# Confusion concerning the application of Gauss's law using symmetry arguments

Suppose we want to prove Coulomb's law from Gauss's law. The thing to do is to invoke a spherical Gaussian surface centered at a point charge, notice that there's symmetry and use that to calculate $$\int \boldsymbol{E}\cdot d\boldsymbol{A}$$. My concern is that while it's clear that the field must point in the radial direction, how is it that we know (using a rigorous argument) that it points radially outward for a positive charge and radially inward for a negative charge?

• There's a related question at physics.stackexchange.com/q/183164 Plus the fact that the convention for closed surface integrals is that $d\mathbf{A}$ is an area element multiplied by an outward-pointing unit vector, but I presume that's not your worry.
– user197851
Oct 30 '18 at 16:22