By B. Hague D.SC., PH.D., F.C.G.I. (auth.)
The vital adjustments that i've got made in getting ready this revised variation of the e-book are the next. (i) Carefuily chosen labored and unworked examples were extra to 6 of the chapters. those examples were taken from classification and measure exam papers set during this college and i'm thankful to the college courtroom for permission to exploit them. (ii) a few extra subject at the geometrieaI program of veetors has been included in bankruptcy 1. (iii) Chapters four and five were mixed into one bankruptcy, a few fabric has been rearranged and a few additional fabric extra. (iv) The bankruptcy on int~gral theorems, now bankruptcy five, has been elevated to incorporate an altemative evidence of Gauss's theorem, a treatmeot of Green's theorem and a extra prolonged discussioo of the category of vector fields. (v) the single significant switch made in what at the moment are Chapters 6 and seven is the deletioo of the dialogue of the DOW out of date pot funetioo. (vi) A small a part of bankruptcy eight on Maxwell's equations has been rewritten to offer a fuller account of using scalar and veetor potentials in eleetromagnetic conception, and the devices hired were replaced to the m.k.s. system.
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Additional info for An Introduction to Vector Analysis For Physicists and Engineers
50). Now examine the motion of a fluid and eonsider what may happen to an infinitesimal element within it. The element ean have three kinds of motion simultaneously. (i) It may be moving with a linear velocity of translation as a whole. (ii) If the fluid is deformable it may ehange its shape. (iii) It may be in rotation. At any instant the little element may be regarded as a rigid body; the eurl of the veloeity of the fluid at the point where the element is situated gives twice its angular velocity.
As an example, we consider the area ahed in fig. 27(b) with sides dx, dy, its normal being along the axis of Z. Since the reetangle is very small, the value of the component of V at the middle of any side may reasonably be taken as the average value along that side; the arrows show the directions in which the components act. Since Vz, V", Vz are funetions of the co-ordinates (x, y, z) of the middle of the rectangle, the average values along the four sides ah, be, de, ad, 51 THE OPERA TOR V AND ITS USES respectively, are VII 1 oVII +i ax- dx , The line integral around the contour abcd is, therefore, [( V" 1 oV" dx) + 'ia; 1 oV" )] dy - ( V" - iax-dx + x x [( Vx - "21 oV dy ) - ( Vx + "21 oV dy )] dx, ay ay that is, _ OVx) dx dy.
The Divergence of a Vector Field. In fig. 26 let V be the veetor at the centre of an infinitesimal element of volume with sides dx, dy Vt V, - I el + .!.. elV, dz 2 ell v, 2" äi dy I~ v,+! el, dy I av, Vz -zazdl Figure 26 Divergence of a veetor point function and dz parallei to the axes of x, y and z. The veetor V has components Vx , Vy and Vz in the direetions of these axes. To fix his ideas the reader may think of V as a veetor giving the velocity of a moving fluid in magnitude and direetion.