why is facilitated diffusion not a form of active transport
Facilitated diffusion Facilitated diffusion (also known as facilitated transport or passive-mediated transport) is the process of spontaneous passive transport (as opposed to active transport) of molecules or ions across a biological membrane via specific transmembrane integral proteins.[1] Being passive, facilitated transport does not directly require chemical energy from ATP hydrolysis in the transport step itself; rather, molecules and ions move down their concentration gradient reflecting its diffusive nature.
Diffusion where   W  {\displaystyle W}  is intensity of any local source of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the no-flux boundary conditions can be formulated as   ( J ( x ) , ν ( x ) ) = 0  {\displaystyle (\mathbf {J} (x),\nu (x))=0}  on the boundary, where   ν  {\displaystyle \nu }  is the normal to the boundary at point   x  {\displaystyle x} .