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chapt01 - CHAPTER 1 VECTOR ANALYSIS Vector analysis is a...

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CHAPTER 1 VECTOR ANALYSIS Vector analysis is a mathematical subject which is much better taught by math- ematicians than by engineers. Most junior and senior engineering students, how- ever, have not had the time (or perhaps the inclination) to take a course in vector analysis, although it is likely that many elementary vector concepts and opera- tions were introduced in the calculus sequence. These fundamental concepts and operations are covered in this chapter, and the time devoted to them now should depend on past exposure. The viewpoint here is also that of the engineer or physicist and not that of the mathematician in that proofs are indicated rather than rigorously expounded and the physical interpretation is stressed. It is easier for engineers to take a more rigorous and complete course in the mathematics department after they have been presented with a few physical pictures and applications. It is possible to study electricity and magnetism without the use of vector analysis, and some engineering students may have done so in a previous electrical engineering or basic physics course. Carrying this elementary work a bit further, however, soon leads to line-filling equations often composed of terms which all look about the same. A quick glance at one of these long equations discloses little of the physical nature of the equation and may even lead to slighting an old friend. Vector analysis is a mathematical shorthand. It has some new symbols, some new rules, and a pitfall here and there like most new fields, and it demands concentration, attention, and practice. The drill problems, first met at the end of Sec. 1.4, should be considered an integral part of the text and should all be 1

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worked. They should not prove to be difficult if the material in the accompany- ing section of the text has been thoroughly understood. It take a little longer to ``read'' the chapter this way, but the investment in time will produce a surprising interest. 1.1 SCALARS AND VECTORS The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number. The x ; y , and z we used in basic algebra are scalars, and the quantities they represent are scalars. If we speak of a body falling a distance L in a time t , or the temperature T at any point in a bowl of soup whose coordinates are x ; y , and z , then L ; t ; T ; x ; y , and z are all scalars. Other scalar quantities are mass, density, pressure (but not force), volume, and volume resistivity. Voltage is also a scalar quantity, although the complex representation of a sinusoidal voltage, an artificial procedure, produces a complex scalar , or phasor , which requires two real numbers for its representation, such as amplitude and phase angle, or real part and imaginary part.
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