Advanced Calculus fi..

Vector DifferentialCalculus3.1 INTRODUCTION+ 8 lt5aIn Section 2.13 the path of a moving point P is described by giving its positionvector r = ?$ as a function of time r. It is shown how this vector function can bedifferentiated to give the velocity vector of the moving point. This operation canproperly be termed a part of vector differential calculus.In this chapter there is again a natural physical model-a fluid in motion. Ateach point of the fluid, one has a velocity vector v, the velocity of the "fluid particle"located at the given point. Such vectors are defined for all points of the fluid andtogether form what is termed a vectorjield. This is illustrated in Fig. 3.1. The fieldmay change with time or remain the same (stationary flow).One can again trace the paths of the individual particles, the "stream lines," anddetermine the velocity vector v = drldt for each. However, one can also considerthe velocity field at a given time as describing a vector v, which is a function ofx, y, and z, that is, of position in space. The vector function v(x, y, z) can thenbe differentiated with respect to x, y, and z; that is, one can consider the rate andmanner in which v varies from point to point in space.The description of the variation of v turns out to require not merely partialderivatives, but special combinations of these, the divergence and curl. At eachpoint of the field a scalar, div v, the divergence of v, and a vector, curl v, will bedefined. The divergence measures the net rate at which matter is being transportedaway from the neighborhood of each point, and the conditiondiv v E 0