H Vdr around the boundary of the square with vertices (1, 0), (0, 1), (1, 0), (0, 1), if

If A = 2i j k, B = 2i 3j + k, C = j + k, find (A B)C, A(B C), (A B) C, A (B C), (A B) C, A (B C).

Find the work done by the force B acting on an object which undergoes the displacement C.

Find the total work done by forces A and B if the object undergoes the displacement C. Hint: Can you add the two forces first?

Let O be the tail of B and let A be a force acting at the head of B. Find the torque of A about O; about a line through O perpendicular to the plane of A and B; about a line through O parallel to C.

Let A and C be drawn from a common origin and let C rotate about A with an angular velocity of 2 rad/sec. Find the velocity of the head of C.

In Problem 5, draw B with its tail at the head of A. If the figure is rotating as in Problem 5, find the velocity of the head of B. With the same diagram, let B be a force; find the torque of B about the head of C, and about the line C.

A force F = 2i 3j + k acts at the point (1, 5, 2). Find the torque due to F (a) about the origin; (b) about the y axis; (c) about the line x/2 = y/1 = z/(2).

A vector force with components (1, 2, 3) acts at the point (3, 2, 1). Find the vector torque about the origin due to this force and find the torque about each of the coordinate axes.

The force F = 2i j 5k acts at the point (5, 2, 1). Find the torque due to F about the origin and about the line 2x = 4y = z.

In Figure 3.5, let r be another vector from O to the line of F. Show that r F = r F. Hint: r r is a vector along the line of F and so is a scalar multiple of F. (The scalar has physical units of distance divided by force, but this fact is irrelevant for the vector proof.) Show also that moving the tail of r along n does not change n r F. Hint: The triple scalar product is not changed by interchanging the dot and the cross. 11

Write out the twelve triple scalar products involving A, B, and C and verify the facts stated just above (3.3).

(a) Simplify (A B) 2 [(A B) B] A by using (3.9). (b) Prove Lagranges identity: (AB)(CD)=(AC)(BD)(AD)(BC).

Prove that the triple scalar product of (A B), (B C), and (C A), is equal to the square of the triple scalar product of A, B, and C. Hint: First let (BC) = D, and evaluate (A B) D. [See Am. J. Phys. 66, 739 (1998).]

Prove the Jacobi identity:

In the figure u1 is a unit vector in the direction of an incident ray of light, and u3 and u2 are unit vectors in the directions of the reflected and refracted rays. If u is a unit vector normal to the surface AB, the laws of optics say that 1 = 3 and n1 sin 1 = n2 sin 2, where n1 and n2 are constants (indices of refraction). Write these laws in vector form (using dot or cross products).

In the discussion of Figure 3.8, we found for the angular momentum, the formula L = mr ( r). Use (3.9) to expand this triple product. If r is perpendicular to , show that you obtain the elementary formula, angular momentum = mvr.

Expand the triple product for a = ( r) given in the discussion of Figure 3.8. If r is perpendicular to (Problem 16), show that a = 2r, and so find the elementary result that the acceleration is toward the center of the circle and of magnitude v2/r.

Two moving charged particles exert forces on each other because each one creates a magnetic field in which the other moves (see Problem 4.6). These two forces are proportional to v1 [v2 r] and v2 [v1 (r)] where r is the vector joining the particles. By using (3.9), show that these forces are equal and opposite (Newtons third law) if and only if r (v1 v2) = 0. Compare Problem 14.

The force F = i + 3j + 2k acts at the point (1, 1, 1). (a) Find the torque of the force about the point (2, 1, 5). Careful! The vector r goes from (2, 1, 5) to (1, 1, 1). (b) Find the torque of the force about the line r = 2i j + 5k + (i j + 2k)t. Note that the line goes through the point (2, 1, 5).

The force F = 2i 5k acts at the point (3, 1, 0). Find the torque of F about each of the following lines. (a) r = (2i k) + (3j 4k)t. (b) r = i + 4j + 2k + (2i + j 2k)t.

Verify equation (6.8); that is, find f in spherical coordinates as we did for cylindrical coordinates. Hint: What is ds in the direction? See Chapter 5, Figure 4.5.

V = (y + z)i + (x z)j + (x2 + y2 )k

RR F n d where F = (y2 x2)i + (2xy y)j + 3zk and is the entire surface of the tin can bounded by the cylinder x2 + y2 = 16, z = 3, z = 3.

RR r n d over the entire surface of the hemisphere x2 + y2 + z2 = 9, z 0, where r = xi + yj + zk.

RR Vn d over the curved part of the hemisphere in Problem 24, if V = curl(yixj).

RR (curl V) n d over the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at (2, 2, 2), where V = (2 y)i + xzj + xyzk.

Problem 26, but integrate over the open surface obtained by leaving out the face of the cube in the (x, y) plane.

H F dr around the circle x2 + y2 + 2x = 0, where F = yi xj

H Vdr around the boundary of the square with vertices (1, 0), (0, 1), (1, 0), (0, 1), if V = x2i + 5xj.

C (x2y)dx+ (x+y3 )dy, where C is the parallelogram with vertices at (0, 0), (2, 0), (1, 1), (3, 1).

R (y2 x2) dx + (2xy + 3)dy along the x axis from (0, 0) to (5, 0) and then along a circular arc from (5, 0) to (1, 2). Hint: Use Greens theorem.