N A B PP W MATH 2310 - Section B - Final Exam Review Questions Fall 2023 December 4, 2023 The following questions are being given to assist you with your preparation for the final exam. Some of the answers have also been provided. On the exam, of course, you must show your work as to how these answers were derived. 1. Locate the relative maxima, minima, and saddle points of the function: f(z,y) = log(z? + y* + 1). Answer: (z,y) = (0,0) is the only critical point. It is a local minimum. Find the maxima and minima of the function f(z,y) = 72 +y?—z—y+1 in the disk D defined by 2 +¢* < 1. Answer: Absolute minimum occurs at: (1/2,1/2), and absolute maximum occurs at (—v/2/2, — » 3. Verify that the vector field: V(z,y,2) = (y2—=z3)/(z*+ y?) is irrotational when (z,y) # (0,0). Answer: Curl of V: (VxV)= 4. Compute the line integral of v = 6 + yz?§ + (3y +2)2 along the following triangular path: e e Answer: 8/3 5. Check the dxvergenoe theorem for the function: v = 2 cos 67 + r2 cos 9 — 2 cos sin ¢, using as your vol- ume the the following figure (which is one octant of the sphere of radius R: 2/2). 6. oo Answer: TR*/4 Compute the gradient and Laplacian of the function T = r(cos + sinf cos ¢). Check the Laplacian by con- verting T to Cartesian coordinates. Test the fundamen- tal theorem for gradients for this function, using the path shown below from (0,0,0) to (0,0,2): Answer: Both sides of the FTG should give 2. . Prove the following product rules for derivatives: (a) V-(AxB)=B:(VxA)-A-(VxB). (b) V x (fA) = f(V x A) - A x (Vf), where A, B are vector fields and f is a scalar function. . In lecture, we calculated a surface integral over a cube with 5 sides, the “bottom” side being open. For the same function, calculate the surface integral over the bottom of the bax. Does the surface integral depend only on the boundary line for this function?
