a). $$F(x,y,z)= (zy )i +(y^3+xz) j + (1/z+xy) k$$ and $y$ is the quarter circle with center $(1,1,1)$ going from $(1,0,1)$ to $(0,1,1)$.
I know that I need to get a parameterization for the function, however I don't really know where to head next
I am kind of at a loss for solving this line integral
can anyone help me with this?
On a neighbourhood of the prescribed path this vector field is conservative as its curl vanishes. Try finding a scalar field $f$ such that $F=\nabla f$ and then the integral becomes $f(B)-f(A)$ where $B$ is the terminal point and $A$ is the initial point of the path.
To obtain a scalar field $f$ with $F=\nabla f$, we want $$\begin{cases}\frac{\partial f}{\partial x}=F_1=zy\\\frac{\partial f}{\partial y}=F_2=y^3+xz\\\frac{\partial f}{\partial z}=F_3=\frac{1}{z}+xy\end{cases}$$ so we get $$\begin{cases}f(x,y,z)=\int zy~dx+h_1(y,z)=xyz+h_1(y,z)\\f(x,y,z)=\int y^3+xz~dy+h_2(x,z)=\frac{1}{4}y^4+xyz+h_2(x,z)\\f(x,y,z)=\int \frac{1}{z}+xy~dz+h_3(x,y)=\ln(z)+xyz+h_3(x,y)\end{cases}$$ which gives $f(x,y,z)=xyz+\frac{1}{4}y^4+\ln(z)$.