Advanced Calculus fi..

198 Advanced Calculus, Fifth EditionA being a positive constant scalar. Thus we have changed scale in the ratio A : 1,and l/A is our new unit of distance. For a point moving on a path we have hithertoassigned a velocity vector v = (v, , v,, v,) and thought of this as a definite geometricobject, represented by a directed line segment. But the components of v, as a velocityvector, depend on the coordinate system; that is, here, where we are considering achange of scale, these components depend on the unit of distance chosen. In the(x, y, Z) coordinates we would assigndxUx = - vydy d 2= - Uz = -dt ' dt ' dt 'but in the (f, J, Z) coordinates we would assignr sBy virtue of Eqs. (3.69), dfldt = A dxldt, so that ex = Av,, and in general,Thus in the new coordinates (f , J , Z) we assign to v the new components (Ex,@,, U,),which are h times the previous components.Now a vector v can also be obtained as the gradient vector of a function f:v = grad f, so thatv,a fa f a f= -, vy=-, vz=-.ax ay a2If we change scale by (3.69), we would still like v to be the gradient off. But in thenew coordinates, f becomes f (f/A, Y/A, .?/A) = f(f , Y, Z), and this has gradientHence nowThus when we change scale by (3.69), the three components of the gradient vectorare divided by A. This result is to be expected, since the gradient vector measuresthe rate of change of f with respect to distance (in various directions), and we arechanging the unit of distance. If for example, A = 2, then the new unit of distanceis half the old one, and the amount of change in f per unit of distance is half asmuch as before, whereas velocity components are twice as large as before, since onecovers twice as many units of distance per unit of time.The two different rules (3.70) and (3.7 1) show that we really have two differentways of assigning components to vectors. In the case of (3.70), one is dealing witha "contravariant vector," and in the case of (3.71), one is dealing with a "covariantvector." In obtaining the two types of components we have emphasized the changein unit of distance. However, in changing coordinates we can always keep in mindthe original unit of distance (as a standard of reference), so that all distances canultimately be restated in the original units. This concept of a standard unit of distancewill be important in the development to follow.