The Equals Sign in Gauss' Law
The Equals Sign (=)
The Equals Sign (=) in Equation 1, or Gauss' Law, means the quantities on the left and right side are identical. This may sound trivial, but what exactly are the quanties on either side of the equals sign?
The Equality Sign in Gauss' law is a scalar equals - which means the quantities on either side are single numbers. This is in contrast to the equalities in Ampere's or Faraday's Law, where the equality is a vector equality.
So we have equality of a scalar quantity, but it should also be noted that the equality forces not just the values to be equal, but also the units. And the units of Gauss' Law are Coulombs per meter cubed [C/m^3]. This is electric charge per volume.
Hence, if the divergence of the electric flux density at any point is +5 Coulombs per cubic meter, then the electric charge volume density must also equal +5 Coulombs per cubic meter.