new[{x_ , y_}]:= {(((x-y))/2) , (((x+y))/2)};
originalColor = {1.0 , 0.0 , 0.0};
newColor[c_]:= Reverse[c];
apply[t_][fun_][r_]:=(1 - t) r + t fun[r];
animate[new_][t_][x0_,y0_ ,Δx_ , Δy_]:= Module[{vx , vy , points},
points = Flatten[Table[{x , y} , {x , x0 , x0 + Δx , Δx/10.} , {y , y0 , y0 + Δy , Δy/10.}] , 1];
vx = D[new[{x , y}] , x]/.{x->x0 , y->y0};vy = D[new[{x , y}] , y]/.{x->x0 , y->y0};
Graphics[
{
{RGBColor[apply[t][newColor][originalColor]],Point/@(apply[t][new]/@points)},
{Black , 
Text["((Subscript[x, 0]),(Subscript[y, 0]))" , {x0 , y0}],
Text["((Subscript[x, 0])+Δx,(Subscript[y, 0]))" , {x0 + Δx , y0}],
Text["((Subscript[x, 0]),(Subscript[y, 0])+Δy)" , {x0 , y0 + Δy}],
Text["((Subscript[x, 0])+Δx,(Subscript[y, 0])+Δy)" , {x0 + Δx , y0 + Δy}],
Text["((Subscript[x, 0])',(Subscript[y, 0])')" , new@{x0 , y0}],
Text["(Δx((∂x')/(∂x)),Δx((∂y')/(∂x)))" , ((new@{x0 , y0} + vx Δx)+(new@{x0 , y0}))/2.],
Text["(Δy((∂x')/(∂y)),Δy((∂y')/(∂y)))" , ((new@{x0 , y0} + vy Δy)+(new@{x0 , y0}))/2.]
},
{Blue , 
Arrow[{new@{x0 , y0} , new@{x0 , y0} + vx Δx}],
Arrow[{new@{x0 , y0} , new@{x0 , y0} + vy Δy}]
},
{EdgeForm[{Red,Thick}],FaceForm[Opacity[0.]],Rectangle[{x0 , y0} , {x0 + Δx , y0 +Δy}]},
{EdgeForm[{Blue,Thick}],FaceForm[Opacity[0.]],Polygon[{new[{x0 , y0}] , new[{x0 , y0}] + vx Δx , new[{x0 , y0}] + vx Δx + vy Δy , new[{x0 , y0}] + vy Δy}]}
},PlotRange->All,AspectRatio->1 , ImageSize->Large , PlotLabel->Row[{"x'=",new[{x , y}][[1]] , " , y'=",new[{x , y}][[2]]}]]
];
Row[{Manipulate[
animate[new][t][x0,y0 ,Δx,Δy ] , {t , 0 , 1},{{x0 , 0.5} , -1 , 1},{{y0 , 0.5} , -1 , 1},{{Δx , 0.5} , 0.01 , 1} , {{Δy , 0.5} , 0.01 , 1}] , Manipulate[
animate[((1/3)#+Reverse[#])&][t][x0,y0 ,Δx,Δy] , {t , 0 , 1},{{x0 , 0.5} , -1 , 1},{{y0 , 0.5} , -1 , 1},{{Δx , 0.5} , 0.01 , 1} , {{Δy , 0.5} , 0.01 , 1}]}]