Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 177Figure 3.3 Example 1. ~IW5Figure 3.4 Example 2.that is, each vector field is equivalent to a triple of scalar functions of the threevariables x, y , z.dLet v = xi + yj + zk. Here v = 0 at the origin; at other points, vEXAMPLE 1is a vector pointing away from the origin, as illustrated in Fig. 3.3.EXAMPLE 2 Let v = - yi + x j. Here the vectors can be interpreted as the velocityvectors of a rigid rotation about the z axis. This is illustrated in Fig. 3.4.EXAMPLE 3 Let F denote the gravitational force exerted on a particle of mass mat P: (x, y, z) by a mass M concentrated at the origin. Newton's law of gravitationgivesMmraF = -k--,* - Ir2 rwhere r = 3 and k is a universal constant. Here r/r is a unit vector, so F has