16.35 please 1094CHAPTER 16 Vector Calculus0=()1 mintdo ow (LI) dow002 02vd 0/1 =016.3 EXERCISES1. The figure shows a curve C and a contour map of a function fwhose gradient is continuous. Find f Vf. dr.XTvelengnoney 20029. F(x, y) = (y² cos x + cos y)i + (2y sin x - x sin y) jne savent with bloft obrol auounimos & of 1stqedo eid 10. F(x, y) = (lny + y/x)i + (lnx + x/y) jsYAbrsni sdi 21 A(stboronwishant (1)X012Cy100132. A table of values of a function f with continuous gradient isgiven. Find Jc Vf. dr, where C has parametric equationsx = 1² + 1y = t³ + t83+27(0Comparing this equation with Equation 16, we see thatP(A) + K(A) = P(B) + K(B)reason the vector field is called conservative.which says that if an object moves from one point A to another point B under the influ-ence of a conservative force field, then the sum of its potential energy and its kineticenergy remains constant. This is called the Law of Conservation of Energy and it is theon instemos A.A - (Onborant boI15or pelenk 50403020 Wib165WOND6024729silnopoA.D 19.11.Niog s te ((1))3. F(x, y) = (xy + y²)i + (x² + 2xy) j(A)A4. F(x, y) = (y² - 2x) i + 2xy j5. F(x, y) = y²e³ i + (1 + xy)e² j6. F(x, y) = ye* i + (e* + e³) j7.F(x, y) = (ye* + sin y) i + (e* + x cos y) j8. F(x, y) = (2xy + y 2)i + (x² - 2xy-³) j,21 or 21The figure shows the vector field F(x, y) = (2xy, x²) andthree curves that start at (1, 2) and end at (3, 2).potential funch ((a) Explain why JcF. dr has the same value for all three17curves.do do do ant ve anah, how sidil(71) 26/10]0 ≤ t ≤ 1 Men(htagX1-\V tadi balinovtherIFTObey>0liens ar il(b) What is this common value?y3.0on sulder70+1en we have2sa bado 12. F(x, y) = (3 + 2xy²) i + 2x²y j,مد73706/103200Jediof DYERVISA90 LX418d) ou 14. F(x, y) = (1 + xy)exy i + x²e* j,//12-18 (a) Find a function f such that F = Vf and (b) use●3-10 Determine whether or not F is a conservative vector field.part (a) to evaluate f F dr along the given curve C.If it is, find a function f such that F = Vf.(8)X = VL13. F(x, y) = x²y³ i + x³y²j,ols blail cool erh yd snob show C: r(t) = (t³ - 2t, t³ + 2t), 0≤t≤1bns SASATB ygrone3TC is the arc of the hyperbola y = 1/x from (1, 1) to (4,1)Xstalinatoq sut C: r(t) = cos ti + 2 sin tj, 0≤ t ≤ π/2aved aw1.-15. F(x, y, z) = yzi+xzj + (xy + 2z) k,C is the line segment from (1, 0, -2) to (4, 6, 3)

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3-10 Determine whether or not F is a If it is, find a function f such that F = Vf. 3. F(x, y) = (xy + y²)i + (x² + 2xy) j conservative vector field. 4. F(x, y) = (y² - 2x) i + 2xy j 10.0 tai vaners els no taupo atvenols bleil ecol styd en 5. F(x, y) = y²eyi + (1 + xy)exy j