(Solved): Pls show the processes 5. Consider the vector field \( \boldsymbol{F}=\left(a ...

???????Pls show the processes

5. Consider the vector field \( \boldsymbol{F}=\left(a x y+x e^{b y}+1\right) i+\left(x^{2}+y^{2}-\frac{1}{2} x^{2} e^{b y}\right) \boldsymbol{j} \), where \( a \) and \( b \) are constants chosen so that \( \boldsymbol{F} \) is conservative. (a) Find \( a \) and \( b \). (b) Find \( f \), where \( \boldsymbol{F}=\nabla f \). (c) Evaluate \( \int_{\mathcal{C}} \boldsymbol{F} \cdot d \boldsymbol{r} \), where \( \mathcal{C} \) is the curve \( x(t)=\cos ^{3} t, y(t)=\sin ^{5} t, 0 \leqslant t \leqslant \pi / 2 \). Ans: (a) \( a=2, b=-1 \) Ans: (b) \( f(x, y)=x^{2} y+x+\frac{1}{3} y^{3}+\frac{1}{2} x^{2} e^{-y}+C \) Ans: \( (\mathrm{c})-7 / 6 \) 6. Let \( \boldsymbol{F}(x, y, z)=\left\langle x^{3}+\sin (z y), y^{3}+\arctan \left(x z^{42}\right), 3 z\right\rangle \) and \( S \) be the closed surface (oriented with outward pointing normal) consisting of the circular cone \( z=1-\sqrt{x^{2}+y^{2}} \) closed below by the plane \( z=0 \). Evaluate the surface integral \[ \iint_{S} \boldsymbol{F} \cdot d \boldsymbol{S} \] Ans: \( \frac{13}{10} \pi \) 7. Consider the vector field \[ \boldsymbol{F}(x, y, z)=\left\langle\sin \left(e^{y z}\right), \cos \left(e^{x z}\right), e^{x^{2}}+e^{y^{2}}+e^{z^{2}}\right\rangle \] (a) Find curl \( \boldsymbol{F} \). (b) Find the flux of \( \boldsymbol{G}:=\boldsymbol{c u r l} \boldsymbol{F} \) across \( S \), where \( S \) is the part of the paraboloid \( z=9-x^{2}-y^{2} \), with \( z \geqslant 0 \) oriented with a downward pointing normal. Ans: (a) \( \nabla \times \boldsymbol{F}=\left\langle 2 y e^{y^{2}}+x e^{x z} \sin \left(e^{x z}\right), y e^{y z} \cos \left(e^{y z}\right)-2 x e^{x^{2}},-z e^{x z} \sin \left(e^{x z}\right)-z e^{y z} \cos \left(e^{y z}\right)\right\rangle \)