MATH 202 Exam 1 SAMPLE005, Spring 2015

Math 202 - Quiz I --- Sample 005 1) Use (Gauss) Divergence Theorem carefully to find the outward flux 𝑺 𝑭. 𝒏 𝒅𝝈 of the vector field z2 y2 x2 where ,y 5z 9 ,x 1/ F         across i) the surface   2 ii) the surface surrounding the region 2 1    between two spheres. 2) Let F = < 5y, 8y+2z , 4z > and let S be the open paraboloid 𝑧 = 9 − 𝑥2 + y2 & 𝑧 ≥ 5. Use Divergence Theorem to find 𝑺 𝑭. 𝒏 𝒅𝝈 (where n is the outer normal vector to our surface ) 2’) Let F = < 4x, 8y , 4z > and let S be the cylinder x2 + y2 = 4 which is closed on top through z =3; but open on bottom through the plane x+2y+z=2. Use Divergence Theorem to find 𝑺 𝑭. 𝒏 𝒅𝝈 (where n is the outer normal vector to our surface ) 3) Use Stokes’ Theorem to find the outward flux of CURL(F) 𝑺 𝐂𝐮𝐫𝐥 𝑭 . 𝒏 𝒅𝝈 of the vector field    z x y 6 , 4 , F across the upper part of the sphere 25 z y x 2 2 2    whose boundary C lies on the plane z = 4. 4) Let F = < 5y, 8x, z > and let S be the open paraboloid 𝑧 = 10 − 𝑥2 + y2 & 𝑧 ≥ 6 with boundary C. Find the counter clockwise circulation 𝐶 𝐹. 𝑑𝑟 (i) directly (by parametrizing C), and (ii) by Stokes’ Theorem. (iii) Find 𝑺 𝐂𝐮𝐫𝐥 𝑭 . 𝒏 𝒅𝝈 by Stokes’ Theorem, and (iv) solve part (iii) directly. 4’) 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   y x (counterclockwise). 5) Solve the IVP .... ;y )1( 2y )y xe (x xyy' 2 5x 2 9/4     6) Change the following (non-exact) D.E to exact then STOP! 0 ) 7 (8 3     xdy dx x e y x 7) Solve the DE 4xy) (3x 2xy) 'y (7y 2 2    8) Solve the IVP 5) ysin(xy xy '   ; y(3)=0 9) Solve y’= sinx) (x 2 3 3x e) (cos x   9) (y 2 