Ex.1021 p.1
`
`

`
`CALCULUS
`
`AND
`
`ANALYTIC
`
`GEOMETRY
`
`FOURTH EDITION
`
`GEORGE B. THOMAS, JR.
`Dcpurlmcnt of jfalhz-1na.’ic.s
`J!assac.5.usc!£s Insfilnfe of Technology
`
`A
`VV
`
`ADDISON-WESLEY PUBLISHING COMPANY
`Re:=.ri:ng,
`I\I;1.~'s.1:lL'.|.~"ul‘.s
`Menln Park, Culifurniu - L-;-rxrir‘-n - Dnn Milisz 0nL'1ri0
`
`Ex.1021 p.2
`
`

`
`This book is in the
`Addison-Wesley Series in Mathematics
`
`Second printing, December 1969
`
`Copyright © 1968, Philippines copyright 1968 by Addison-
`Wealey Publishing Company, Inc-. All rights reserved. No
`part of
`this publication may be reprodiiced, stored in a
`retrievzfl system, or t.r.<msmi1.ted,
`in any form or by any
`means, electronic, mechanical, pholrrcopying, recording, or
`utherwise, without the prior written permission of the pub-
`lisher. Printed in the United States of America. Published
`simultaneously in Canada.
`Librs.r_v of Congress Camlng
`Card No. 68-17568.
`
`Ex.1021 p.3
`
`

`
`11.1 THE POLAR COORDINATE SYSTEM
`
`We know that a point can be located in 3 plane by
`giving its abscissa and ordinate relative ‘to at given
`coordinate system. Such r~ and y-coordinates are
`called Cartesian coordinates, in honor of the French
`niathematician-philosopher Pic-né Descartes‘ (1.396-
`1650), who is credited with discovering this method
`of fixing the position of a. point in :1 plane.
`
`/,nP|.' 6'2
`
`‘.
`.
`/ Y.
`0.:
`
`-T T
`
`Another useful way to locate ‘d. point in 3 plane
`by polar coordinaJ.'es -{see Fig. 11.1). First. we fix
`an origir. 0 and an zfnitial rayf from 0. The point P
`has polar coordinates r, 6, \\'lLl\
`
`= directed distance from 0 to P,
`
`(la)
`
`and
`
`0 = directed angle from initial ray to OP.
`
`(lb)
`
`the angle 6 is po.sit'it:e- when
`As in trigonoinetry.
`measured counterclockwise and negative when mea-
`sured clockwise (Fig. 11.1}. But the angle associated
`ith a. given point ‘: not unique (Fig. 11.2]. For
`instance, the _D0lI‘.L 2 units from the origin. along the
`ray 6 = 30°, has polar coordinates r -= ‘3_, 6 -= 30°.
`It also has coordinates r =— '2. = —33()"‘. or r = 2,
`6 = 390°.
`There are occnsions when we wish to allow r to be
`negative.
`'l‘liat’s why we
`"directed distance"
`
`“ For an interesnng biographical account together wizh
`an excerpt from Descartes’ mvn WI'lt1I)g>. see World of
`.\Ir.:tI'.~<~rr.c.!t'cs, Vol. 1, pp. 235-253.
`T.-X ray is a. linlf-line c0:'1sisLir.5.’, of a vertex and points
`of A line on one side of tlzo vr-rtex. For vxnmzgle, the
`origin and positive z—nx.is is 9. my. The points on 1112': line
`y = 2.: -— 3 with: Z 1 is anotherray; 1'-.S's"&1’Tv€X is L1. 53-
`
`CHAPTER 11
`
`Ex.1021 p.4
`
`

`
`.335’ :oordln¢|es
`
`11.2 T:.- 5' = 30° is the same as the ray :9 = -—-330°.
`
`__, 0 = 13:)"
`
`IL: The rays 9 = 30° and 9 = 210° make a line.
`
`7 l‘ r_ -2, 3o°;.
`
`11.4 The terminal ray 6 = -.r_.»‘6 and its negative.
`
`in Eq. (la). The ray 9 = 30° and the ray 5 = 210°
`together make up 3. complete line through 0 (see
`Fig. 11.3). The point P(2. 210°) 2 units front 0 on
`the ray 6 = 210° has polar coordinates r = 2,
`9 = "10°.
`It can be reached by 5: person standing
`at 0 and facing out along the initial ray, if he first
`turns 210° counterclockwise, and then goes forward
`
`2 units. He would reach the same point by tumin;
`only 30" countercioci<\vise :'rc-111 the initial my ani
`than going !5czck-.z'arrz'
`'2 units. So we say tizzrt.
`tl‘-’
`point, also has polar coord‘.n-ates r = ——:2. 0 = 30°.
`Whenever the angle betwcezt two rays is 180°, tr.-,
`rays atztvually m:i‘.~:e a straight line. We then s:J._v the.’
`either rag:
`is the negative of the other. Points If’
`the ray 9 = u have polar coordinates -;.~'_, oi)
`\\"‘."
`r 2 0.
`l’o‘.nts on the nega~‘.i\‘c mgr.
`I. = a — 181:”
`have coordinates 1:2-. Ct:|
`\'.'itl'1 r 3 0. The origin
`r = 0.
`(See Fig. 11:} for the I‘£'.}‘ 6 = 30° and
`no:-g;ative.. A word of caution: The "negative"
`the ray 6 .— 30‘ is the ray 64
`.— 30°
`180° = 21*"
`and not the rag.‘ 6 -— -30°.
`“.\'eg:‘~.tive" refers to 1':
`directed distance r.'J
`There is a great atlvmzzagc in being able to us-
`both polar and Cartesian coordinates at once.
`T
`do this. we use :3 common origin and take the ini1._
`my as the positive .r-axis, and take the ray 9 = E4 '
`as the positive y-axis. The coordinates, shovm
`Fig. 11.5, are then related by the equations
`
`x=rcos0, y=:-sins.
`
`;
`
`These are the equations that define sin 5 and Cu: ~
`when r is positive. They are gist: valid if .~‘ is net.-
`tive, because
`
`cos ;9 + 180°) —«
`
`cos 9,
`
`sin ('3 -- 180°] = —sin 9.
`
`so positive r’.< on the [9 — 180°)-ray corrcsponti '.
`negative -r’s associated with the 0-ray. When -r = .
`then .7: — y = 0. and P is the origin.
`If we impose the condition
`
`.-- —— a
`
`{a constant}-.
`
`then the locus of P is st circle with center 0
`radius a. and P describes the circle once as 6 \‘{1.''''.‘
`from 0 to 360° (see Fig. 11.6). Or. the other h:-.;_.
`if we let r vn.r_v and hold 6 fixed, say
`9 = 30¢.
`
`the locus of P is the straight line shown in Fig. 12 -.
`
`"
`
`Polar and C
`
`6 The circle r
`
`7‘~'e adopt the c
`;i£;'.bC1" .._w < ,
`
`ti-:_~= 0!y=0i
`T
`
`‘c ;e origin, :5 =
`_ _.'.-ie same poin
`tterenl‘. ways in
`lr point
`(2, 30°)
`'
`resent-ations:
`(
`* L‘. ~—]50°). Th
`- '46 two formul
`
`(2, 30° + n .
`‘-2, 210° -1- n
`
`- _’ we represent
`_ulas
`
`(2, %7r+
`i? +
`
`Ex.1021 p.5
`
`

`
`I
`
`The polar coordinate system
`
`The fact that the same point may be represented
`in several dificrent ways in polar coordinates makes
`added care necessary in certain situations.
`For
`example, the point (2a, 1r) is on the curve
`
`r =4a3cos6
`2
`
`(6)
`
`even though its coordinates as given do not satisfy
`the equation, because the same point is represented
`by (-20, 0) and these coordinates do satisfy the
`equation. The same point (241, 1r) is on the curve
`
`r = a(1
`
`cos 0).
`
`(7)
`
`and hence this point should be included among the
`points of intersection of the two curves represented
`by Eqs. (6) and (7). But if we solve the equations
`simultaneously b_v first substituting cos 0 = r’/4a’
`from (6)
`into (7) and then solving the resulting
`quadratic equation
`
`for
`
`(2)345)-4=°
`
`7'
`U
`
`= -2-.é.-2\/5,
`
`(8)
`
`11.6 The circle r - a is the locus P.
`
`We adopt the convention that 1' may be any real
`zumber, —:r. < r < -r.. Then r = 0 corresponds
`‘.01’ = 0.. y = 0 in Eqs. (2), regardless of 0. That is,
`
`r = 0,
`
`0 any value.
`
`(5)
`
`is the origin, 2: = 0, y = 0.
`The same point ma}-‘ be represented in several
`dilferent. ways in polar coordinates. For example,
`‘he point
`(2, 30°), or (2,,7r,x'6), has the following
`representations:
`(2, 30°),
`(2, —330°),
`C -2, 210°‘.
`-2, -150’). These and all others are summarized
`:n the two formulas
`
`(2, 30° + 7:. 360°),
`(-2, 210° —- n 360°),
`
`}n=0,:i:1,i2,...;
`
`or, if we represent the angles in radians, in the two
`formulas
`
`we do not obtain the point (2a, 1r) as a point of inter-
`section. Thc reason is simple enough: The point is
`not on the curves “simultaneously " in the sense of
`being reached at the “same time,” since it is reached
`in the one case when 6 = 0 and in the other case
`
`It is as though two ships describe
`when 6 = 1r.
`paths that intersect at 3. point, but _the ships do not
`collide because they reach the point of intersection
`at different
`times!
`The curves represented by
`Eqs. (6) and (7) are shown in Fig.
`l1.9(c). They
`are seen to intersect at the four points
`
`l
`
`1
`
`(0:
`
`(20, 1|’),
`
`(T1: 01).»
`
`(T1: —01)i
`
`(98)
`
`where
`
`r, = (-2 — 2\/§)a,
`*1
`cos91=1—;=3-—2\/§.
`
`(gb)
`
`mr),
`1r
`(2,
`-4 2
`2;
`(—2,§1r—{-Qmr),
`
`"T:
`
`O
`
`‘L
`.
`'=1**2
`
`....
`
`Only the last two of the points (911) are found from
`the simultaneous soltition; the first two are disclosed
`only by the graphs of the curves.
`
`Ex.1021 p.6
`
`

`
`Vectors and parametric equations
`
`Then, applying (3). We have
`
`and
`
`Therefore
`
`1605
`
`( .311’)
`,
`isin9—j cos 0.
`
`, ..
`
`.
`
`£7‘ a out + (a0_‘:u2
`= a{i cos0+jsin 01- afiii sin 9 -— j cos 9}
`= a (cos 9 + 9 sin 9)i -1- a sin 0 — 6 cos 93-].
`
`We equate this with zie yj and, since corresponding
`components. must be equal, we obtain the parametric
`equations
`
`2: = a (cos0-,- Bsin 0},
`y=cr[sir\.9—9cos9).
`
`6
`K
`
`'.
`
`EXERCEEEE.
`
`In Exercises 1 through 10, express each of the vectors in
`the form ai + bj.
`Indicate all quantities graphically.
`
`1. PT;, if P1 is the point :11, 3) and P2 is the point
`(2. -1)
`
`« 2. E if 0 is the origin and P3 is the midpoint of tilt:
`vector 1% joining [’;(2, -1) and P22} —4, 3}
`. The vector from the point .-1-‘.2, 3} to the origin
`
`The sum of the vectors I5‘ and (.7), given the four
`points .40, —-1), B(2, 0),
`(-1, 3}, and D-(-2, 2)-
`A unit vector making an angle 01' 30° with the posi-
`tive 2-axis
`
`. The unit vector olntained by rotatingj through 120°
`in the clockwise direction
`
`. A unit vector having the same direction as the
`vector 3i — 4j
`. A unit vector tangent to the curve :41 = x’-’ at the
`point (2, ~1_‘.-
`. A unit. vector normal to the curve y = :9 at the
`point I’:I2. 4} and pointing from P toward the con-
`cave side of the curve {that is, an "inner" normal)
`
`3
`
`.
`
`_
`
`to the involute of 9. circ:
`10. A uni: vector tangent
`whose parametric equations are giver. in Eq. (6)
`
`Find the lengths of each of the following vectors and
`angle that each makes with the positive I-axis.
`
`ll. '
`
`14.
`
`'
`
`'
`
`12. 2i — 31'
`
`13. V3 i +5
`
`.'
`
`15. 5i4 l2j
`
`16. —5i — 12;
`
`17. Use vcctor methods to determine parametric eq';.=r
`tions for :he trochoid of Fig. 12.6, by taking
`
`R = 07’ = ofi + R — C7’.
`
`'
`
`. Lot :1, B. C‘, D be the vcrtices, in order, of a quadr-
`lateral. Let
`.-1’, B’, C’. D’ be the midpoints of 1":
`sides AB, BC, CD, and DJ, in order. Prove Il1';‘
`.-l’B’C’D’ is a parallelogram‘.
`Hm. First show that TB’ = 5?’ = $.40.
`. Ilsitzg vectors. show that :he diagonals 0:‘ a parallel.-
`gram bisect each other.
`.l:'e.!koa'. Let
`:1 be one vertex and let .1! and N he
`the midpoir.-ts of the diagonals. Then show Ilia‘
`= _a.\'.
`
`12.4 SPACE COORDINATES
`
`Cartesian coordinates
`
`,
`
`‘-
`
`In Fig. 12.17, a system of mutually orthogonal cv:-
`ordinate axes. Oz, 01;. and Oz,
`is indicated.
`Tl:-:
`S}‘Sl.€I11
`is called rigiz!-itandeti
`if a right-threade:
`screw pointing along 02 will advance when the blad-I
`of the screw driver is twisted from Or to 0y throng:
`an angle, say, of 90°.
`In the right-handed systen '
`shown, the y- and 2-axes lie in the plane of the papa:
`and the z-axis points out from the paper.
`Tl.-‘
`Cartesian coordinates of a point Pnfz, y, 2) in spat‘-I
`may be read from the scales along the coordinate
`axes by passing planes through P perpendicular 1
`each axis. All points on the x-axis have their ;.-
`and z-coordinates both zero; that is, they have the
`fomi
`(1, 0,0). Points in a plane perpendicular r
`the z-axis, say, all have the same value for thei:
`z-coordinate.
`Thus,
`for example,
`2 = 5 is a;
`equation satisfied by every point
`(1. y, 5) l_x»-ing i:
`
`:'.3.ne pe -
`2L‘-fie the xy-
`
`sx '-::sect in t
`:v_:.acterized
`
`.r_:r.ates (r,
`;:r'.icular_.
`cg.
`’
`there i
`
`Ex.1021 p.7
`
`

`
`Spaee coordinates
`
`2
`
`(D. 0. 2)
`
`2 =' Kfihfilialll
`i.
`I
`I
`
`,
`HI. 1;: 21-
`
`y - constant.
`
`2.17 Cartesian coordinates.
`
`12.13 Cylindrical coordinates.
`
`1 plane perpendicular to the 2-axis and 5 units
`move the zy-plane. The three planes
`
`:t=2, y=3, z=5
`
`:-.ersect in the point P(2, 3, 5). The yz-plane is
`zzaracterized by 2 = 0. The three coordinate planes
`_- = o, y = o, z = o divide the space into eight cells,
`sled octants. That octant
`in which the points
`:. y, 2) have all three coordinates positive is called
`t:.e_firs1 octam, but there is no conventional number-
`2g of the remaining seven octants.
`
`’
`
`’
`
`qlindrical coordinates
`
`is frequently convenient to use cylindrical co-
`Z‘.
`3:-Jlinates (r, 0,2) to locate a. point in space.
`In
`amicular, cylindrical coordinates are convenient
`then there is an axis of symmetry in a physical
`-_v-Jblem. Essentially, cylindrical coordinates are
`".51 the polar coordinates (r, 9}, used instead of
`:. y)
`in the plane 2 = 0, coupled with the z-
`.~:-:~rdinate (see Fig. 12.18). Cylindrical and Cartes-
`_=:. coordinates are related by the familiar equations
`
`:c=rcos0,
`
`r"'=:z:2+y’,
`
`tan 0 = y/::,
`y = rsin 9,
`2-’-Z.
`
`Z = COHSLEXDI
`
`f - (‘(ll'lfiC:'fl‘lI
`
`If we hold 1' = constant and let 9 and z vary, the
`locus of P(r, 0, z) is then a right circular cylinder of
`radius r and axis along Oz. The locus r = O is just
`the z-axis itself. The locus 0 =_ constant is a plane
`containing the 2:-axis and making an angle 0 with the
`xz—plane (Fig. 12.19).
`
`rectors and ‘Ir
`:-axis.
`
`s/§i-H
`—si—— 125
`
`rametric c—;_ J .
`' taking
`
`513.
`
`ter, of a qua.;---
`idpoints of 1;:
`er. Prove :3.-.
`
`= arc.
`s of a paralla-
`
`t M and N 2»
`hen show ILS
`
`o rthogonal cr-
`
`'
`
`_ y, z) in space y
`he coordinate
`
`rpendicular It
`have their 9-
`they have the
`rpendicular t-:
`'alue for their
`z = 5 is ar.
`
`T
`
`E
`
`v
`
`y, 5) lying ll.
`
`Ex.1021 p.8