there are 4 questions which all are answered I just want someone to double check them and give corrections (do please dont type it) thanks.

div ( v ) = ( d /ax, a/ ay, 2 / 72). (4- 2 x42 - xe cos (y ), 42z , e cas ( y ) ) Taking the partial desivatives , we get : dax ( 4 - 2nyz - xe= cos (y)) = - 242 - e2 cos(y) 8 / sy ( y? 2 ) = 2 42 8 / 32 ( e ? cos(y ) ) = - e cos(y ) Therefore , the divergence of the velocity field is : div ( v ) = - 242 - e" cos(y) + 2yz - e cosly) = - 2e- cos ( y ) Now, applying the divergence theorem, we have: Flux = SSS V div ( v ) dy Since the surface S is not closed , we need to consider the I flux across both sides of the Surface . However , ' since the outward unit normal vector is oriented away from the origin , the flux across the pestion 1 s oriented toward the origin will be negative. Hence , we only need to calculate the flux across the portion of s oriented away the ofugin.We can use spherical coordinates to palameterize the surface s : K = 9 sin ( D) cos (e ) 4 = 9 sin ( 8) sin (6 ) 2 = 9 cos ( 0 ) where a ranges from o to2, pranges from atout, and 6 9angles from O to 251 . To simplify the calculations , we substitute these palametaizations into the divergence . div ( v ) = -2e" (9 cos ( 9 ) ) cos ( as in ( 9 ) since ) ) Now, we need to evaluate the suface integral : Flux = SSSvinds = sss ( 4 - 2ny2 -x22 cos( y ) , y2z . el cos(y ) ) . nas = SAS (- 2 0 (cos( 9> > Eos( 9 sin (9) sin (6)>, 22, e2cos(y))-nds =sss - ze cacas ( 0) # ( 9 sin () sin (6) ). " ds Now , we integrate over the surface S using the parameterization in spherical coordinates . Flux : SM SML - 2 1 9 cos (9)) cos ( 9 sin (9) since)),2 2 sin (@) . dodo This is the integral that represents the flux of the across the surface s.3 [ # 116 - V 2 + 8 SOM" ( 4 ) + 5 [ * JOY 1 3 8 x 5 7 - 0 | + 5 [ 1 6 - 0 ] = 3 x 8 57 + 40 = 12 1 + 40 - 177.7 )Suppose that (, f(z, y)dA = 74 and 9(z, y)dA = 32. Further assume that k = 7. (A) The value of (B) It must be the case that f(x, y) > g(x, y/) for all (x, y/) in R. O True O False and Find ( [ (32 + 5y)dA where D = ((z,y) | 2' + 17 0, oriented away from the origin. The flux of the fluid across S is: (Suggestion: Use the Divergence Theorem. Note that the surface S is not closed.) and If & is the solid tetrahedron with vertices (0, 0, 0), (2, 0, 0), (0, 6, 0) and (0, 0, 3) then:To evaluate the SSS,dy , when E is the solid tetrahedron with restices ( c, 0. 0 ), ( 2 , 0, 0 ) , (0 , 6 , 0 ) and ( 0. 0, 3 ) , we can use the concept of the pp volume of a tetrahedron . The volume of tetechedson with vertices ( o , 0 ,0 ) , ( a , 0, 0 ). ( 0, 6 , 0 ) and ( 0, 0 , c ) is given by V = ( 1/ { ) * a x b x c In our case the length a , b and c are as follows a = 2 ( difference in a-coordinates between vertices ( ",0 , ( ) and ( 2 1 0 , 0 ) b = 6 ( y - coordinates "1. (0, 0, 0 ) and (0 , 6, 0 ) @ C = 3 C 17 2 - coordinates " ( 0, 0, 0 ) and (o, 0 , 3 ) Substituting the values into the formula , we get : P * 2 * 6 * 3 = 6 Therefore , the value of the triple integral SSS , dV is equal to 6