Vector analysis; a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs ... - Tapa blanda

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1909 Excerpt: ...by PotFkv,,, The first partial derivative of the potential with respect to xt is therefore The value of this limit may be determined by a simple device (Fig. 36). Consider the potential at the point (x1 + Axv yv z) due to a certain body T. This is the same as the potential at the point (.xv yv z1 FlG-36 due to the same body T displaced in the negative direction by the amount A xv For in finding the potential at a point P due to a body T the absolute positions in space of the body T and the point P are immaterial. It is only their positions relative to each other which determines the value of the potential. If both body and point be translated by the same amount in the same direction the value of the potential is unchanged. But now if T be displaced in the negative direction by the amount A x, the value of V at each point of space is changed from xv yv 2a), where A xa = A xv Axtr=QJ J J M ru J J J M rls 9 xi 2 when A a;i approaches zero as its limit the regions mand m', which are at no point thicker than A x, approach zero; M' and M both approach if as a limit. t There are cases in which this reversal of the order in which the two limits are taken gives incorrect results. This is a question of double limits and leads to the mazes of modern mathematical rigor. If the derivative of Kis to exist at the surface bounding T the values of the function V must diminish continuously to zero upon the surface. If Pchauged suddenly from a finite value within the surface to a zero value outside the derivative 3 VI 3 i1 would not exist and the triple integral would be meaningless. For the same reason V is supposed to be finite and continuous at every point within the region T. Then if it be assumed that the region T is finite and that V vanishes upon the surface bounding T...