$\def\hl#1#2{\bbox[#1,1px]{#2}} \def\box#1#2#3#4#5{\color{#2}{\bbox[0px, border: 2px solid #2]{\hl{#3}{\color{white}{\color{#3}{\boxed{\underline{\large\color{#1}{\text{#4}}}\\\color{#1}{#5}\\}}}}}}} \def\verts#1{\left\vert#1\right\vert} \def\Verts#1{\left\Vert#1\right\Vert} \def\pra#1{\left(#1\right)} \def\R{\mathbb{R}} \def\N{\mathbb{N}} \def\Z{\mathbb{Z}}$
$\box{black}{black}{} {Question} {(a)\text{ Let F$\pra{x,y,z}=(y+z,\alpha x+z,x+\beta y)$. For what values of $\alpha$,}\\ \text{$\beta$ is F conservative. For those cases, find $f$ s.t. F$=\nabla f$}\\ (b)\text{ Let C=$\left\{(t\cos(t),t\sin(t),t^2):0\le t\le\frac{\pi}{2}\right\}$. Compute $\int_CF\cdot d\text{x}\hspace{21.3ex}$}}\\ \box{black}{red}{} {Theorem 1.} {\text{Let $U$ be an open subset of $\R^n$ for $n\ge2$, and let $G:U\to\R^n$ be a continuous vector}\\ \text{field. Then the following are equivalent:}\\ \text{1. There exists a function $f:U\to\R$ of class $C^1$ such that $G=\nabla f$.}\\ \text{2. $\int_CG\cdot d\text{x}$ for any closed piecewise smooth oriented curve $C$ in $U$.}\\ \text{3. $\int_{C_1}G\cdot d\text{x}=\int_{C_2}G\cdot d\text{x}$ for any two piecewise smooth oriented curves $C_1,C_2$ in $U$ that}\\ \text{ both start at the same point $P\in U$ and end at the same point $q\in U.$} }\\ \box{black}{green}{} {Def.(Conservative Vector Field)} {\text{A continuous vector field $G:U\to\R^n$ is said to be conservative if any one of these cond-}\\ \text{itions is satisfied.}}\\ \box{black}{red}{} {Theorem 2.} {\text{If G is a conservative vector field of class $C^1$ on an open set $U\subset\R^3$, then curl G$=0.\hspace{4ex}$}}\\ \box{black}{red}{} {Theorem 3.} {\text{If G$:U\to\R^3$ is a $C^1$ vector field, $U$ is a convex, and curl G$=0$, then G is conservative.$\hspace{1.3ex}$}}$
My attempts
$(a)$ From Theorem $2.$ and $3.$ we know that
If F has convex domain, then curl F$=0$ if and only if F is conservative
As F$:U=\R^3\to\R^3$ where $U$ is convex. and \begin{align} \text{curl F}=&\nabla\times F\\ =&(\beta-1,0,\alpha-1) \end{align} i.e. curl F$=0$ if $\beta=1$ and $\alpha=1$. Therefore F$(x,y,z)=(y+z,x+z,x+y)$ have \begin{align} f(x,y,z)=&\int_a^xF_1(t,b,c)dt+\int_b^yF_2(x,t,c)dt+\int_c^zF_3(x,y,t)dt\\ =&\int_a^xb+c~dt+\int_b^yx+c~dt+\int_c^zx+y~dt\\ =&~x y + x z + y z-a b - a c - b c \end{align} Where $(a,b,c)\in U=\R^3$, take $(a,b,c)=(0,0,0)$ then $f(x,y,z)=x y + x z + y z$ s.t. F$=\nabla f$
$(b)$ Then by Fundamental Theorem of Line Integrals \begin{align} \int_C \text{F}\cdot d\text{x}=&\int_C \nabla f\cdot d\text{x}\\ =&f\pra{\frac{\pi}{2}\cos\pra{\frac{\pi}{2}},\frac{\pi}{2}\sin\pra{\frac{\pi}{2}},\pra{\frac{\pi}{2}}^2}-f(0,0,0)\\ =&\cos\pra{\frac{\pi}{2}}\sin\pra{\frac{\pi}{2}}\pra{\frac{\pi}{2}}^2+\cos\pra{\frac{\pi}{2}}\pra{\frac{\pi}{2}}^3+\sin\pra{\frac{\pi}{2}}\pra{\frac{\pi}{2}}^3\\ =&\pra{\frac{\pi}{2}}^3 \end{align} Is my solution correct?