math 3 lecture notes

math 3 lecture notes - Lecture Notes for MAS204: CALCULUS...

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Unformatted text preview: Lecture Notes for MAS204: CALCULUS III M.A.H. MacCallum These notes are based on independent previous versions by M.J. Thompson, with figures by C.D. Murray, and by P. Saha. School of Mathematical Sciences, Queen Mary, University of London September-December 2007 It is one of the most unnatural features of science that the abstract language of mathematics should provide such a powerful tool for describing the behaviour of systems both inanimate, as in physics, and living, as in biology. Why the world should conform to mathematical descriptions is a deep question. Whatever the answer, it is astonishing. Lewis Wolpert (1992) Students on the course MAS204 Calculus III are welcome to download, print and photocopy these notes, in whole or in part, for their personal use. The notes are intended to supplement rather than to replace the lectures. c circlecopyrt M.A.H. MacCallum 2007, 2006, 2001, P. Saha 2005, M. J. Thompson 1999. Chapter 1 Introductory material This chapter gives a quick review of some of those parts of the prerequisite courses (Calculus I and II and Geometry I) which we will actually use, adding some extra material. Those parts which are revision will be done with few examples. 1.1 Curves and surfaces We shall need a number of geometrical shapes for use in examples, so we need the equations for them. The main ones are so-called conic sections in two dimensions, and related three-dimensional surfaces. First we discuss curves in two dimensions. You should already know that x 2 + y 2 = a 2 (1.1) is the equation of a circle centred at the origin, ( , ) , and radius a (Thomas, 1.2). Given x 2 + 6 x + y 2 + 8 x = we can carry out a process called completing the square to write it as ( x + 3 ) 2 + ( y + 4 ) 2 = 25 which we now recognize as a circle radius 5, centre ( 3 , 4 ) : this circle passes through the origin. Similar methods can be used to recognize the other standard curves if they are given relative to origins different from the ones used in the most standard forms below (cf. Thomas 1.5). The equation x 2 a 2 + y 2 b 2 = 1 (1.2) is the equation of an ellipse (a sort of squashed circle), centred at ( , ) and with semi-major axes a and b (Thomas 1.5). The equation y = x 2 (1.3) 1 is the simplest form of the equation of a parabola. A somewhat more general form (to which all others can be transformed by change of coordinates) is y = ax 2 + b . One standard form of the equation of a hyperbola is y 2 x 2 = a 2 , (1.4) or more generally cy 2 kx 2 = a 2 , (1.5) where c > 0, k > 0. Another is xy = b 2 . One can see the relationship between xy = b 2 and (1.4) by noting that ( y + x ) 2 ( y x ) 2 = 4 xy so xy = b 2 is using axes at 45 to those used in (1.4) (and 4 b 2 = a 2 )....
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math 3 lecture notes - Lecture Notes for MAS204: CALCULUS...

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