ClearAll["Global`*"]
u[x_, y_] := {f[x, y], g[x, y]}
Div[u[x, y], {x, y}]

f^(1,0)(x,y)+g^(0,1)(x,y)

(f^(1,0)(x,y)+g^(0,1)(x,y))+f^(1,2)(x,y)+f^(3,0)(x,y)+g^(0,3)(x,y)+g^(2,1)(x,y)