Verify that Stokes' Theorem is true for the vector field F 6yzi – 3yj + rk and the surface S the part of the paraboloid z = 18 – x2 y? that lies abc the plane z = 2, oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS curl F = <0,1-6y Σ IL curl F· dS = Σ dy da where Y1 = 2pi 9 curl F· dS = Σ Now compute F. dr M M M M

a) Draw 4 pictures showing all possible combinations of a vector field and a curve b) Compute the circulation and the flux

In Cartesian coordinates, a vector field takes the form F= 2xzi+ 2yzj+ (X2+y²)k a. State whether F is conservative, and give a one-sentence justification for your statement b. Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a, b, c). This path has the parametric representation x=at, y=bt, z=ct (0sts1). c. Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F=-VU.