I am trying to find a loop integral of a vector field $F$ on a closed curve but I have no idea how to show it since the polynomials are not specified.
Given that polynomials $P(x)$, $Q(y)$ and $R(z)$ and $$F(x,y,z) = f(|r|)(P(x),Q(y),R(z))$$
where $|r| = \sqrt{x^2 +y^2+ z^2}$ and $f: \mathbb{R}_{+} \rightarrow \mathbb{R} $ is any smooth function.
The aim is to find a loop integral of $F$ evaluated on any simple closed curve $\gamma$ on the sphere $S_{R}^{2}$ and $R>0$ be arbitrary.
Let $\gamma$ be a closed curve on the sphere $S^2_{R_0}$ for some arbitrary $R_0 > 0$ (I write $R_0$ to distinguish this radius from the polynomial named $R$). Let $k = f(R_0).$ Note that $k$ is a simple number that can never change unless we change the value of $R_0$ (and hence have a different sphere and presumably also a different curve).
Then for any point $(x,y,z)$ on the curve $\gamma,$
$$ F(x,y,z) = k(P(x), Q(y), R(z)).$$
Hence the integral of $F(x,y,z)$ over $\gamma$ is the same as the integral of $k(P(x), Q(y), R(z)).$ But $$ k(P(x), Q(y), R(z)) = k(P(x), 0, 0) + k(0, Q(y), 0) + k(0, 0, R(z)). $$
Integrate the three functions on the right-hand side separately over $\gamma,$ then add the results.