Let be f and g scalar functions on $\mathbb{R}^3$ and $f_{{x}_i}$ denotes partial differentiation of f with respect to the i-th Cartesian coordinate, when this equality is true:
$$\int_{S^2(R)} f_{x_{i}}g(x)dx=-\int_{S^2(R)} f(x)g_{x_{i}}dx$$
where $S^2(R)$ is the sphere of radius $R$ centered at origin.
What conditions does it true for f and g??