Advanced Calculus fi..

magnitudeand the force is inversely proportional to the square of the distance.The notion of vector field can be specialized to two dimensions. Thus a vectorfield v in a domain D of the xy-plane is given byv = v,(x, y)i + v,(x, y)j, (3.3)where v,(x, y) and v,(x, y) are two scalar functions of x and y defined in D. Twodimensionalfields arise in applications mainly in connection with planarproblems,that is, problems concerning a vector field v in space such that v is always parallel tothe xy-plane and v is independent of z; that is, v, = 0 and v, and v, depend only onx and y, as in (3.3). From (3.3), one can construct the vectors v first in the xy-planeand then, using exactly the same vectors, in any plane parallel to the xy-plane.iEXAMPLE 4Let F be the fieldThis is illustrated in Fig. 3.5 and can be interpreted as the electric force field due totwo infinite straight wires, perpendicular to the xy-plane at (1, 0) and (- 1, 0), homogeneouslyand oppositely charged with electricity.aYFigure 3.5 Example 4.