Question: Please help me ASAP. Plz don't spend time on explanation. Please just share your work and highlight the answer. Appreciate, Here is the formula sheet
Please help me ASAP. Plz don't spend time on explanation. Please just share your work and highlight the answer. Appreciate,
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.Surface Integral SS F . AdA = SS F . dA Plane z = a: n = th; dA = dxdy Spherical surface: fi = + 1(x, y,z); dA = a2 sin $ dodo; z = a coso; x = a sin $ cos0 ; y = a sin $ sin 0 Cylindrical surface: f = _ _(x, y,0); dA = a dz d0; x = a cos 0 ; y = asin 0 General surface (explicit): z = f(x, y), dA = fidA = 4 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
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