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`Page 1 of 8
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`Samsung Exhibit 1021
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`Page 1 of 8
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`Samsung Exhibit 1021
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`

`

`GEORGE B. THOMAS, JR.
`Depurlmcnt of 31'alhtmalics
`J!assaci.usc!£s Institute of Technology
`
`CALCULUS
`
`AND
`
`ANALYTIC
`
`GEOMETRY
`
`FOURTH EDITION
`
`A
`VV
`
`ADDISON—WESLEY PUBLISHING COMPANY
`Rearimg, Mussulmsuus
`Menlcu Park, Cufifxrrniu - L-mrir-n - Dnn Mills: 0mm
`
`'
`
`‘
`
`- v I
`
`I ,wmva'.‘ .‘fg-a...‘
`
`-
`
`.
`
`'-
`
`'
`
`'
`
`‘
`
`" -
`
`. I..— ___..
`
`
`
`‘
`
`I
`
`I
`I
`
`;
`i
`T
`
`I»
`v.
`
`3
`
`i.
`
`Q.
`3"
`
`i
`
`3
`E
`
`‘:
`
`Page 2 of 8
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`Page 2 of 8
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`

`

`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
`pan. of
`this publication may be reproduced. stored in a
`retrieval system, or transmitted,
`in any form or by any
`mums, electronic, mechanical, pholtruopying, recording, or
`otherwise, without the prior written permission or" the pub-
`lisher. Prim/ed in the United States of America. Published
`simultaneously in Canada.
`Library of Congress Camlng
`Card No. 68-17568.
`
`
`
`Page 3 of 8
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`Page 3 of 8
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`

`

`
`
`11.1 THE POLAR COORDINATE SYSTEM
`
`We know that a point can be located in a plane by
`giving its abscissa and ordinate relative to a. given
`coordinate system. Such x- and y-coordinateg are
`called Cartesian coordinates, in honor of the French
`ntathenmtician-philosopher Rene Descartes“ (1596—
`1650), who is credited with discovering this method
`of fixing the position of a. point in a plane.
`
`[API .- é:
`
`
`\
`/ l";‘
`
`
`
`l'l.l
`
`y = 2.: —- 3 with: Z 1 is another ray; its vertex is L1, 5).
`
`Another useful way to locate a point in a plane
`by polar coordinates (see Fig. 11.1). First. we fix
`an origin 0 and an initial mm from 0. The point P
`has polar coordinates r, a, with
`
`r = directed distance from 0 to P,
`
`(la)
`
`and
`
`0 = directed angle from initial ray to 0P.
`
`(lb)
`
`the angle 8 is positive when
`trigonometry.
`As in.
`measured counterclockwise and negative when mea-
`sured clockm'se (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 polar coordinates r -= 2, 6 = 30°.
`It also has coordinates r —— ‘2. = —33()"‘. or r = 2,
`6 = 390°.
`There are occusions when we wish to allow r to be
`negative. That’s why we say "directed distance‘
`
`“ For an interesting biographical account together with
`an excerpt from Descartes’ own writing. see “'0er of
`Mathematics, Vol. 1, pp. 235—253.
`TA ray is a. half—line consisting of a vertex and points
`of A line on one aldc oi (lav vortex. For example, the
`origin and positive z-nxis is a ray. The points on the line
`
`POLAR
`
`3OORDINATES
`
`CHAPTER 11
`
`Page 4 of 8
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`Page 4 of 8
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`

`

`362
`
`?aa :oordlnales
`
`‘_
`
`11.1
`
`
`
`11.2
`
`'7" — 5' = 30° is the same as the ray :9 = ~330°.
`
`r, 0:150'
`
`
`
`
`
`
`— ' ‘1—2,30°:-
`
`11.4. The terminal ray 6 = «[6 and its negative.
`
`in Eq. (1a). The ray 6 = 30° and the ray 5 r 210°
`together make up a. complete line through 0 (see
`i-‘ig. 11.3). The point P(2. 210°) 2 units from O on
`the ray 6 = 210° has polar coordinates :- = 2,
`9 = 910? 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
`
`'2 units. He would reach the 531110 point by turning
`only 30': counterclockwise from the initial my art
`then going backward '2 units. So we say that. th’
`point also has polar coordinates r = ——:2. 0 = 30°.
`thnever the angle between two rays is 180°, t'r.—,
`rays actually make a straight line. We then so.)- the:
`either ray is the negative of the other. Points If’
`the ray 19 = u have polar coordinates (r, a) “"1"
`r 2 0. Points on the. negative my. b = a — 18C”
`have coordinates it. a) with r S 0. The origin
`r =- 0.
`(See Fig. 11:1 for the my 6 = 2‘10c and
`negative. A word of caution: The "negative"
`the ray 6 .— 30c is the ray 6 .— 30°
`180° = 21*"
`and not the my 6 -— —-30°. “Negative” refers to 1':
`directed distance r.)
`There is a great advantage in being able to us-
`both polar and Cartesian cmrrdinates at once.
`T
`do this. we use a common origin and take the init._
`my as the positive x-axis. and take the my 9 = 5‘? V
`as the positive y-znzis. The coordinates, shovm
`Fig. 11.5, are then related by the. equations
`
`x=rcosl9, y=rsin0.
`
`;
`
`‘
`These are the equations that define sin 6 and CU:
`when r is positive. They are also valid if r is neg--
`tive, because
`
`COS L9 + 180°) —‘
`
`cos 9,
`
`sin ('3'
`
`-- 180°] = —sin 9.
`
`so positive r’s on the :‘9 — 180°:t-ray corrcsponti '.
`negative r’s associated with the 0-ray. When 7' = .
`then .7: — y = 0. and P is the origin.
`If we impose the condition
`
`r —— a
`
`{a constant).
`
`then the locus of P is a circle with center 0
`radius a. and P describes the circle once as 6 var-2
`from 0 to 360° (see Fig. 11.6}. Or. the other h:-.; .
`if we let r vary and hold 6 fixed, say
`
`9 :
`
`'2
`
`the locus of P is the straight line shown in Fig. 1‘.
`
`-.
`
`Page 5 of 8
`
`'
`
`5 Polar and C:
`
`a The circle r =
`
`
`
`
`7‘59 adopt the c
`;i£;‘_bcr‘
`.._w <7
`
`"" -‘ = 0,y= Oir.
`.r =
`i
`
`
`
`‘2 Le origin, :5 = 1
`Tzie same point
`I
`iferent ways in
`
`If point
`(‘2, 30°)
`
`v'
`resent-etions:
`(
`
`in". *-]50°). The
`
`- Le two formula
`
`
`
`(2: 30° + n:
`—2, 210° -’- it:
`
`
`' 4 “’9 represent
`‘ _ulas
`.
`
`(2, £7? +
`1'"
`%
`t
`
`IO‘l
`N)S
`
`
`
`Page 5 of 8
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`

`

`
`
`
`
`.__+_._<
`
`.I
`
`‘0
`
`~a—--x
`I
`
`t\“l/
`
`11.5 Polar and Cartesian coordinates.
`
`
`
`H.‘ The circle r - a is the locus P.
`
`We adept the convention that r may be any real
`:umber‘, —-oe < r < x. Then r = 0 corresponds
`to a: = 0. y = 0 in Eqs. (2), regardless 01‘0. That is,
`
`r = 0,
`
`0 any value.
`
`(5)
`
`is the origin, 2: = 0, y = 0.
`The same point may be represented in several
`difi'erent. ways in polar coordinates. For example,
`'he point
`(2, 30°), or (2,,7rf6), has the following
`representations:
`(130°),
`(2, —330°),
`(—2,210°‘.
`—2, —150°). These and all others are summarized
`:n the two formulas
`
`(2, 30° + n 360°),
`(—2, 210° ; n 360°),
`
`n=0,:l:1,12,...;
`
`or, if we represent the angles in radians, in the two
`formulas
`
`".1
`
`l
`
`The polar coordim system
`
`The fact that the same point mayr be represented
`in several different ways in polar coordinates makes
`added care necessary in certain situations.
`For
`example, the point (2a, 11') is on the curve
`
`r2 = 4a: cos 8
`
`(6)
`
`even though its coordinates as given do not satisfy
`the equation, because the same point is represented
`by (—2t1,0) 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 01' intersection of the two curves represented
`by Eqs. (6) and {7). But if we solve the equations
`simultaneously by first substituting cos 9 2 1-3/4th2
`from (6)
`into (7) and then solving the resulting
`quadratic equation
`
`for
`
`(2>14(2)—4=°
`
`7.
`a
`
`= —2 s.- 2\/§,
`
`(8)
`
`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 9 = 1r.
`paths that intersect at a 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. 11.9(c). They
`are seen to intersect at the four points
`
`(or
`
`(2a: 7r):
`
`(r11 01):
`
`('1: —ol)r
`
`where
`
`71 =
`
`*- Zfihll
`1*,
`cosol=l—Z=3-2\/§.
`
`(9b)
`
`mr).
`7r
`(2.-
`é'”
`(—,{r1r+2mr),
`
`—0 1+2
`"“ '=**
`
`....
`
`Only the last two of the points (9a) are found from
`the simultaneous soltition; the first two are disclosed
`only by the graphs of the curves.
`
`by turning
`ll ray and
`t that the
`0 = 30°.
`
`5 180°, the
`n say that
`Points 0:
`r, a) with
`a + 180:
`: origin is
`3" and its
`:ative” of
`)° = 210:
`’ers to the
`
`)le to use
`)nce. Tr.
`:he initia.‘
`r o = 90=
`shown ii.
`
`spond 1r-
`n r = O.
`
`(3
`
`.‘ 0 and
`0 varies
`er hand.
`
`{-l-
`
`:g. 11.4.
`
`
`
`Page 6 of 8
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`Page 6 of 8
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`

`

`to the involute of a 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 x-axis.
`
`11.
`
`i—:j
`
`12. 2i — 3i
`
`13. viii—ii
`
`14. —2i+3j
`
`15. 5i412j
`
`16. —5i -- 12]
`
`17. Use vector methods to determine parametric eqLEr
`tions for the trochoid of Fig. 12.6, by taking
`
`R = 07’ = OH + .U—C — 67’.
`
`Let A, B. C. D be the vertices, in order, of a quad:—
`lateral. Let
`.-l’. B’, C’, D' be the midpoian of 1":
`sides AB, BC, CD, and 0.1, in order. Prove Ihi‘
`:1’B’C’D’ is a parallelogram.
`you. First show that 73' = 1757 = $.70.
`19. Using vecrors. show that :he diagonals of a parallel-.-
`gram bisect each other.
`.lr'eihod. Let
`:1 be one vertex and let M and N he
`the midpoint-Ls of the diagonals. Then show [its
`m = :5)".
`
`12.4 SPACE COORDINATES
`
`Cartesian coordinates
`
`390
`
`Vectors and parametric equations
`
`12.4
`
`Then, applying (3), we have
`
`sad
`
`3w
`...
`311'
`.
`.
`u2=1cos +7 TJSIH 0—?
`isine—j cos 0.
`
`Therefore
`
`a; 2 out + (aoluz
`= a{i cosO-i—j sin 6} — afiii sin 9 -— j cos 9}
`= a (cos 9 + 9 sin 9in + a Lsin 0 — 6 cos 0'2-j.
`
`We equate this with zie yj and, since corresponding
`components must be equal, we obtain the parametric
`equations
`
`a: = a (0030'.— Bsin 0},
`y=clsin9—9c059).
`
`B
`K
`
`'.
`
`EXERCEEE:
`
`In Exercises 1 through 10, express each of the vectors in
`the form ai + bj.
`Indicate all quantities graphically.
`
`1. 13172, if P1 is the point (1, 3) and P2 is the point
`(2. —1)
`
`I 2. 07;, if 0 is the origin and P3 is the midpoint. of the
`vector 1’1—1’2; joining [’1-12, -1) and 1’22} ~11, 3}
`3. The vector from the point .=l-;2, 3) to the origin
`
`4. The sum of the vectors I}? and LT), given the four
`points. .40, —-l}, 8(2, 0), C(—1,3}, and D(—2, 2)
`5. A unit vector making an angle of 30° with the posi-
`tive z-axis
`
`6. The unit vector ohtained by rotatingj through 120°
`in the clockwise direction
`
`NI
`
`. A unit vector having the same direction as the
`vector 3i — 4j
`8. A unit vector tangent to the curve y = :2 at the
`point (2, 4}
`9. A unit vector normal to the curve y = x2 at the
`point P2. 4} and pointing from P toward the con-
`cave side of the curve {that is, an "inner" normal)
`
`In Fig. 12.17, a System of mutually orthogonal cv:
`ordinate axes. Oz, 03;. and Oz,
`is indicated.
`le-z
`system is called right-handed if a right-threade:
`screw pointing along ()2 will advance when the blad-Z ,
`of the screw driver is twisted from Or to 0y throug:
`an angle, say, of 90°.
`In the right-handed syster-
`shown, the y- and 2-axes lie in the plane of the papa:
`and the z-axis points out from the paper. Th;
`Cartesian coordinates of a point Pufz, y, z) in spat-I
`may be read from the scales along the coordinate
`axes by passing planes through P perpendicular :
`each axis. All points on the x-axis have their 4.-
`and z~coordinates both zero; that is, they have the
`iomi (1,0,0). Points in a plane perpendicular r
`the 2-axis, say. all have the same value for their
`z-coordinate.
`Thus,
`for example,
`2 = 5 is
`:1;
`equation satisfied by every point
`(z. y, 5) lying i:
`
`'
`
`‘V
`
`.
`
`.
`
`
`
`IJ'. (1|, z
`
`Hull:
`
`2 ' 7 Cartesis
`
`:ane perpi
`:c-fl‘e the xy—l
`
`:x'csect in t1
`tarscterized l
`
`I' = 0: y = 0:
`{Li-d octants.
`
`2) have :
`,:
`Li _-2 rs: octant.
`a: :-1‘ the rem
`
`3F'mdricalcoor
`
`.1‘ 5 frequent
`.t-zzzates (r,£
`
`xrticular, cy
`':_.:1 there is
`:ofolem. Es
`
`the pola:
`"1:".
`in the
`:
`j.)
`'rdinate (set
`r
`'.:_ coordinate
`
`z 2
`
`y:
`
`
`
`Page 7 of 8
`
`Page 7 of 8
`
`

`

`12.4
`
`|
`
`Space coordinates
`
`391
`
`2 =’ constant
`i
`,
`I
`I
`
`t
`,
`i". y: 2:
`
`y - constant.
`
`ZA
`
`‘
`(0. 0. z;
`
`l
`|
`|
`l
`
`i:
`
`I
`
`'
`
`J
`
`l), :}
`
`-
`
`.‘nstuut
`
`>-—
`
`I
`
`{.r, n, 0} z’
`
`I
`
`A I
`l/
`
`‘s
`9’“ ‘ PG“ 1" z]
`“--
`
`. 0}
`(I), -'
`‘l ‘ \i
`.3 y
`
`
`12.19
`
`1'
`
`1:, z.-. 03
`
`2.17 Cartesian coordinates.
`
`1 plane perpendicular to the z-axis and 5 units
`move the zy—plane. The three planes
`
`23:2, y=3, =5
`
`:ersect in the point P(2, 3, 5). The yz-plane is
`:zaracterized by z = 0. The three coordinate planes
`: = O, y = 0, z = O divide the space into eight celIS,
`sled octants. That octant
`in which the points
`:. y, 2) have all three coordinates positive is called
`
`11:"first admit, but there is no conventional number-
`
`2g of the remaining seven octants.
`
`qthdrical coordinates
`2‘.
`is frequently convenient to use cylindrical co-
`riinates (r, 0,2) to locate a point in space.
`In
`articular, cylindrical coordinates are convenient
`then there is an axis of symmetry in a physical
`tablem. Essentially, cylindrical coordinates are
`":1 the polar coordinates (r, 0), 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
`
`as: recso,
`
`r22 x2+y’,
`
`tan 0 = 11/13
`9 = 7 5m 9:
`z = z.
`
`(I)
`
`1 2." Cylindrical coordinates.
`
`5
`
`|
`'
`
`‘
`
`.1
`
`‘
`
`5
`
`Z = constant
`
`f '- constant
`
`l
`
`j
`
`!
`{
`
`If we hold r = constant. and let 9 and z vary, the
`locus of P(r, a, z) is then a right circular cylinder of
`radius r and axis along Oz. The locus r = 0 is just
`the z-axis itself. The locus 0 =_ constant is a plane
`containing the e-axis and making an angle a with the
`arc-plane (Fig. 12.19).
`
`
`
`rectors and 'Ir
`:-axis.
`
`\/§i+j
`—5i—— 125
`
`.rametric e;- s.»
`' taking
`
`513.
`
`tor, of a quart-
`.idpoints of 1;:
`er. Prove 23:.
`
`= are.
`is of it pushes.-
`
`:t M and N :v
`'hen show 1'2;
`
`’
`
`orthogonal cr— '
`.dicated. The -
`
`right-threade;
`when the blad:
`
`to 0y throug:
`landed syster;
`ie of the pape:
`: paper.
`The
`,3], z) in space ‘
`the coordinate
`
`T
`:rpendicular t;
`have their 9— ‘
`they have the
`rpendicular t:
`ralue for their
`z = 5 is at.
`
`v
`
`.y, 5) lying in
`
`Page 8 of 8
`
`Page 8 of 8
`
`