The document contains 11 multi-part math problems involving vector calculus concepts like divergence, flux, and curl. It provides the problems, calculations, and answers for finding things like the angle between two vectors, components of a vector field, and evaluating vector expressions at given points and over surfaces using theorems like divergence and Stokes' theorem.

1. Problems 1. If 퐴 =푎 푥+3푎 푧 and 퐵 =5푎 푥+2푎 푦−6푎 푧, find the angle between 퐴 and 퐵 . Answer: 120.6° 2. Find the angles that the vector 퐴 =6푎 푥−12푎 푦+4푎 푧 makes with the x, y , and z axes. Answer: θx = 64.62° θy = 149.0° θz = 73.4° 3. Show that vectors a=(4,0,-1), b=(1,3,4), and c = (-5,-3,-3) form the sides of a triangle. Is this a right angle triangle? Calculate the area of the triangle. Answer: Yes, 10.5. 4. A vector field is specified as: 퐵 =24푥푦푎 푥+12(푥2+2)푎 푦+18푧2푎 푧. Given two points, P(1, 2, -1) and Q(-2, 1, 3), find: (a) 퐵 at P; (b) a unit vector in the direction of 퐵 at Q; (c) a unit vector directed from Q toward P; (d) the equation of the surface on which 퐵 =60 Answer: (a) (48,36,18) (b) (-0.26, 0.39,0.88) (c) (0,59, 0.20, -0.78) (d) 100= 16푥2푦2 +4푥4+16푥2+16+9푧4 5. Two uniform vector fields are given by 퐸 =−5푎 휌+10푎 휙+3푎 푧 and 퐹 =푎 휌+2푎 휙−6푎 푧 Calculate: (a) 퐸 ×퐹 (b) The vector component of E at P(5,π/2,3) parallel to the line x = 2, z = 3 (c) The angle E makes with the surface z = 3 at P
Answer: (a) 74.06 (b) −5푎 휌 or −5푎 푦 (c) 15.02°

2. 6. Express vector 퐵 = 10 푟 푎 푟+푟cos휃푎 휃+푎 휙 in Cartesian and cylindrical coordinates. Find 퐵 (-3,4,0) and 퐵 (5 ,π/2,-2). Answer: 퐵 =−2푥푎 푥+푎 푦 퐵 =2.467푎 휌+푎 휙+1.167푎 푧 7. Transform the following vectors to spherical coordinates:
Answer:
8. Given that 퐹 =푥2푎 푥−푥푦푎 푦−푦2푎 푧, calculate the circulation of 퐹 around the (closed) path shown in the following figure. Answer: -1/6 9. Determine the divergence of the following vector fields and evaluate them at the specified points.
Answer:
10. Determine the flux of 퐷 =휌2cos2휙푎 휌+푧sin휙푎 휙 over the closed surface of the cylinder ρ = 4, 0 < z < 1. Verify the divergence theorem for this case. Answer: 64π

3. 11. If 퐴 =휌cos휙푎 휌+sin휙푎 휙 , evaluate 퐴 .푑푙 around the path shown in the following figure. Confirm this using Stokes's theorem. Answer: 4.941