To every scalar field s(x,y) there corresponds a 'constant' vector field x = A s(x,y) and y = B s(x,y), where A,B are direction cosines. The vector field is only partially constant since only the directions, and not the magnitudes, which are equal to |f(x,y)|, of the field vectors are constant. The scalar field corresponds to the magnitudes of a vector field, that can be specified by only the magnitudes of the field vectors since the directions are constants A,B. Is this correct? This came up when evaluating a line integral f . dr , and splitting it up in the dx,dy components of dr. Can the dot product of a scalar field and a vector field define a scalar field as well as a vector field (since a(x,y)=(ax,ay))?