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

The crucial adjustments that i've got made in getting ready this revised version of the booklet are the next. (i) Carefuily chosen labored and unworked examples were further 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 collage court docket for permission to take advantage of them. (ii) a few extra topic at the geometrieaI software 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 further. (iv) The bankruptcy on int~gral theorems, now bankruptcy five, has been increased to incorporate an altemative facts of Gauss's theorem, a treatmeot of Green's theorem and a extra prolonged discussioo of the class 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significant switch made in what at the moment are Chapters 6 and seven is the deletioo of the dialogue of the DOW out of date pot funetioo.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 provide a fuller account of using scalar and veetor potentials in eleetromagnetic idea, and the devices hired were replaced to the m.k.s. system.

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Example text

Now take a elosed path, eonsisting of path 1 from A to B followed by path 2 from B to A. ds = O. 5) Thus, when a veetor field can be expressed as the gradient of a scalar field, the line integral of the veetor taken between two points is independent of the path followed and is equal to the difference between the values of the scalar at its ends; further, the line integrai round any elosed path in sueh a veetor field is zero. A veetor field Vs derived from a scalar S by the relation Vs = grad S is sometimes called a sealar potential field, S being the potential of Vs.

Positive total ftux diverges from the endosed volume while negative ftux converges up on it. Should the amount of flux entering by some elements be balaneed by that leaving by other elements in such a way that the total ftux is zero. then either there are no sourees or sinks of ftuid within the endosed volume or the sum of their strengths is zero. Similar ideas apply to other ftuxes. g. of electric or magnetic induction. of heat, etc. We now work out a few miscelIaneous examples on veetor algebra and its applications.

3. partial Difl'erentiatiOll. The simple properties of difl'erentiation as applied to veetors in §§ 1 and 2 can be extended to partial derivatives when a veetor is a function of more than one independent scalar variable. e. of a vector field. If y and z remain constant while x increases, the partial derivative oV/ox denotes the rate of increase of V with respeet to x. Likewise changing y and z alone we obtain the partial derivatives õV/oy and õV/oz, denoting the rates ofincrease with respeet to y and z, respeetively.

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