# Problems

# Section 3.1: Vector Algebra

1. *3.1 Vector A starts at point (1,−1,−3) and ends at point (2,−1,0). Find a unit vector in the direction of A.

2. 3.2 Given vectors A=x ˆ 2−y ˆ 3+z ˆ, B=x ˆ 2−y ˆ+z ˆ 3, and C=x ˆ 4+y ˆ 2−z ˆ 2, show that C is perpendicular to both A and B.

3. *3.3 In Cartesian coordinates, the three corners of a triangle are P 1=(0,4,4), P 2=(4,−4,4), and P 3=(2,2,−4). Find the area of the triangle.

4. 3.4 Given A=x ˆ 2−y ˆ 3+z ˆ 1 and B=x ˆ B x+y ˆ 2+z ˆ B z:

   1. Find B x and B z if A is parallel to B.

   2. Find a relation between B x and B z if A is perpendicular to B.

5. 3.5 Given vectors A=x ˆ+y ˆ 2−z ˆ 3, B=x ˆ 2−y ˆ 4, and C=y ˆ 2−z ˆ 4, find the following:

   1. *a.A and a ˆ

   2. b.The component of B along C

   3. c.θ A C

   4. d.A×C

   5. *e.A⋅(B×C)

   6. f.A×(B×C)

   7. g.x ˆ×B

   8. *h.(A×y ˆ)⋅z ˆ

6. 3.6 Given vectors A=x ˆ 2−y ˆ+z ˆ 3 and B=x ˆ 3−z ˆ 2, find a vector C whose magnitude is 9 and whose direction is perpendicular to both A and B.

7. 3.7 Given A=x ˆ(x+2 y)−y ˆ(y+3 z)+z ˆ(3 x−y), determine a unit vector parallel to A at point P=(1,−1,2).

8. 3.8 By expansion in Cartesian coordinates, prove:

   1. The relation for the scalar triple product given by Eq. (3.29).

   2. The relation for the vector triple product given by Eq. (3.33).

9. *3.9 Find an expression for the unit vector directed toward the origin from an arbitrary point on the line described by x=1 and z=−2.

10. 3.10 Find an expression for the unit vector directed toward the point P located on the z axis at a height h above the x–y plane from an arbitrary point Q=(x,y,−5) in the plane z=−5.

11. *3.11 Find a unit vector parallel to either direction of the line described by

    2 x+z=4

12. 3.12 Two lines in the x–y plane are described by the following expressions:

    Line 1 x+2 y=−6 Line 2 3 x+4 y=8

    Use vector algebra to find the smaller angle between the lines at their intersection point.

13. *3.13 A given line is described by

    x+2 y=4.

    Vector A starts at the origin and ends at point P on the line such that A is orthogonal to the line. Find an expression for A.

14. 3.14 Show the following given two vectors A and B:

    1. The vector C defined as the vector component of B in the direction of A is given by

       C=a ˆ(B⋅a ˆ)=A(B⋅A)|A|2,

       where a ˆ is the unit vector of A.

    2. The vector D defined as the vector component of B perpendicular to A is given by

       D=B−A(B⋅A)|A|2

15. *3.15 A certain plane is described by

    2 x+3 y+4 z=16

    Find the unit vector normal to the surface in the direction away from the origin.

16. 3.16 Given B=x ˆ(z−3 y)+y ˆ(2 x−3 z)−z ˆ(x+y), find a unit vector parallel to B at point P=(1,0,−1).

17. *3.17 Find a vector G whose magnitude is 4 and whose direction is perpendicular to both vectors E and F, where E=x ˆ+y ˆ 2−z ˆ 2 and F=y ˆ 3−z ˆ 6.

18. 3.18 A given line is described by the equation:

    y=x−1.

    Vector A starts at point P 1=(0,2) and ends at point P 2 on the line, at which A is orthogonal to the line. Find an expression for A.

19. 3.19 Vector field E is given by

    E=R ˆ 5 R cos θ−θ ˆ 12 R sin θ cos ϕ+ϕ ˆ 3 sin ϕ.

    Determine the component of E tangential to the spherical surface R=2 at point P=(2,30°,60°).

20. 3.20 When sketching or demonstrating the spatial variation of a vector field, we often use arrows, as in Fig. P3.20, wherein the length of the arrow is made to be proportional to the strength of the field and the direction of the arrow is the same as that of the field’s. The sketch shown in Fig. P3.20, which represents the vector field E=r ˆ r, consists of arrows pointing radially away from the origin and their lengths increasing linearly in proportion to their distance away from the origin. Using this arrow representation, sketch each of the following vector fields:

    1. E 1=−x ˆ y

    2. E 2=y ˆ x

    3. E 3=x ˆ x+y ˆ y

    4. E 4=x ˆ x+y ˆ 2 y

    5. E 5=ϕ ˆ r

    6. E 6=r ˆ sin ϕ

    # Figure P3.20 Arrow representation for vector field E=r ˆ r (Problem 3.20).

21. 3.21 Use arrows to sketch each of the following vector fields:

    1. E 1=x ˆ x−y ˆ y

    2. E 2=−ϕ ˆ

    3. E 3=y ˆ (1/x)

    4. E 4=r ˆ cos ϕ

# Sections 3.2 and 3.3: Coordinate Systems

1. 3.22 Convert the coordinates of the following points from Cartesian to cylindrical and spherical coordinates:

   1. *a.P 1=(1,2,0)

   2. b.P 2=(0,0,2)

   3. c.P 3=(1,1,3)

   4. *d.P 4=(−2,2,−2)

2. 3.23 Convert the coordinates of the following points from cylindrical to Cartesian coordinates:

   1. P 1=(2,π/4,−3)

   2. P 2=(3,0,−2)

   3. P 3=(4,π,5)

3. 3.24 Convert the coordinates of the following points from spherical to cylindrical coordinates:

   1. *a.P 1=(5,0,0)

   2. b.P 2=(5,0,π)

   3. c.P 3=(3,π/2,0)

4. 3.25 Use the appropriate expression for the differential surface area d s to determine the area of each of the following surfaces:

   1. a.r=3; 0≤ϕ≤π/3; −2≤z≤2

   2. b.2≤r≤5; π/2≤ϕ≤π; z=0

   3. *c.2≤r≤5; ϕ=π/4; −2≤z≤2

   4. d.R=2; 0≤θ≤π/3; 0≤ϕ≤π

   5. e.0≤R≤5; θ=π/3; 0≤ϕ≤2 π

   Also sketch the outline of each surface.

5. 3.26 Find the volumes described by the following:

   1. *a.2≤r≤5; π/2≤ϕ≤π; 0≤z≤2

   2. b.0≤R≤5; 0≤θ≤π/3; 0≤ϕ≤2 π

   Also sketch the outline of each volume.

6. 3.27 A section of a sphere is described by 0≤R≤2, 0≤θ≤90°, and 30°≤ϕ≤90°. Find the following:

   1. The surface area of the spherical section.

   2. The enclosed volume.

   Also sketch the outline of the section.

7. 3.28 A vector field is given in cylindrical coordinates by

   E=r ˆ r cos ϕ+ϕ ˆ r sin ϕ+z ˆ z 2.

   Point P=(2,π,3) is located on the surface of the cylinder described by r=2. At point P, find:

   1. The vector component of E perpendicular to the cylinder.

   2. The vector component of E tangential to the cylinder.

8. 3.29 At a given point in space, vectors A and B are given in spherical coordinates by

   A=R ˆ 4+θ ˆ 2−ϕ ˆ,B=−R ˆ 2+ϕ ˆ 3.

   Find:

   1. The scalar component, or projection, of B in the direction of A.

   2. The vector component of B in the direction of A.

   3. The vector component of B perpendicular to A.

9. *3.30 Given vectors

   A=r ˆ(cos ϕ+3 z)−ϕ ˆ(2 r+4 sin ϕ)+z ˆ(r−2 z),B=−r ˆ sin ϕ+z ˆ cos ϕ,

   find

   1. θ A B at (2,π/2,0)

   2. A unit vector perpendicular to both A and B at (2,π/3,1)

10. 3.31 Find the distance between the following pairs of points:

    1. P 1=(1,2,3) and P 2=(−2,−3,−2) in Cartesian coordinates.

    2. P 3=(1,π/4,3) and P 4=(3,π/4,4) in cylindrical coordinates.

    3. P 5=(4,π/2,0) and P 6=(3,π,0) in spherical coordinates.

11. 3.32 Determine the distance between the following pairs of points:

    1. *a.P 1=(1,1,2) and P 2=(0,2,3)

    2. b.P 3=(2,π/3,1) and P 4=(4,π/2,3)

    3. c.P 5=(3,π,π/2) and P 6=(4,π/2,π)

12. 3.33 Transform the vector

    A=R ˆ sin 2 θ cos ϕ+θ ˆ cos 2 ϕ−ϕ ˆ sin ϕ

    into cylindrical coordinates and then evaluate it at P=(2,π/2,π/2).

13. 3.34 Transform the following vectors into cylindrical coordinates and then evaluate them at the indicated points:

    1. a.A=x ˆ(x+y) at P 1=(1,2,3)

    2. b.B=x ˆ(y−x)+y ˆ(x−y) at P 2=(1,0,2)

    3. *c.C=x ˆ y 2/(x 2+y 2)−y ˆ x 2/(x 2+y 2)+z ˆ 4 at P 3=(1,−1,2)

    4. d.D=R ˆ sin θ+θ ˆ cos θ+ϕ ˆ cos 2 ϕ at P 4=(2,π/2,π/4)

    5. *e.E=R ˆ cos ϕ+θ ˆ sin ϕ+ϕ ˆ sin 2 θ at P 5=(3,π/2,π)

14. 3.35 Transform the following vectors into spherical coordinates and then evaluate them at the indicated points:

    1. a.A=x ˆ y 2+y ˆ x z+z ˆ 4 at P 1=(1,−1,2)

    2. b.B=y ˆ(x 2+y 2+z 2)−z ˆ(x 2+y 2) at P 2=(−1,0,2)

    3. *c.C=r ˆ cos ϕ−ϕ ˆ sin ϕ+z ˆ cos ϕ sin ϕ at P 3=(2,π/4,2)

    4. d.D=x ˆ y 2/(x 2+y 2)−y ˆ x 2/(x 2+y 2)+z ˆ 4 at P 4=(1,−1,2)

# Sections 3.4 to 3.7: Gradient, Divergence, and Curl Operators

1. 3.36 Find the gradient of the following scalar functions:

   1. a.T=3/(x 2+z 2)

   2. b.V=x y 2 z 4

   3. c.U=z cos ϕ/(1+r 2)

   4. d.W=e−R sin θ

   5. *e.S=4 x 2 e−z+y 3

   6. f.N=r 2 cos 2 ϕ

   7. g.M=R cos θ sin ϕ

2. 3.37 For each of the following scalar fields, obtain an analytical solution for ∇T and generate a corresponding arrow representation.

   1. a. T=10+x, for −10≤x≤10

   2. *b. T=x 2, for −10≤x≤10

   3. b. T=100+x y, for −10≤x≤10

   4. c. T=x 2 y 2, for −10≤x,y≤10

   5. e. T=20+x+y, for −10≤x,y≤10

   6. f. T=1+sin(π x/3), for −10≤x≤10

   7. *g. T=1+cos(π x/3), for −10≤x≤10

   8. h. T=15+r cos ϕ, for 0≤r≤10 0≤ϕ≤2 π.

   9. i. T=15+r cos 2 ϕ, for 0≤r≤10 0≤ϕ≤2 π.

3. *3.38 The gradient of a scalar function T is given by

   ∇T=z ˆ e−3 z

   If T=10 at z=0, find T(z).

4. 3.39 Follow a procedure similar to that leading to Eq. (3.82) to derive the expression given by Eq. (3.83) for ∇ in spherical coordinates.

5. *3.40 For the scalar function V=x y 2−z 2, determine its directional derivative along the direction of vector A=(x ˆ−y ˆ z) and then evaluate it at P=(1,−1,4).

6. 3.41 Evaluate the line integral of E=x ˆ x−y ˆ y along the segment P 1 to P 2 of the circular path shown in Fig. P3.41.

   # Figure P3.41 Problem 3.41.

7. 3.42 For the scalar function T=1 2 e−r/5 cos ϕ, determine its directional derivative along the radial direction r ˆ and then evaluate it at P=(2,π/4,3).

8. *3.43 For the scalar function U=1 R sin 2 θ, determine its directional derivative along the range direction R ˆ and then evaluate it at P=(5,π/4,π/2).

9. 3.44 Each of the following vector fields is displayed in Fig. P3.44 in the form of a vector representation. Determine ∇⋅A analytically and then compare the result with your expectations on the basis of the displayed pattern.

   1. A=−x ˆ cos x sin y+y ˆ sin x cos y, for −π≤x,y≤π

      # Figure P3.44(a)

      Figure P3.44(a) Full Alternative Text

   2. A=−x ˆ sin 2 y+y ˆ cos 2 x, for −π≤x,y≤π

      # Figure P3.44(b)

      Figure P3.44(b) Full Alternative Text

   3. A=−x ˆ x y+y ˆ y 2, for −10≤x,y≤10

      # Figure P3.44(c)

      Figure P3.44(c) Full Alternative Text

   4. A=−x ˆ cos x+y ˆ sin y, for −π≤x,y≤π

      # Figure P3.44(d)

      Figure P3.44(d) Full Alternative Text

   5. A=x ˆ x, for −10≤x≤10

      # Figure P3.44(e)

      Figure P3.44(e) Full Alternative Text

   6. A=x ˆ x y 2, for −10≤x,y≤10

      # Figure P3.44(f)

      Figure P3.44(f) Full Alternative Text

   7. A=x ˆ x y 2+y ˆ x 2 y, for −10≤x,y≤10

      # Figure P3.44(g)

      Figure P3.44(g) Full Alternative Text

   8. A=x ˆ sin π x 10+y ˆ sin π y 10, for −10≤x,y≤10

      # Figure P3.44(h)

      Figure P3.44(h) Full Alternative Text

   9. A=r ˆ r+ϕ ˆ r cos ϕ, for 0≤r≤10 0≤ϕ≤2 π.

      # Figure P3.44(i)

      Figure P3.44(i) Full Alternative Text

   10. A=r ˆ r 2+ϕ ˆ r 2 sin ϕ, for 0≤r≤10 0≤ϕ≤2 π.

       # Figure P3.44(j)

       Figure P3.44(j) Full Alternative Text

10. 3.45 The scalar function V is given by

    V=2 sin x+x y 2+y e 2 z.

    Determine ∇V and then evaluate it at P(0,1,2).

11. 3.46 The scalar function V is given by

    V=2 z x 2+y 2.

    1. Determine ∇V in Cartesian coordinates.

    2. Convert the result of part (a) from Cartesian to cylindrical coordinates.

    3. Convert the expression for V into cylindrical coordinates and then determine ∇V in those coordinates. Compare the results of parts (b) and (c).

12. *3.47 Vector field E is characterized by the following properties: (a) E points along R ˆ; (b) the magnitude of E is a function of only the distance from the origin; (c) E vanishes at the origin; and (d) ∇⋅E=12, everywhere. Find an expression for E that satisfies these properties.

13. 3.48 For the vector field E=x ˆ x z−y ˆ y z 2−z ˆ x y, verify the divergence theorem by computing

    1. the total outward flux flowing through the surface of a cube centered at the origin and with sides equal to 2 units each and parallel to the Cartesian axes, and

    2. the integral of ∇⋅E over the cube’s volume.

14. 3.49 For the vector field E=r ˆ 10 e−r−z ˆ 3 z, verify the divergence theorem for the cylindrical region enclosed by r=2, z=0, and z=4.

15. *3.50 A vector field D=r ˆ r 3 exists in the region between two concentric cylindrical surfaces defined by r=1 and r=2 with both cylinders extending between z=0 and z=5. Verify the divergence theorem by evaluating the following:

    1. ∮S D⋅d s

    2. ∫υ∇⋅D d υ

16. 3.51 For the vector field D=R ˆ 3 R 2, evaluate both sides of the divergence theorem for the region enclosed between the spherical shells defined by R=1 and R=2.

17. 3.52 For the vector field E=x ˆ x y−y ˆ(x 2+2 y 2), calculate

    1. ∮C E⋅d l around the triangular contour shown in Fig. P3.52(a).

       # Figure P3.52 Contours for (a) Problem 3.52 and (b) Problem 3.53.

    2. ∫S(∇×E)⋅d s over the area of the triangle.

18. 3.53 Repeat Problem 3.52 for the contour shown in Fig. P3.52(b).

19. 3.54 Verify Stokes’s theorem for the vector field

    B=(r ˆ r cos ϕ+ϕ ˆ sin ϕ)

    by evaluating

    1. ∮C B⋅d l over the semicircular contour shown in Fig. P3.54(a), and

       # Figure P3.54 Contour paths for (a) Problem 3.54 and (b) Problem 3.55.

    2. ∫S(∇×B)⋅d s over the surface of the semicircle.

20. 3.55 Repeat Problem 3.54 for the contour shown in Fig. P3.54(b).

21. 3.56 Verify Stokes’s theorem for the vector field

    A=R ˆ cos θ+ϕ ˆ sin θ

    by evaluating it on the hemisphere of unit radius.

22. 3.57 Verify Stokes’s theorem for the vector field

    B=(r ˆ cos ϕ+ϕ ˆ sin ϕ)

    by evaluating:

    1. ∮C B⋅d ℓ over the path comprising a quarter section of a circle, as shown in Fig. P3.57, and

       # Figure P3.57 Problem 3.57.

    2. ∫S(∇×B)⋅d s over the surface of the quarter section.

23. 3.58 Determine if each of the following vector fields is solenoidal, conservative, or both:

    1. *a.A=x ˆ x 2−y ˆ 2 x y

    2. b.B=x ˆ x 2−y ˆ y 2+z ˆ 2 z

    3. c.C=r ˆ(sin ϕ)/r 2+ϕ ˆ(cos ϕ)/r 2

    4. *d.D=R ˆ/R

    5. e.E=r ˆ 3−r 1+r+z ˆ z

    6. f.F=(x ˆ y+y ˆ x)/(x 2+y 2)

    7. g.G=x ˆ(x 2+z 2)−y ˆ(y 2+x 2)−z ˆ(y 2+z 2)

    8. *h.H=R ˆ (R e−R)

24. 3.59 Find the Laplacian of the following scalar functions:

    1. a.V=4 x y 2 z 3

    2. b.V=x y+y z+z x

    3. *c.V=3/(x 2+y 2)

    4. d.V=5 e−r cos ϕ

    5. e.V=10 e−R sin θ

25. 3.60 Find the Laplacian of the following scalar functions:

    1. V 1=10 r 3 sin 2 ϕ

    2. V 2=(2/R 2) cos θ sin ϕ

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