The graph of the function y= f(x) is a curve in with points denoted by (x, y). The graph of the function z= f(x,y) is a surface in with points denoted by (x, y, z). The graph of the vector function (u,v)= f(x,y) is in with points denoted by (x, y, u, v).

But you are right about evaluating F at the "given point": at , , , NOT (0, 0) as implied by . Are you sure you have copied that correctly? If it were f(x,y)= (sin(x-y), cos(x+y)) then would be correct.

If we think of the curve y= f(x) as a "level curve" of F(x,y)= y- f(x)= 0, then the two dimensional vector is normal to the curve. If we think of the surface z= f(x,y) as a "level surface" of F(x,y,z)= z- f(x,y), then the three dimensional vector is normal to the level surface. Finally, if we think of the graph (u,v)= f(x,y) as a level curve of , then will be normal to it.