What does Fi stand for in mutivariable smooth function?

In the books, the smooth function's definition is: "F is smooth at a if all its component Fi is smooth at a". when Rn>RmF(x1,...,xn)=F1(x1,...,xn)+...+Fm(x1,...,xn)

I don't understand what does real number R presents in this definition. is Rn>Rm, the same as x is limited in the range [n,m]?

and How does the F(x1,...,xn)=F1(x1,...,xn)+...+Fm(x1,...,xn) comes up? And how to apply it with an example?

when writing F(x,y)= . Is F(x) the gradient level? (already derivate of f(x,y) ). What's more, I don't understand the way he get the value of Fi(-y and x) and the number of i=2.

The example I looked (and confused by) is attached below.

Is there a smooth function (e.g., with continuous second derivatives) f(x, y) such that Vf = xyi - 2(x + y)j?F: R- R FIX , y ) = Fi(x , y ) = F2 ( x , y )= O CS