Please help me ASAP. Plz don't spend time on explanation. Please just share your work and highlight the answer. Appreciate,

Please only work on (b)

Here is the formula sheet that you may need:

Gradient vector Vf = 21 1 + 95 j + 95 k Directional Derivative (Daf)po = (ds) apo )= ( Vf ) po . a Double Integral If f(x.y)dxdy = [f f(r cos,r sin @)rarde Mass = mass Jf x8 (x,y)dxdy; y = mass /[ yo(x, y) dxdy; 1x = [JyzodA; lo = [[ (x2 + y2)8dA Change- of-variable theorem If, f (x, y)dady = [ f(x(u, v), y(u, v)Idet /| dudv ax det/ = au av ay ay au av Triple Integral If, f(x.y,z ) av Cylindrical Coordinates: x = r cos 0 ; y = rsin0 ; z = z dxdydz = dz rdr d0, if you integrate z first, then r and 0 Spherical Coordinates: z = p coso ; x = p sino cos0 ; y = p sind sine dxdydz = p2 sin p dpdode Mass = SIS, 6(x, y, z)dV ; Iz = Slo (x2 + y?)odv ; x = Mass Slo xodv Curl F = M(x, y, z) i + N(x, y, z) j + P(x, y,z) k VX F = a ax dy az ap _ON) it (az - ax) it ( ox - ay) k IM N pl Divergence V . F = OM + ON + ap Vector Line Integral C: r(t) = x(t)i + y(t) / + z(t) k; F = M(x, y, z)i + N(x, y, z)j + P(x, y, z) k F . Tds = [ F .dro) at = [ F . di(t) = [Mdx + Ndy + Pdz Fundamental theorem of calculus for line integral If F = Vf, SF . dr = f(P,) -f(Po) Po and Pi are the start and end points of curve C.APSC248-Formula Sheet| 2021W SF . dA dxdy x, y,z); dA = a2 sin p dode; z = acoso; x = a sin & cos0 ; y = a sin d sin 0 - (x, y,0); dA = adz de; x = a cos0 ; y = a sin 0 s 21 = + ( - fx , - fy, 1/dxdy Question 2 (20 points) (a) Show that the vector field F = (16x - 8y)i + (-8x + 8y)] is conservative. Obtain its potential function f (x, y) where F = Vf. (15 points) (b) Let V = Vv (a conservative field), where v = 2x3y + xy3 + x + 2y. Consider the curve C:T(t) = et costi + et sint j, with starting point Po at t = It/2, and end point P, at t = It. Compute S V . dr using the fundamental theorem of calculus. (5 points) Note: More space is given on the next page.Surface Integral SS F . ndA = SS, F . dA Plane z = a: n = th; dA = dxdy Spherical surface: n = 1=(x, y,z); dA = a2 sino dodo; z = a coso; x = asin q cose ; y = a sin q sin Cylindrical surface: n = =(x, y,0); dA = a dz d0; x = a cos0 ; y = asin0 General surface (explicit): z = f(x, y), dA = ndA = +(-fx, -fy, 1)dxdy Green's Theorem (closed curve in 2D-plane) SF . ar = ff (vx F) . kdA = ], (ox ay) ._ OM ) axdy Divergence Theorem (closed region or solid and the surface of the closed region with unit normal n pointing outwards) ffe . ndA = [ff dive av = JJJv .Fav If region D is enclosed by surfaces S, and $2, then JU, dive av = [[ F. midA + [[ F. RedA Stokes Theorem (closed curve in 3D and any surface it bounds) fF . ar = [ ( vxF) . n dA Curve C and surface S need to be oriented compatibly using right-hand-rule. Second derivative test for local extreme values (unconstrained) Suppose that f (x, y) and its first and second partial derivatives are continuous throughout a disk centered at point (a, b) and that fx(a, b) = 0 and fy(a, b) = 0. Then 1. f has a local maximum at (a, b) if fxx 0 at (a, b) 2. f has a local minimum at (a, b) if fxx > 0 and fxxfyy - fry > 0 at (a, b) 3. f has a saddle point at (a, b) if fxxfyy - fly