1. Using the definitions in Eqs. 1.1 and 1.4, and appropriate
diagrams, show that the dot product and cross product are distributive,
a) when the three vectors are coplanar; b) in the general case.
Reference equation 1.1 Dot product of two vectors. The dot product of
two vectors is defined by A · B ≡ AB cos θ, Reference equation 1.4
Cross product of two vectors. The cross product of two vectors is
defined by ... Get solution
2. Is the cross product associative? ... If so, prove it; if not, provide a counterexample (the simpler the better). Get solution
3. Find the angle between the body diagonals of a cube. Get solution
4. Use the cross product to find the components of the unit vector ... perpendicular to the shaded plane in Fig. 1.11. Reference figure 1.11 ... Get solution
5. Prove the BAC-CAB rule by writing out both sides in component form. Get solution
6. Prove that [A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C? Get solution
7. Find the separation vector ...from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude...and construct the unit vector... . Get solution
8. (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that ... (b) What constraints must the elements (Ri j ) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (for all vectors A)? Reference equation 1.30... Reference equation 1.29 ... Get solution
9. Find the transformation matrix R that describes a rotation by ...about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin. Get solution
10. (a) How do the components of a vector5 transform under a translation of coordinates ... Fig. 1.16a)? (b) How do the components of a vector transform under an inversion of coordinates ... Fig. 1.16b)? (c) How do the components of a cross product (Eq. 1.13) transform under inversion? [The cross-product of two vectors is properly called a pseudovector because of this “anomalous” behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical mechanics. (d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.) ... Reference equation 1.13 ... Get solution
11. Find the gradients of the following functions: (a) f (x, y, z) = x2 + y3 + z4. (b) f (x, y, z) = x2 y3z4. (c) f (x, y, z) = ex sin(y) ln(z). Get solution
12. The height of a certain hill (in feet) is given by h ( x , y ) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point? Get solution
13. Let ... be the separation vector from a fixed point (x ′, y′, z′ ) to the point (x, y, z), and let ... be its length. Show that ... Get solution
14. Suppose that f is a function of two variables (y and z) only. ... Reference equation 1.29 ... Get solution
15. Calculate the divergence of the following vector functions: ... Get solution
16. Sketch the vector function ... and compute its divergence. The answer may surprise you. . . can you explain it? Get solution
17. In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine ...and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that ... ... Refrence: Eq. 1.29 ... ... Get solution
18. Calculate the curls of the vector functions in Prob. 1.15. Refrence:prob 1.15 ... Get solution
19. Draw a circle in the xy plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of .... According to Eq. 1.41,then, what is the direction of ∇ ×v? Explain how this example illustrates the geometrical interpretation of the curl. Refrence: Eq 1.41: ... Get solution
20. Construct a vector function that has zero divergence and zero curl everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!) Get solution
21. Prove product rules (i), (iv), and (v).Reference rules(i), (iv), and (v) (i) ∇( f g) = f ∇g + g∇ f, (iv) ∇ . (A × B) = B . (∇ × A) − A · (∇ × B), (v) ∇ × ( f A) = f (∇ × A) − A × (∇ f ), Get solution
22. (a) If A and B are two vector functions, what does the expression (A · ∇)B mean? (That is, what are its x, y, and z components, in terms of the Cartesian components of A, B, and ∇?) (b) Compute (ˆr · ∇)ˆr, where ˆr is the unit vector defined in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate (va · ∇)vb. Reference: Eq. 1.21. ... Reference: Prob. 1.15. Calculate the divergence of the following vector functions: ... Get solution
23. (For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of (A · ∇)B. Refrence: (ii) and (vi). ... ... Reference: Prob. 1.22: (a) If A and B are two vector functions, what does the expression (A · ∇)B mean? (That is, what are its x, y, and z components, in terms of the Cartesian components of A, B, and ∇?) (b) Compute (ˆr · ∇)ˆr, where ˆr is the unit vector defined in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate (va · ∇)vb. Reference: Eq. 1.21: Prove product rules (i), (iv), and (v). (i) ∇( f g) = f ∇g + g∇ f, (iv) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), Get solution
24. Derive the three quotient rules. Get solution
25. (a) Check product rule (iv) (by calculating each term separately) for the functions ... (b) Do the same for product rule (ii). (ii) ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A, two for divergences: (c) Do the same for rule (vi). (vi) ∇ × (A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A). Get solution
26. Calculate the Laplacian of the following functions: ... Get solution
27. Prove that the divergence of a curl is always zero. Check it for function v a in Prob. 1.15. Get solution
28. Prove that the curl of a gradient is always zero. Check it for function (b) in Prob. 1.11. Reference Prob. 1.11. Find the gradients of the following functions: (a) f (x, y, z) = x2 + y3 + z4. (b) f (x, y, z) = x2 y3z4. (c) f (x, y, z) = ex sin(y) ln(z). Get solution
29. Calculate the line integral of the function ...from the origin to the point (1,1,1) by three different routes: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1). (b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1). (c) The direct straight line. (d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)? Get solution
30. Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let “upward” be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)? [Note: For the closed surface, the positive direction is “outward,” and hence “down,” for the bottom face.] Calculate the surface integral of ... over five sides (excluding the bottom) of the cubical box (side 2) in Fig. 1.23. Let “upward and outward” be the positive direction, as indicated by the arrows. Figure 1.23 ... Get solution
31. Calculate the volume integral of the function T = z2 over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). Get solution
32. Check the fundamental theorem for gradients, using T = x2 + 4xy + 2yz3, the points a = (0, 0, 0), b = (1, 1, 1), and the three paths in Fig. 1.28: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1); (b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1); (c) the parabolic path z = x2; y = x. Fig. 1.28 ... Get solution
33. Test the divergence theorem for the function ... ... Take as your volume the cube shown in Fig. 1.30, with sides of length 2. Figure 1.30 ... Get solution
34. Test Stokes’ theorem for the function ... ..., using the triangular shaded area of Fig. 1.34. Fig. 1.34 ... Get solution
35. Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open. ... Reference example 1.11 Suppose ... Check Stokes’ theorem for the square surface shown in Fig. 1.33. ... Get solution
36. (a) Show that ... (b) Show that ... Get solution
37. Find formulas for r, θ, ϕ in terms of x, y, z (the inverse, in other words, of Eq. 1.62). Eq. 1.62. x = r sin θ cos &ϕ, y = r sin θ sin ϕ;, z = r cos θ. Get solution
38. Express the unit vectors ... (that is, derive Eq. 1.64). Check your answers several ways .... Also work out the inverse formulas, giving .... ... Get solution
39. (a) Check the divergence theorem for the function v1 = r 2 ˆr, using as your volume the sphere of radius R, centered at the origin. (b) Do the same for v2 = (1/r 2)ˆr. (If the answer surprises you, look back at Prob. 1.16.) Reference: Prob. 1.16. Sketch the vector function ... and compute its divergence. The answer may surprise you. . . can you explain it? Get solution
40. Compute the divergence of the function ... Check the divergence theorem for this function, using as your volume the inverted hemispherical bowl of radius R, resting on the xy plane and centered at the origin (Fig. 1.40). Figure 1.40 ... Get solution
41. Compute the gradient and Laplacian of the function T = r (cos θ + sin θ cos ϕ). Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path shown in Fig. 1.41, from (0, 0, 0) to (0, 0, 2). Eq. 1.42 ... Get solution
42. Express the cylindrical unit vectors ... (that is, derive Eq. 1.75). “Invert” your formulas to get ...Equation 1.75 ... Get solution
43. (a) Find the divergence of the function ... (b) Test the divergence theorem for this function, using the quarter-cylinder (radius 2, height 5) shown in Fig. 1.43. (c) Find the curl of v. ... Get solution
44. Evaluate the following integrals: ... Get solution
45. Evaluate the following integrals: ... Get solution
46. (a) Show that ... [Hint: Use integration by parts.] (b) Let θ(x) be the step function: ... Show that dθ/dx = δ(x). Get solution
47. (a) Write an expression for the volume charge density ρ(r) of a point charge q at r ′. Make sure that the volume integral of ρ equals q. (b) What is the volume charge density of an electric dipole, consisting of a point charge −q at the origin and a point charge +q at a? (c) What is the volume charge density (in spherical coordinates) of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centered at the origin? [Beware: the integral over all space must equal Q.] Get solution
48. Evaluate the following integrals: ... Get solution
49. Evaluate the integral ... (where V is a sphere of radius R, centered at the origin) by two different methods, as in Ex. 1.16. Reference: Ex. 1.16. Evaluate the integral ... where V is a sphere13 of radius R centered at the origin. Get solution
50. (a) Let ... Calculate the divergence and curl of F1 and F2. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. 15In physics, the word field denotes generically any function of position (x, y, z) and time (t). But in electrodynamics two particular fields (E and B) are of such paramount importance as to preempt the term. Thus technically the potentials are also “fields,” but we never call them that. (b) Show that ... can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function. Get solution
51. For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference Theorem 1 Theorem 1 Curl-less (or “irrotational”) fields. The following conditions are equivalent (that is, F satisfies one if and only if it satisfies all the others): 14In some textbook problems the charge itself extends to infinity (we speak, for instance, of the electric field of an infinite plane, or the magnetic field of an infinite wire). In such cases the normal boundary conditions do not apply, and one must invoke symmetry arguments to determine the fields uniquely. (a) ∇ × F = 0 everywhere. (b)... is independent of path, for any given end points. (c) ... for any closed loop. (d) F is the gradient of some scalar function: F = −∇V. Get solution
52. For Theorem 2, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference Theorem 2, Divergence-less (or “solenoidal”) fields. The following conditions are equivalent: (a) ∇ · F = 0 everywhere. (b) ? F · da is independent of surface, for any given boundary line. (c) ...for any closed surface. (d) F is the curl of some vector function: F = ∇ × A. Get solution
53. (a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job. (b) Which can be expressed as the curl of a vector? Find such a vector. Reference: Problem 1.15. Calculate the divergence of the following vector functions. ... Get solution
54. Check the divergence theorem for the function ... using as your volume one octant of the sphere of radius R (Fig. 1.48). Make sure you include the entire surface. figure 1.48 ... Get solution
55. Check Stokes’ theorem using the function ... (a and b are constants) and the circular path of radius R, centered at the origin in the xy plane. Get solution
56. Compute the line integral of ... along the triangular path shown in Fig. 1.49. Check your answer using Stokes’ theorem. figure 1.49 ... Get solution
57. Compute the line integral of ... around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates). Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes’ theorem. figure 1.50 ... Get solution
58. Check Stokes’ theorem for the function ... sing the triangular surface shown in Fig. 1.51. Figure 1.51 ... Get solution
59. Check the divergence theorem for the function ... using the volume of the “ice-cream cone” shown in Fig. 1.52 (the top surface is spherical, with radius R and centered at the origin). figure 1.52 ... Get solution
60. Here are two cute checks of the fundamental theorems: (a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem (v = ∇T , in this case). Show that the result is consistent with what you already knew about second derivatives. (b) Combine Corollary 2 to Stokes’ theorem with the divergence theorem. Show that the result is consistent with what you already knew Get solution
61. Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: ... Get solution
62. The integral ... is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinary (scalar) area, obviously. (a) Find the vector area of a hemispherical bowl of radius R. (b) Show that a = 0 for any closed surface. [Hint: Use Prob. 1.61a.] (c) Show that a is the same for all surfaces sharing the same boundary. (d) Show that ... Reference problem 1.61aAlthough the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: ... where the integral is around the boundary line. [Hint: One way to do it is to draw the cone subtended by the loop at the origin. Divide the conical surface up into infinitesimal triangular wedges, each with vertex at the origin and opposite side dl, and exploit the geometrical interpretation of the cross product (Fig. 1.8).] (e) Show that ... for any constant vector c. [Hint: Let T = c · r in Prob. 1.61e.] figure 1.8 ... Reference problem 1.61a Get solution
63. (a)Find the divergence of the function ... First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem as in Eq. 1.85. Is there a delta function at the origin, as there was for ˆr/r 2? What is the general formula for the divergence of r n ˆr? ...unless n = −2, in which case it is 4πδ3(r); for n −2, the divergence is ill-defined at the origin. ] (b)Find the curl of r n ˆr. Test your conclusion using Prob. 1.61b. Reference: Eq. 1.84. ... Reference: Eq. 1.85 .... Get solution
64. In case you’re not persuaded that ∇2(1/r ) = −4πδ3(r) (Eq. 1.102 with ...for simplicity), try replacing r by ...and watching what happens as ... Specifically, let ... ... Reference:Eq. 1.102. ... Get solution
2. Is the cross product associative? ... If so, prove it; if not, provide a counterexample (the simpler the better). Get solution
3. Find the angle between the body diagonals of a cube. Get solution
4. Use the cross product to find the components of the unit vector ... perpendicular to the shaded plane in Fig. 1.11. Reference figure 1.11 ... Get solution
5. Prove the BAC-CAB rule by writing out both sides in component form. Get solution
6. Prove that [A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C? Get solution
7. Find the separation vector ...from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude...and construct the unit vector... . Get solution
8. (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that ... (b) What constraints must the elements (Ri j ) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (for all vectors A)? Reference equation 1.30... Reference equation 1.29 ... Get solution
9. Find the transformation matrix R that describes a rotation by ...about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin. Get solution
10. (a) How do the components of a vector5 transform under a translation of coordinates ... Fig. 1.16a)? (b) How do the components of a vector transform under an inversion of coordinates ... Fig. 1.16b)? (c) How do the components of a cross product (Eq. 1.13) transform under inversion? [The cross-product of two vectors is properly called a pseudovector because of this “anomalous” behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical mechanics. (d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.) ... Reference equation 1.13 ... Get solution
11. Find the gradients of the following functions: (a) f (x, y, z) = x2 + y3 + z4. (b) f (x, y, z) = x2 y3z4. (c) f (x, y, z) = ex sin(y) ln(z). Get solution
12. The height of a certain hill (in feet) is given by h ( x , y ) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point? Get solution
13. Let ... be the separation vector from a fixed point (x ′, y′, z′ ) to the point (x, y, z), and let ... be its length. Show that ... Get solution
14. Suppose that f is a function of two variables (y and z) only. ... Reference equation 1.29 ... Get solution
15. Calculate the divergence of the following vector functions: ... Get solution
16. Sketch the vector function ... and compute its divergence. The answer may surprise you. . . can you explain it? Get solution
17. In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine ...and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that ... ... Refrence: Eq. 1.29 ... ... Get solution
18. Calculate the curls of the vector functions in Prob. 1.15. Refrence:prob 1.15 ... Get solution
19. Draw a circle in the xy plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of .... According to Eq. 1.41,then, what is the direction of ∇ ×v? Explain how this example illustrates the geometrical interpretation of the curl. Refrence: Eq 1.41: ... Get solution
20. Construct a vector function that has zero divergence and zero curl everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!) Get solution
21. Prove product rules (i), (iv), and (v).Reference rules(i), (iv), and (v) (i) ∇( f g) = f ∇g + g∇ f, (iv) ∇ . (A × B) = B . (∇ × A) − A · (∇ × B), (v) ∇ × ( f A) = f (∇ × A) − A × (∇ f ), Get solution
22. (a) If A and B are two vector functions, what does the expression (A · ∇)B mean? (That is, what are its x, y, and z components, in terms of the Cartesian components of A, B, and ∇?) (b) Compute (ˆr · ∇)ˆr, where ˆr is the unit vector defined in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate (va · ∇)vb. Reference: Eq. 1.21. ... Reference: Prob. 1.15. Calculate the divergence of the following vector functions: ... Get solution
23. (For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of (A · ∇)B. Refrence: (ii) and (vi). ... ... Reference: Prob. 1.22: (a) If A and B are two vector functions, what does the expression (A · ∇)B mean? (That is, what are its x, y, and z components, in terms of the Cartesian components of A, B, and ∇?) (b) Compute (ˆr · ∇)ˆr, where ˆr is the unit vector defined in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate (va · ∇)vb. Reference: Eq. 1.21: Prove product rules (i), (iv), and (v). (i) ∇( f g) = f ∇g + g∇ f, (iv) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), Get solution
24. Derive the three quotient rules. Get solution
25. (a) Check product rule (iv) (by calculating each term separately) for the functions ... (b) Do the same for product rule (ii). (ii) ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A, two for divergences: (c) Do the same for rule (vi). (vi) ∇ × (A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A). Get solution
26. Calculate the Laplacian of the following functions: ... Get solution
27. Prove that the divergence of a curl is always zero. Check it for function v a in Prob. 1.15. Get solution
28. Prove that the curl of a gradient is always zero. Check it for function (b) in Prob. 1.11. Reference Prob. 1.11. Find the gradients of the following functions: (a) f (x, y, z) = x2 + y3 + z4. (b) f (x, y, z) = x2 y3z4. (c) f (x, y, z) = ex sin(y) ln(z). Get solution
29. Calculate the line integral of the function ...from the origin to the point (1,1,1) by three different routes: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1). (b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1). (c) The direct straight line. (d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)? Get solution
30. Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let “upward” be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)? [Note: For the closed surface, the positive direction is “outward,” and hence “down,” for the bottom face.] Calculate the surface integral of ... over five sides (excluding the bottom) of the cubical box (side 2) in Fig. 1.23. Let “upward and outward” be the positive direction, as indicated by the arrows. Figure 1.23 ... Get solution
31. Calculate the volume integral of the function T = z2 over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). Get solution
32. Check the fundamental theorem for gradients, using T = x2 + 4xy + 2yz3, the points a = (0, 0, 0), b = (1, 1, 1), and the three paths in Fig. 1.28: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1); (b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1); (c) the parabolic path z = x2; y = x. Fig. 1.28 ... Get solution
33. Test the divergence theorem for the function ... ... Take as your volume the cube shown in Fig. 1.30, with sides of length 2. Figure 1.30 ... Get solution
34. Test Stokes’ theorem for the function ... ..., using the triangular shaded area of Fig. 1.34. Fig. 1.34 ... Get solution
35. Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open. ... Reference example 1.11 Suppose ... Check Stokes’ theorem for the square surface shown in Fig. 1.33. ... Get solution
36. (a) Show that ... (b) Show that ... Get solution
37. Find formulas for r, θ, ϕ in terms of x, y, z (the inverse, in other words, of Eq. 1.62). Eq. 1.62. x = r sin θ cos &ϕ, y = r sin θ sin ϕ;, z = r cos θ. Get solution
38. Express the unit vectors ... (that is, derive Eq. 1.64). Check your answers several ways .... Also work out the inverse formulas, giving .... ... Get solution
39. (a) Check the divergence theorem for the function v1 = r 2 ˆr, using as your volume the sphere of radius R, centered at the origin. (b) Do the same for v2 = (1/r 2)ˆr. (If the answer surprises you, look back at Prob. 1.16.) Reference: Prob. 1.16. Sketch the vector function ... and compute its divergence. The answer may surprise you. . . can you explain it? Get solution
40. Compute the divergence of the function ... Check the divergence theorem for this function, using as your volume the inverted hemispherical bowl of radius R, resting on the xy plane and centered at the origin (Fig. 1.40). Figure 1.40 ... Get solution
41. Compute the gradient and Laplacian of the function T = r (cos θ + sin θ cos ϕ). Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path shown in Fig. 1.41, from (0, 0, 0) to (0, 0, 2). Eq. 1.42 ... Get solution
42. Express the cylindrical unit vectors ... (that is, derive Eq. 1.75). “Invert” your formulas to get ...Equation 1.75 ... Get solution
43. (a) Find the divergence of the function ... (b) Test the divergence theorem for this function, using the quarter-cylinder (radius 2, height 5) shown in Fig. 1.43. (c) Find the curl of v. ... Get solution
44. Evaluate the following integrals: ... Get solution
45. Evaluate the following integrals: ... Get solution
46. (a) Show that ... [Hint: Use integration by parts.] (b) Let θ(x) be the step function: ... Show that dθ/dx = δ(x). Get solution
47. (a) Write an expression for the volume charge density ρ(r) of a point charge q at r ′. Make sure that the volume integral of ρ equals q. (b) What is the volume charge density of an electric dipole, consisting of a point charge −q at the origin and a point charge +q at a? (c) What is the volume charge density (in spherical coordinates) of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centered at the origin? [Beware: the integral over all space must equal Q.] Get solution
48. Evaluate the following integrals: ... Get solution
49. Evaluate the integral ... (where V is a sphere of radius R, centered at the origin) by two different methods, as in Ex. 1.16. Reference: Ex. 1.16. Evaluate the integral ... where V is a sphere13 of radius R centered at the origin. Get solution
50. (a) Let ... Calculate the divergence and curl of F1 and F2. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. 15In physics, the word field denotes generically any function of position (x, y, z) and time (t). But in electrodynamics two particular fields (E and B) are of such paramount importance as to preempt the term. Thus technically the potentials are also “fields,” but we never call them that. (b) Show that ... can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function. Get solution
51. For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference Theorem 1 Theorem 1 Curl-less (or “irrotational”) fields. The following conditions are equivalent (that is, F satisfies one if and only if it satisfies all the others): 14In some textbook problems the charge itself extends to infinity (we speak, for instance, of the electric field of an infinite plane, or the magnetic field of an infinite wire). In such cases the normal boundary conditions do not apply, and one must invoke symmetry arguments to determine the fields uniquely. (a) ∇ × F = 0 everywhere. (b)... is independent of path, for any given end points. (c) ... for any closed loop. (d) F is the gradient of some scalar function: F = −∇V. Get solution
52. For Theorem 2, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference Theorem 2, Divergence-less (or “solenoidal”) fields. The following conditions are equivalent: (a) ∇ · F = 0 everywhere. (b) ? F · da is independent of surface, for any given boundary line. (c) ...for any closed surface. (d) F is the curl of some vector function: F = ∇ × A. Get solution
53. (a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job. (b) Which can be expressed as the curl of a vector? Find such a vector. Reference: Problem 1.15. Calculate the divergence of the following vector functions. ... Get solution
54. Check the divergence theorem for the function ... using as your volume one octant of the sphere of radius R (Fig. 1.48). Make sure you include the entire surface. figure 1.48 ... Get solution
55. Check Stokes’ theorem using the function ... (a and b are constants) and the circular path of radius R, centered at the origin in the xy plane. Get solution
56. Compute the line integral of ... along the triangular path shown in Fig. 1.49. Check your answer using Stokes’ theorem. figure 1.49 ... Get solution
57. Compute the line integral of ... around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates). Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes’ theorem. figure 1.50 ... Get solution
58. Check Stokes’ theorem for the function ... sing the triangular surface shown in Fig. 1.51. Figure 1.51 ... Get solution
59. Check the divergence theorem for the function ... using the volume of the “ice-cream cone” shown in Fig. 1.52 (the top surface is spherical, with radius R and centered at the origin). figure 1.52 ... Get solution
60. Here are two cute checks of the fundamental theorems: (a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem (v = ∇T , in this case). Show that the result is consistent with what you already knew about second derivatives. (b) Combine Corollary 2 to Stokes’ theorem with the divergence theorem. Show that the result is consistent with what you already knew Get solution
61. Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: ... Get solution
62. The integral ... is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinary (scalar) area, obviously. (a) Find the vector area of a hemispherical bowl of radius R. (b) Show that a = 0 for any closed surface. [Hint: Use Prob. 1.61a.] (c) Show that a is the same for all surfaces sharing the same boundary. (d) Show that ... Reference problem 1.61aAlthough the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: ... where the integral is around the boundary line. [Hint: One way to do it is to draw the cone subtended by the loop at the origin. Divide the conical surface up into infinitesimal triangular wedges, each with vertex at the origin and opposite side dl, and exploit the geometrical interpretation of the cross product (Fig. 1.8).] (e) Show that ... for any constant vector c. [Hint: Let T = c · r in Prob. 1.61e.] figure 1.8 ... Reference problem 1.61a Get solution
63. (a)Find the divergence of the function ... First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem as in Eq. 1.85. Is there a delta function at the origin, as there was for ˆr/r 2? What is the general formula for the divergence of r n ˆr? ...unless n = −2, in which case it is 4πδ3(r); for n −2, the divergence is ill-defined at the origin. ] (b)Find the curl of r n ˆr. Test your conclusion using Prob. 1.61b. Reference: Eq. 1.84. ... Reference: Eq. 1.85 .... Get solution
64. In case you’re not persuaded that ∇2(1/r ) = −4πδ3(r) (Eq. 1.102 with ...for simplicity), try replacing r by ...and watching what happens as ... Specifically, let ... ... Reference:Eq. 1.102. ... Get solution