S
`
`Thomas
`
`Ex.1020 p.1
`
`Ex.1020 p.1
`
`

`
`GEORGE E. THOMAS, JR.
`
`Dcparlmcnl of .5[c!hcrnc1ic.s
`
`.‘»Iassac.‘.use-.!:.s‘ 1n.s::'fule of Technology
`
`CALCULUS
`
`AND
`
`ANALYTIC
`
`GEOMETRY
`
`FOURTH EDITION
`
`-.-.-aw:-v-
`
`A
`VV
`
`ADDISON-WESLEY PUBLISHING COMPANY
`
`I\I:1.~:s.1'.'l'.'.1.~'L-[Is
`R.i?.“.(‘i!TlK,
`Menlcu Park, C-ulifurniu ' L-mrir-n - Drm Mills, ()n’mri0
`
`-a=§v’_'>..-.'~ -.
`
`"
`
`' "
`
`‘
`
`Ex.1020 p.2
`
`Ex.1020 p.2
`
`

`
`This book is in the
`Addison-Wesley Series in Mathematics
`
`Second printing. December 1969
`
`Copyright © 1968, Philippines copyright 1968 by Addison-
`Wesley Publishing Company, Inc. All rights reserved. No
`part of
`this publication ma._v be rcprodimed, stored in a
`retriew-3.1 system, or transrriitted,
`in any form or by any
`means. electronic. mechanical, phow-cop_ving, recording, or
`otherwise, without. the prior written permission of the pub-
`iisher. Printed in the United States of America. Published
`simultaneoxisly in Canada.
`Libra;-_v of Congress Catalog
`Card No. 68-17568.
`
`
`
`Ex.1020 p.3
`
`Ex.1020 p.3
`
`

`
`
`
`POLAR
`
`}0ORDINATES
`
`CHAPTER 11
`
`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 1- and 3;-coordinates are
`called Ca.rtesian coordinates, in honor of the French
`mathematician-philosopher Ptc-né Descartes‘ (1595-
`l650), who is credited with discovering this method
`of fixing the position of 3. point in J. plane.
`
`/AFN‘ E‘?
`
`(J ‘I'll;as T _
`
`i2illi_:.
`
`!':‘._'.’
`
`|'I.'l
`
`Another useful way to locate it point in :1 plane
`is by polar conrd:'na!e.s- {see Fig. 11.1). First. we fix
`an I.)?‘J,'[:'Z.I‘: 0 and an z'm'tia.! rayf from 0. The point 1-’
`has polar coordinates r, 6, ts-itii
`
`= directed distance from 0 to P,
`
`(la)
`
`and
`
`6 = directed angle from initial ray to OP.
`
`(lb)
`
`,'.Iosi!it-‘c’ whezi
`the angle 6 is
`As in trigononietry.
`measured counterclockwise and negative when incan-
`sured clockwise (Fig. 11.1}. But the angle associated
`with a given point is not unique (Fig. 11.2]. For
`instance, the point. 2 units from the origin. along the
`ray 6 = 30°, has poiar coordinates r ——-
`‘.2, 6 = 30°.
`It also has coordinates r =— :2, = -330"; or 2- = 2.
`6 = 390°.
`There are occasions when we wish to allow r to be
`negative.
`'I‘lm'.’s why we
`"directed distance"
`
`“ For an iriteresmig biographicai account together with
`an excerpt from Descartes’ own i\'I'ltl[)g.-. see World of
`J[atr'mr.a.!i::s, Vol. 1, pp. 235-25-'3.
`T.-\ ray is a lmlf-line consisting of a vertex and points
`of 3 line on one :~i'l-.: of r.}-.:- vr-rt-ex. For t-xamzgle, the
`origin and positive z-axis is st ray. The points on zlie line
`14 = 2: — 3 with: Z 1 is another ray; its vertex is 11. 5?-
`
`Ex.102O p.4
`
`Ex.1020 p.4
`
`

`
`
`
`5 Polar and C.-
`
`6 The circle r =
`
`We adopt the c
`:1£;;ber' Wm < .1
`
`ii‘:
`
`05 y = 0 if
`I‘ :
`
`_
`.
`.
`_
`u
`..e or1g1n,:7: = |
`_ Tue same poim
`iferent ways in
`1* point
`(2, 30°)
`~-Tesent-ations:
`(
`L‘. -150°). The
`e
`'_'e two formula
`
`t
`
`_
`
`(2. 30° + n:
`‘-2, 210° -1- n:
`
`‘I. we represent
`‘
`b _.ulas
`
`352
`
`.=e-is :ooi-dlnates
`
`IL!
`
`
`
`11.2 T:.- ~22‘? = 30°ist'he same asthe t'a)'l7 = —330°.
`
`-I, ¢i .= ijI_'I-
`
`
`
`-210°:
`
`a=' 1-3‘
`wk
`
`2 units. He would reach the same point by tumin;
`only 30’ counterciocI<wise :'ron1 the initial ray an?
`then going farzclcivarrl
`'2 units. So we say that. th
`point also has polar coordinates r = —-:2. 0 = 30°.
`Whenever the angle b€[‘s'-'('.‘(-II‘.
`two rays is 1530’, fr.-
`rays atrtually in-.i‘.~;e a straight line. We then say th=:-.'
`either I‘8.}'
`is the :1eg_atEve of the other. Points I?’
`the ray 6 = a have polar coordinates tr. as} w'.'‘
`r 2 0. Points on the negative rug.-. 0 = a -— 184?’
`have coordinates ii‘. a) with ." 5 0. The origin
`r = 0.
`{See Fig. 11.4 for the r:-.3‘ 6 = 30° and
`negative. A word of caution: The "negative"
`the ray 6 .— 30'‘ is the ray 6 - 30°
`180° -= 21"‘
`and not the my 9 ~— —30°.
`".\'egs~.t.ive" refers to 1':
`directed distance r.)
`There is :1 great z'~.(l.'»';1I‘.?.a[It‘.‘ in being; able to us-
`buth polar and Cartesian coordinates at once.
`7
`do this. we use a ccvnxmon origin and take the init-
`my as the poi-zitivc .1‘-axis, and take the ray = C‘?
`as the positive 9'-axis. The coordinrztes, shovm
`Fig. 11.5, are then related by the equations
`
`'
`
`11.3 The rays 9 = 30° and 0 = 210° make at
`
`l‘:r.--2.
`
`1-: rcosfi,
`
`y
`
`,- sin 9_
`
`-_
`
`6
`
`
`
`—
`
`-f— 1, 3-21*’:
`
`\ ~-
`
`\_ I: _2] 306:]
`
`n.4 The terminal ray 0 = n-£6 and its negative.
`
`in I-.‘c:_. (la). The ray 6 = 30° and the ray 5 = 210°
`together make up :4. complete line through 0 [see
`Fig. 11.3). The point P(2. 210°} 2 units from 0 on
`the ray 0 = 210° has polar coordinates
`r = 2,
`9 = 210°.
`It can be reached by a person standing
`at 0 and facing out along the initial ray, if he first
`turns 210° counterclockwise, and then goes forward
`
`These are the equations that define sir; 6 and cm:
`when r is positive. They are r:‘;.~:o valid if .-- is net.-
`tive, because
`
`cos [9 —r 180°) —-
`
`cos 6,
`
`sin [Ct
`
`-- 180°] = —sin 9.
`
`.
`so positive r’s on the :39 — 180°)-re.;c corrcspomi
`negative r’s associated with the 9-ray. When ‘r : .
`then .7: — y = (l. and P is the origin.
`If we impose the condition
`
`r -= a
`
`{a constant}.
`
`4
`,
`‘
`
`then the locus of P is a. circle with center 0
`radius a. and P describes the circle once as 6 \-a.'—_-
`from O to 360° isee Fig. 11.6]. Or. the other ha; .
`if we let r \'rL!'.'~‘ and hold 6 lixcd. 5:1."
`
`9 2'
`
`-.
`
`:
`
`the locus of P is the straight line shown in Fig. 1'.
`
`»_
`
`(2,
`(-2I
`
`%1r+
`lvlo
`in’ L‘
`
`‘K
`1.
`
`
`
`Ex.102O p.5
`
`Ex.1020 p.5
`
`

`
`
`
`11.1
`
`|
`
`The polar coordinate system
`
`363
`
`The fact that the same point. may be represented
`in several different ways in polar coordinates makes
`added care necessary in certain situations.
`For
`example, the point (2a_. 1r) is on the curve
`
`r2 = 4a’ cos 6
`
`(6)
`
`even though its coordinates as given do not satisfy
`the equation, because the same point is represented
`by (—2a, 0) and these coordinates do satisfy the
`equation. The same point (2a_. 1r) is on the curve
`
`r = a(1
`
`cos 9).
`
`(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 by first substituting cos 0 = r’/4a”
`from (6)
`into (7) and then solving the resulting
`quadratic equation
`
`(2Y~4(2)~4=°
`
`= -2 2 2\/§,
`
`(8)
`
`T Z
`
`for
`
`y
`
`11.6 The circle r — a is the locus P.
`
`We adopt the convention that 7‘ may be any real
`zumber, — ac < r < :c. Then 7- = 0 corresponds
`to I = 0. y = 0 in Eqs. (2), regardless of 0. That is,
`
`r = 0,
`
`9 any value.
`
`(5)
`
`is the origin, as = 0, 3/ = O.
`The mic point niay be represented in several
`different ways in polar coordinates. For example,
`‘he point {2_.30°), or (‘.>,,7r_/6), has the following
`representations:
`(2,30°),
`(2, —330°),
`(-2, 21 °),
`-2, -150°). These and all others are summarized
`:n the two formulas
`
`(2, 30° —: n 360°),
`(-2, 210° =- n 360°),
`
`)n=0,:l:1,:l:2,...;
`
`or, if we represent the angles in radians, in the two
`fonnulas
`
`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 “simultaneou:sly" in the sense of
`being reached at the “same time, "" since it is reached
`in the one case when 0 = 0 and in the other case
`
`It is as though two ships describe
`when 9 = 1r.
`paths that intersect at a point, hul._1.he ships do not
`collide because they reach the point) of intersection
`at difiercnt
`times!
`The curves represented by
`Eqs. (6) and (7) are shown in Fig. ll.9(c). They
`are seen to intersect at the four points
`
`I
`
`I
`
`(0:
`
`(20: 1r):
`
`(7.1: 01).‘
`
`(T1: —0l.)1
`
`where
`
`T] =
`— 2\/§)a,
`cos01=l—%=3——2\/5.
`
`(gb)
`
`n = 0. :':1, _-*_-2
`
`Only the last two of the points (92) are found from
`the simultaneous solution; the firs: two are disclosed
`only by the graphs of the curves.
`
`oy turning
`11 ray and
`.' that the
`0 = 30°.
`
`5 180°, the
`n say that
`Points on
`
`r, a) witl:
`0: + 180’
`
`2 origin is
`3° and its
`gative” of
`P = 21o=
`’ers to the
`
`;le to use
`moe. T-.
`:he initial
`~' 0 = 90‘
`shown ix.
`\'
`
`(2)
`
`md cos 5
`
`' is nega-
`
`spond tr-
`ll 1‘ =
`
`Ex.1020 p.6
`
`Ex.1020 p.6
`
`

`
`390
`
`Vectors and parametric equations
`
`12.4
`
`Then, applying (3), we have
`
`and
`
`ll] =icos0+jsin0
`
`"2
`
`icos(0+3—.::)+isin(9—¥)
`
`isin9—jcos0.
`
`Therefore
`
`C7’ as am + (a0}u2
`
`= a(i cos9+j sin 61- aflfiisin 9 — j cos 9)
`
`= a (cos 9 + 0 sin 9)i + a Lsin 0 — 6 C05 02-)‘.
`
`We equate this with :ci+ yj and, since corresponding
`components must be equal, we obtain the parametric
`equations
`
`as = a (cosfie tisin 6},
`y=a(sin9—9cos9).
`
`6
`I
`
`E
`
`EXERCEEEEL
`
`to the involu:-3 of a cir-:_— :
`ll). A uni: vector tangent
`whose parametric equations are giver. in F.q. (6)
`
`Find the lengths of each of the foilowing vectors and
`angle that each makes with the positive I-axis.
`
`11.
`
`i—:—j
`
`12. 2i — 3i
`
`13. xǤi+j
`
`14. —-.>i+ 31'
`
`15. 5i — 12]
`
`16. -5: — 121
`
`.~'.'.il11
`
`17. Use vector methods to determine parametric eq“_‘r
`tions for the trochoid of Fig. 12.6, by taking
`
`R=07>=0fi+W—fi.
`
`Let :1, B. C. D be the vertices, in order, of a quad.-.-
`lateral. Let :1’. B’, C’, D’ be the midpoints of ti‘
`Provo tits"
`sides AB. BC’, CD, and DJ, in order.
`--l'B’C'D’ is a parallelogram.
`Hm. First show that 3'3’ = 51:" = gfc.
`
`19. Using vectors. show that the diagonals 0:‘ a parallei:
`gram biscct each other.
`.li'ez'I'zod. Let .4 be one vertex and let. M and N 1''.
`the midpoints of
`the diagonals. Then show Elli‘
`= .a_\'.
`
`.
`
`f
`
`.
`
`'
`
`: lane perpc
`zcorre the my-1
`
`zxvzrsecl; in tl
`:3-:acterized l
`
`' = 0: 3/ = 0:
`:t.'_e--.1 actants.
`
`z) have ;
`.:
`-_’_~— _—Z rst octantp
`ti; 2-1' the rcm
`
`3_r-indricalcoor
`
`‘_= frequent
`.1"
`r.::.ates (r,£
`_':L".icula,r, cy
`'."_4.r.
`there is
`zéfolem. Es
`
`the pola:
`'5'.
`in the
`j.‘.I
`.- rdinate (sen
`-:_ coordinate
`
`I ‘.
`
`y 2
`
`12.4 SPACE COORDINATES
`
`Cartesian coordinates
`
`In Fig. 12.17, a systeni of mutually orthogonal cv.— -
`ordinate axes. 0:. 01;. and Oz,
`is indicated. The
`‘.
`system is called right-handezi
`if a right-threado:
`screw pointing along ()2 will advance when the blaé-3 ,
`of the screw driver is twisted from 0.: to 0y throng:
`an angle, say, of 90°.
`In the right—handed s}'ster_ '
`shown, the y- and z-axes lie in the plane of the pap-1-.‘
`and the .r:—axis points out from the paper.
`Tl.-'
`Cartesian coordinates of a point P-fr, y, 2) in spac-.'
`may be read from the scales along the coordinate
`axes by passing planes through P perpendicular '.
`each axis. All points on the 1:-axis have their ;.-
`and z-coordinates both zero; that is. they have the
`form (2, 0, 0). Points in a plane perpendicular r
`the z-axis, say, all have the same value for their
`2-coordinate.
`Thus.
`for example,
`2 = 5 is
`:1;
`equation satisfied by every point
`[.25. y, 5) lying i:
`
`In Exercises 1 through 10, express each of the vectors in
`the form ai + bj.
`Indicate all quantities graphically.
`
`I 2 E i
`
`1. P1P2, if P1 is the point (1, 3) and P2 is the point
`(2. -1)
`
`. 2. 0P3, if 0 is the origin and P3 is the midpoint» of the
`vector I’1["z joining I’y;2_. -1) and I’-,>:j- »-1, 3)
`
`3. The vector from the point .-ll-;2, 3} to the origin
`
`and CT), given the four
`*1. The sum of the vectors
`points .40, -1), B(2, 0), C(—1, 3], and DI:-2, 2)
`
`5. A unit vector making an angle of 30° with the posi-
`tive 2-axis
`
`6. The unit vector ohtained by rotatingj through 120’
`in the clockwise direction
`
`1. A unit vector having the same direction as the
`vector 3i — 4i
`
`8. A unit vector tangent to the curve y = :r'-’ at the
`point (2, 4,‘.-
`the
`9. A unit vector normal to the curve y = :9 al.
`point. P12. 4} and painting from P toward the con-
`cave side 01' the curve {that ‘Ls, an “inner” normal)
`
`
`
`Ex.1020 p.7
`
`Ex.1020 p.7
`
`

`
`Space coordinates
`
`391
`
`12.13 Cylindrical coordinates.
`
`
`12.19
`
`2 = CODSLQDK.
`
`If we hold r = constant. and let 9 and z vary, the
`locus of P(r, 0, 2) is then at right circular cylinder of
`radius r and axis along 02. The locus r = 0 is just
`the z-axis itself. The locus 0 =_ constant. is a plane
`containing the s—axis and nrmking an angle 0 with the
`xz-plane (Fig. 12.19).
`
`Ex.1020 p.8
`
`24
`
`l
`I
`I
`l
`
`2r ronstzmt
`
`X10. 0. z)
`
`;
`l
`u
`I
`
`\
`(
`UL .43, 2:
`
`i:
`
`2', ll, .-}
`
`_ Lnslunz
`
`..._
`
`‘I
`'
`n’
`v._.r, ll, 0,‘ ’
`
`’
`
`I
`
`./z
`/
`
`y - constant.
`
`I
`I9;-.““.P(5ly V) 3)
`"'-5‘ ‘ (I), y. 0}
`'
`"’-..__
`"h 9'
`
`I
`
`.r
`
`_:.-. 0]
`
`2.17 Cartesian coordinates.
`
`1 plane perpendicular to the z-axis and 5 units
`s.:~3ve the :cy-plane. The three planes
`
`
`
`z=2! 3/=3: z=5
`
`ztersect in the point P(2, 3,5). The yz-plane is
`::a.racterized by 2: = 0. The three coordinate planes
`: = 0, y = 0, z = 0 divide the space into eight cells,
`2-Lied actants. That octant
`in which the points
`:. y, z) have all three coordinates positive is called
`rgefirst octam, but there is no conventional number-
`3 of the remaining seven octants.
`
`qvllndrical coordlnates
`
`is frequently convenient to use cylindrical co-
`:‘,
`at-iinates (r, 0, 2)
`to locate a point in space.
`In
`usrtiicular, cylindrical coordinates are convenient
`-"zen there is an axis of symmetry in a physical
`2-ablem. Essentially, cylindrical coordinates are
`".~Zt
`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
`
`x=rcos0,
`
`r2=z2+y2,
`
`y= rsin0,
`tan 0 = y/3:,
`z = z.
`
`(1)
`
`rte of a r:-.-:1»
`in Eq. -[6
`
`rectors and ‘Ir
`:-axis.
`
`xfii-l—.i
`
`—5i— 12;‘
`
`rametric c—;_ J-
`' taking
`
`(,7.
`
`.er, of a qua;-»
`.idpoints of 1::
`cr. Prove :3;
`
`’
`
`= gfir.
`
`is of a pa.ralie_».—
`
`:t M and X 1»
`‘hen show I’-.5
`
`orthogonal c»--
`.dicated. The
`
`.
`
`right-threude:
`when the blade
`
`to 0y throug:
`ianded s_w-stern.
`ie of the paper
`: paper.
`The
`, y, 2) in space M
`the coordinate
`
`’.
`
`-
`
`3
`
`zrpendicular ts;
`have their 9-
`they have the
`rpendicular t-t
`value for their
`2 = 5 is
`a1".
`
`, y, 5) lying ir.
`
`Ex.1020 p.8