MATH 202 Exam 1, Spring 2014

Exercise 1 (15 pts) Use Divergence Theorem to find the outward flux | [ #-itao > > F S29 +297 +828 of the vector field across the sphere p = 2 Vsphere = recor 3 Volunwe =\5 ( are Ss neve ) wie @ VARA Ske Gow 32 : + = Exercise 2 (15 pts) Let F= Sy t+ 8x5 +zk and S be the paraboloid z=4—-2’?-y’, z2>0 | [Btw he outer normal vector to our surface) by using Diver which is open at the bottom. Find (where Wt ist gence Theorem Exercise 3 Let S be the upper spherical cap formed by cutting ce e+yt2? = 2 with a cone having the equation z = \/x* + y?. Answer the following questions: a) (10 pts) Let C denote the boundary of the surface (the cap) S and let Fe (z- y)e tity K Calculate the circulation of F around the curve C' counter clockwise by parameterizing the _ b) (10 pts) Solve part a) by using Stokes’ Theorem Exercise 4 (10 pts) Solve the IVP: d go —2y = 2° 43, y(1) =2 dx Exercise 5 (10 pts) Solve the DE: e ute dy are a1 ae +e = 0 Exercise 6 (15 pts) Solve the DE:(3z + 1/2)y° + 35+ ys =0 Exercise 7 (15 pts) My Rr e oe ee 4o-mrake Tr exact Phe Solve it: (Say? + 2x7y)dy + (2y? + 8ay")dx = oa