Consider the vector field
$$F=(x^2+y^2,3xy).$$ Compute the line integrals $$\int_{c_1}F⋅dr \text{ and } \int_{c_2}F⋅dr,$$ where $$C_1(t)=(t,t^2) \text{ and } C_2(t)=(t,t)$$ for $0≤t≤1$. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?
I tried substituting the parameters of $C_1$ with $x$ and $y$; $x = t, y = t^2$. Then performing the dot product of $F$ with the derivative of $C_1$ from $0$ to $1$, but that didn't work. $$\int_0^1F⋅dt = \int_0^1[(t^2 + t^4), 3t^3] \cdot [1, 2t] dt$$ I know that F is not conservative because they will not equal each other because they will be different integrals under different paths. I don't understand what to do with the $C_1$.