16.89 please H16.8 EXERCISESved bas2321225711. A hemisphere H and a portion P of a paraboloid are shown.Suppose F is a vector field on R³ whose components havecontinuous partial derivatives. Explain why (8)SSValcurl F. ds = curl F. dsthe pluvely ore 11. (a) Use Stokes' Theorem to evaluate jc F • dr, whereand this theorem to veciTuF(x, y, z) = x²zi + xy²j + z²kffHZA4P(alvis broci so nwond oen szarlo tod le12 Unoibong bne nisigbrola bili to2 bnn manyou 5x4x4m-12y22ZA2 4 ЯОЗНТ OWT GIVA I andpoputing216odtown2-6 Use Stokes' Theorem to evaluate ff, curl F. ds.tion.MitCIL2 y2. F(x, y, z) = x² sin zi + y² j + xy k,S is the part of the paraboloid z = 1 = x² - y² that liesabove the xy-plane, oriented upward3. F(x, y, z) = zei + x cos y j + xz sin y k,S is the hemisphere x² + y² + z² = 16, y ≥ 0, oriented inthe direction of the positive y-axis.12. (a) Use Stokes' Theorem to evaluate f F dr, whereF(x, y, z) = x²yi + x³j + xy k and C is the curve ofintersection of the hyperbolic paraboloid z = y² - x²and the cylinder x² + y² = 1, oriented counterclock-Insie wise as viewed from above.ISLeigalup(b) Graph both the hyperbolic paraboloid and the cylinderSenidorul voHolood sd with domains chosen so that you can see the curve Ca and the surface that you used in part (a).lovisa bo] lo piqe(c) Find parametric equations for C and use them tograph C.4. F(x, y, z) = tan¯¹(x²yz²) i + x²yj + x²z² k, o visiboz, sudgisbelidS is the cone x = √y² + z², 0≤x≤ 2, oriented in thedirection of the positive x-axis6. F(x, y, z) = e¹ i + e™²j + x²zk,S is the half of the ellipsoid 4x² + y² + 4z² = 4 that lies tothe right of thexz-plane, oriented in the direction of theexz-pref1940 bairpositive y-axis7. F(x, y, z)=(x + y²)i + (y + z²)j + (z + x²) k,C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)8. F(x, y, z)= i + (x + yz)j + (xy-√√z)k,C is the boundary of the part of the plane 3x + 2y + z = 1in the first octantSECTION 16.8 Stokes' TheoremGEROUS DIDIAnuomoto7-10 Use Stokes' Theorem to evaluate F. dr. In each case Cis oriented counterclockwise as viewed from above.9. F(x, y, z)= xy i+yzj + zx k, C is the boundary of thepart of the paraboloid z = 1 - x² - y² in the first octantxb (2.00Al10. F(x, y, z) = 2yi+xz j + (x + y) k, C is the curveof intersection of the plane z = y + 2 and the cylinderx² + y² = 1vidand C is the curve of intersection of the planetx + y + z = 1 and the cylinder x2 + y² = 9, orientedcounterclockwise as viewed from above.13. F(x, y, z) = -yi+xj-2 k,sitesinto consideS is the cone z² = x² + y²,0 ≤ z ≤ 4, oriented downwardgelamont bos stins all ne X-omule si all. W 19Line14. F(x, y, z) -2yzi+yj + 3x k,M=5. F(x, y, z) = xyz i + xy j + x²yz k, IDX sinuto puroS is the part of the paraboloid z = 5 - x² - y² that liesS consists of the top and the four sides (but not the bottom)Matabove the plane z 1, oriented upward=of the cube with vertices (±1, ±1, ±1), oriented outward detdoM AM 15.(b) Graph both the plane and the cylinder with domainschosen so that you can see the curve C and the surfacethat you used in part (a).(c) Find parametric equations for C and use them tograph C.AL 700113913-15 Verify that Stokes' Theorem is true for the given vectorfield F and surface S.F(x, y, z) = yi + zj+xk,S is the hemisphere x² + y² + z² = 1, y = 0, oriented inthe direction of the positive y-axis16. Let C be a simple closed smooth curve that lies in the planex + y + z = 1. Show that the line integralS z dx-2x dy + 3ydzdepends only on the area of the region enclosed by C andnot on the shape of C or its location in the plane.Find the work done.17. A particle moves along line segments from the origin to thepoints (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the originunder the influence of the force fieldF(x, y, z) = z² i + 2xy j + 4y² k

Question

Accepted Answer

We have to find the value of k ∫Fdr. Using stokes theorem 
The procedure is given below

F(x, y, z) = xy i+yzj+zxk, C is the boundary of part of the paraboloid z = 1 - x² - y² in the first octant Find