Prove that the curl of a gradient is always zero. Check it

(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 2? What is the general formula for the divergence of n ? [Answer: , unless , in which case it is ; for , the divergence is ill-defined at the origin.] (b) Find the curl of n . Test your conclusion using Prob. 1.61b. Reference: Eq. 1.84. . Reference: Eq. 1.85 .

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.

Problem 3P Find the angle between the body diagonals of a cube.

Prove the BAC-CAB rule by writing out both sides in component form.

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

Problem 2P Is the cross product associative? If so, prove it; if not, provide a counterexample (the simpler the better).

Problem 6P Prove that [A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C?

Problem 7P 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 .

(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 .

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.

(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

The height of a certain hill (in feet) is given by \(h(x, y)=10\left(2 x y-3 x^{2}-4 y^{2}-18 x+28 y+12\right)\) 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?

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 (a) . (b) . (c) What is the general formula for ?

Find the gradients of the following functions: (a) \(f(x,\ y,\ z)=x^2+y^3+z^4\) (b) \(f(x,\ y,\ z)=x^2y^3z^4\) (c) \(f(x,\ y,\ z)=e^x\sin(y)\ln(z)\)

Sketch the vector function , and compute its divergence. The answer may surprise you… can you explain it?

Prove product rules (i), (iv), and (v). Reference rules (i), (iv), and (v)

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 and . According to Eq. 1.41,then, what is the direction of xv? Explain how this example illustrates the geometrical interpretation of the curl. Reference: Eq 1.41:

Suppose that is a function of two variables (y and z) only. Show that the gradient transforms as a vector under rotations, Eq. .29. [Hint: , and the analogous formula for . We that and ; “solve” these equations for y and z (as functions of and ), and compute the needed derivatives , etc.] Reference equation 1.29 (a) (b) . (c) .

Calculate the curls of the vector functions in Prob. 1.15. Reference: Prob 1.15 (a) . (b) . (c) .

Construct a vector function that has zero divergences and zero curls everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!)

In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine and , and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that .] Reference: Eq. 1.29

Problem 24P Derive the three quotient rules.

(a) Check product rule (iv) (by calculating each term separately) for the functions ; . (b) Do the same for product rule (ii). (ii) , two for divergences: (c) Do the same for rule (vi). (vi) .

(a) If A and B are two vector functions, what does the expression mean? (That is, what are its x, y, and z components, in terms of the Cartesian components of A, B, and ?) (b) Compute ( · ), where 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: (a) . (b) . (c) .

Prove that the divergence of a curl is always zero. Check it for function Va in Prob. 1.15. REFERENCE PROBLEM 1.15 Calculate the divergence of the following vector functions: (a) . (b) . (c) .

(For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of (A · )B. Reference: (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 ( · ), where 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).

Calculate the Laplacian of the following functions: (a) . (b) . (c) . (d) .

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) . (b) . (c) .

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)?

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

Calculate the volume integral of the function over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).

Test Stokes’ theorem for the function , using the triangular shaded area of Fig. 1.34. Fig. 1.34

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.

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

Check the fundamental theorem for gradients, using , the points , , 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 ; . Fig. 1.28

(a) Show that (b) Show that

Problem 37P Find formulas for r, ?, ? in terms of x, y, z (the inverse, in other words, of Eq. 1.62). Eq. 1.62. x = r sin ? cos &?, y = r sin ? sin ?;, z = r cos ?.

Express the unit vectors in terms of (that is, derive Eq. 1.64). Check your answers several ways . in terms of (and , ). Reference Eq. 1.64

Problem 39P (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?

Express the cylindrical unit vectors in terms of (that is, derive Eq. 1.75). “Invert” your formulas to get in terms of (and ). Equation 1.75

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

Problem 44P Evaluate the following integrals:

Problem 45P Evaluate the following integrals:

Compute the gradient and Laplacian of the function T = r (cos ? + sin ? cos ?). 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

(a) Show that . [Hint: Use integration by parts.] (b) Let be the step function: . Show that .

Problem 43P (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.

(a) Write an expression for the volume charge density of a point charge at . Make sure that the volume integral of equals . (b) What is the volume charge density of an electric dipole, consisting of a point charge at the origin and a point charge at ? (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.]

Evaluate the following integrals: (a) , where is fixed vector, is its magnitude, and the integral is over all space. (b) , where is a cube of side 2, centered on the origin, and . (c) , where is sphere of radius 6 about the origin, , and is its magnitude. (d) , where , and is a sphere of radius 1.5 centered at (2,2,2).

(a) Let and . Calculate the divergence and curl of and . 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.

Evaluate the integral (where is a sphere of radius , centered at the origin) by two different methods, as in Ex. 1.16. Reference: Ex. 1.16. Evaluate the integral , where is a sphere 13 of radius centered at the origin.

(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.

Problem 52P 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.

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

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.

Problem 51P 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.

Compute the line integral of along the triangular path shown in Fig. 1.49. Check your answer using Stokes’ theorem. Figure 1.49

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

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

Check Stokes’ theorem for the function , using the triangular surface shown in Fig. 1.51. Figure 1.51

Here are two cute checks of the fundamental theorems: (a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem ( , 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.

Problem 61P 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:

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 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 1.61a 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: (a) . [Hint: Let , where is a constant, in the divergence theorem; use the product rules.] (b) . [Hint: Replace by in the divergence theorem.] (c) . [Hint: Let in the divergence theorem.] (d) T . [Comment: This is sometimes called Green’s second identity; it follows from ©, which is known as Green’s identity.] (e) . [Hint: Let in Stokes’ theorem.] 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 in Prob. 1.61e.] Figure 1.8 Reference problem 1.61a

Calculate the divergence of the following vector functions: (a) \(\mathbf{v}_a=x^2\ \hat{\mathbf{x}}+3xz^2\ \hat{\mathbf{y}}-2xz\hat{\mathbf{\ z}}\). (b) \(\mathbf{v}_b=xy\ \hat{\mathbf{x}}+2yz\ \hat{\mathbf{y}}+3zx\ \hat{\mathbf{z}}\). (c) \(\mathbf{v}_c=y^{2\ }\hat{\mathbf{x}}+\left(2xy+z^2\right)\ \hat{\mathbf{y}}+2yz\ \hat{\mathbf{z}}\)