1. (10 pts) In this problem we will check that -00 dr = VI. -DO Let dr. (i) Show that 7T 12 2 7' using the fact that the quartercircle r'ty'0, is a subset of the square [0, a] x [0, a]. (ii) Show that using the fact that the quartercircle and y 2 0, contains the square [0, a] x [0, a].1. Let F: R3 - R3 be the vector field given by F(x, y, z) = ayi + 2y(x + 2)j + (by]+ 23)k. (i) For which values of a and b is the vector field F conservative? (ii) Find a function f: R3 - R such that F = grad f, for these values. (iii) Find the equation of a surface S with the property that for every smooth oriented curve C lying on S, F . ds = 0, C for these values. 2. Let S be the rectangle with vertices (0, 0, 0), (1, 0, 0), (1, 2, 2) and (0, 2, 2). Find the flux of the vector field F: R3 - R3, given by F(c, y, z) = yi - 2'j+xk. through S in the direction of the unit normal vector n, for which n . k > 0.3. Let Ca(P) be the circle of radius a centered at P and oriented counter-clockwise. A smooth rotation free vector field F is defined on the whole of R', except for the points Po = (0, 0), P1 = (4,0), and P3 = (8, 0), and F .ds = -2, F .ds = 1 and F .ds' = 3. JC ( Pa) J Ce ( Pa) JCIO (Fo) Find the following line integrals. (a) F . ds. (b) F . ds. Ja(P) (c) F . ds. 4. (6.3.16) 5. (7.1.4)