

Evaluate the surface integral , where S is the triangle with vertices , and

Question 55

Let $\mathbf { F } ( x , y , z ) = x \mathbf { k }$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ and has upward orientation.

A) $1$
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $2$
E) $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { \sqrt { 3 } } { 2 }$
G) $\frac { 1 } { 2 }$
H) $\frac { 1 } { 4 }$

Question 56

Evaluate the surface integral $\iint _ { S } z d S$ , where S is that part of the cylinder $z = \sqrt { 1 - x ^ { 2 } }$ that lies above the square with vertices $( - 1 , - 1 ) , ( 1 , - 1 ) , ( - 1,1 )$ , and $( 1,1 )$ .

A) $\sqrt { 2 }$
B) $2$
C) $\sqrt { 8 }$
D) $4$
E) $\pi$
F) $2 \pi$
G) $4 \pi$
H) $8 \pi$

Question 57

Use Stokes' Theorem to evaluate where and C is the curve of intersection of the plane and the cylinder .

Question 58

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = ( x , y , z ) = y \mathbf { j } - z \mathbf { k }$ and S is the cube bounded by $x = \pm 1 , \quad y = \pm 1 , \quad z = \pm 1$ .

A) $16$
B) $8$
C) $- 8$
D) $1$
E) $32$
F) $4$
G) $- 4$
H) $0$

Question 59

Evaluate the surface integral $\iint _ { S } x d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

A) $1$
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D)2
E) $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { \sqrt { 3 } } { 2 }$
G) $\frac { \sqrt { 2 } } { 3 }$
H) $\frac { \sqrt { 3 } } { 3 }$

Question 60

Use Stokes' Theorem to evaluate where and C is the triangle with vertices , and

Question 61

A fluid has density 1500 and velocity field . Find the rate of flow outward through the sphere .

Question 62

Evaluate , where and S is the upper half of the sphere , with upward orientation.

Question 63

Evaluate the surface integral for the vector field where S is part of the cone between the planes z = 1 and z = 2 with upward orientation.

Question 64

Evaluate the flux of the vector field through the plane region with the given orientation as shown below.

Question 65

Evaluate where and S is the part of the surface that lies above the rectangle and has upward orientation.

Question 66

Evaluate the surface integral , where S is the part of the sphere that lies above the cone .

Question 67

Evaluate the surface integral , where S is the part of the paraboloid that lies in front of the plane .

Question 68

Compute the surface integral if and S is the piece of the sphere in the second octant .

Question 69

Consider the top half of the ellipsoid parametrized by . Find a normal vector N at the point determined by , and determine if it is upward and/or outward.

Question 70

Evaluate the flux of the vector field through the plane region with the given orientation as shown below.

Question 71

Evaluate the surface integral for the vector field , where S is the hemisphere with upward orientation.

Question 72

Find the flux of the vector field across the paraboloid given by with and upward orientation:

Question 73

Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by .

Question 74

Find the mass of the sphere whose density at each point is proportional to its distance to the plane.

Question 75

Evaluate , where S is the part of the surface that lies between the cylinders and .

Question 76

Evaluate , where and S is the part of the surface below the plane , with upward orientation.

Question 77

Evaluate the surface integral for the vector field where S is the part of the elliptic paraboloid that lies below the square and has downward orientation.

Question 78

Let $\mathbf { F } ( x , y , z ) = z \mathbf { i } + x \mathbf { j } + y \mathbf { k }$ . Find the curl of F.

A) i
B) j
C) $\mathbf { k }$
D) $\mathbf { i } + \mathbf { j } + \mathbf { k }$
E) $- \mathbf { i }$
F) $- j$
G) $- \mathbf { k }$
H) $- \mathbf { i } - \mathbf { j } - \mathbf { k }$

Question 79

Find the mass of a thin funnel in the shape of a cone , is its density
function is .

Question 80

Let $\mathbf { F } ( x , y , z ) = y \mathbf { i } - x \mathbf { j }$ . Find the curl of F.

A) $2i$
B) $2 \mathrm { j }$
C) $2 \mathbf { k }$
D) $2 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
E) $- 2 \mathbf { i }$
F) $- 2 j$
G) $- 2 \mathbf { k }$
H) $- 2 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$