MATH 202 Exam 1 SAMPLE006, Spring 2015

Math 202: Sample 006 1) Use the divergence theorem to find the outward flux of the field F =     xz i  yz j  k across the boundary of the region which is the entire surface of the upper cap cut from the solid sphere 25 2 2 2    z y x by the plane z  3 Big Hint: = ………….. = 2𝑧 (𝑟𝑑𝑧𝑑𝑟𝑑𝜃) 25−𝑟2 3 4 0 2𝜋 0 =……. 2. (Read Carefully) Use Stokes’ theorem to evaluate the counterclockwise circulation of the field F = k j 2 2 3z y y  i  around the boundary of the ellipse in which the plane 6 3 6 2    z y x meets the cylinder 1 2 2  x  y (counterclockwise). BIG HINT: Curl(F) is easy. Next choose your surface S in the plane .6 3z 6y 2x    Now compute n.d𝛿 by the formula ∓ ∇𝑔 |𝑔𝑧| 𝑑𝑦𝑑𝑥. Your projected region in the xy-plane is the region inside 1 2 2   y x 3. (Read Carefully) Use Stokes’ theorem to evaluate the counterclockwise circulation of the field F =    k j 2 4 2 2 2 2 x z x e x sin y    i  around the boundary of the triangle in the first octant cut by the plane 1 2 3    z y x . BIG HINT: Curl(F) is easy. By Stokes’ Thm. , we can choose our surface S in the plane 1 2 3    z y x Now compute n.d𝛿 by the formula ∓ ∇𝑔 |𝑔𝑧| 𝑑𝑦𝑑𝑥. Your projected region in the xy-plane is the triangle in the 1st quadrant bounded by 1 2 y 3 x   and the coordinate axes. 4. Use the divergence theorem to find the outward flux of the field F =      k j sin xy e y xy xz    2 2 i over the surface of the region bounded by the parabolic cylinder 2 1 x z   and the planes z = 0, y = 0, and y + z = 2 Big Hint: = ………….. = 3𝑦 𝑑𝑧𝑑𝑦𝑑𝑥 1−𝑥2 0 2 0 1 −1 =………. 5) Solve the DE (x + 2)siny dx + xcosy dy = 0. 6) a) Change the DE xy x y y x 2 2 ) 1( ) ' 3 1( 53 2     to a linear DE Then STOP!! Big Hint: It is a Bernoulli DE. 6b) Use an integrating factor to change the following D.E to an exact DE. Then STOP! 𝟔𝒙𝒚 𝒅𝒙 + 𝟒𝒚 + 𝟗𝒙𝟐 𝒅𝒚 = 𝟎 6c) Section 2.5 ABC: 13 (Homogenous) 7a) (About the Uniqueness Theorem.) Find all solutions of 𝑦′ = 4 cos 𝑦 sin⁡(𝑦 + 2𝑥) − 6 ; 𝑦 0 = 0. (Hint: Try 𝑦 = 𝑘𝑥 for some 𝑘.) 7b) (About the Uniqueness Theorem.) Consider the differential equation y ‘ = f(x; y). Assume that f(x; y) has continuous partial derivatives (w.r.t x & y) for all values of x and y. Can the two functions y_1 (x) = x^2 & y_2(x) = - x^2 both be solutions to this differential equation? Justify your answer (i.e. if yes, give an example of such an f(x; y) and if no, supply a proof). Big Hint: Impossible because Both solutions have the common propery y(0)= ….. So this contradicts the Uniqueness theorem 1.2 (?)……..