By B. Hague D.SC., PH.D., F.C.G.I. (auth.)

The significant adjustments that i've got made in getting ready this revised version of the publication are the subsequent. (i) Carefuily chosen labored and unworked examples were extra to 6 of the chapters. those examples were taken from category 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 software of veetors has been integrated in bankruptcy 1. (iii) Chapters four and five were mixed into one bankruptcy, a few fabric has been rearranged and a few extra fabric further. (iv) The bankruptcy on int~gral theorems, now bankruptcy five, has been increased 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(v) the one significant swap made in what are actually Chapters 6 and seven is the deletioo of the dialogue of the DOW out of date pot funetioo.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 provide a fuller account of using scalar and veetor potentials in eleetromagnetic concept, and the devices hired were replaced to the m.k.s. system.

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Find also a unit veetor perpendicular to the plane containing these two veetors. 8. Find the area of the parallelogram the diagonats of which are the veetors 3i + j - 2k and i - 3j + 4k, i, j, k being the usual mutually perpendicular unit veetors. 9. Find a veetor r for which (r x a) + r = b, where a, b are given veetors. 10. b) e. 11. a)w; (iii) U x v = ra, b, el a. 12. The veetors u, v, w are non-zero. Show, by taking the veetor product of each side with u, or otherwise, that the general solution of the equation UXy=uxw in v is v = AU + w, where Ais a sealar.

The converse of this result that eurI grad S = 0 will be considered in Chapter 5. 11. The Operator grad div. If V is a veetor field, div V is a scalar field, which, therefore, has a gradient.

Show that the veetors r, f, r are eo-planar if r = 0 when t = 0 and are parallei if, in addition, f = 0 when t = O. 5. Find the derivative with respect to t of [r, t, rl. 6. If u is a funetion of the parameter t and if u and ü are unit veetors, show that u x (ü x ü) = -u. 7. u and vare funetions of t and du dt 4 = ca) x u, dv dt = ca) x v, 40 VEeToa ANALYSIS where w is a given veetor. Show that ;, (il x v) =w x (u x v). 8. u is a function of the time t and u x Il a fixed direction. = O. Show that u has 4: The Operator V and Its Uses 1.