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GEOMETRYSee References 1-3 for additional information.Sets and FunctionsA set is a clearly defined collection of distinct objects o r elements. Theintersection of two sets S and T is the set of elements which belong to S andwhich also belong to T. The union of S and T is the set of all elements whichbelong to S o r to T (or to both, Le., inclusive or).A ficnction is a set of ordered elements such that no two ordered pairs havethe same first element, denoted as (x,y) where x is the independent variableand y is the dependent variable. A function is established when a condition existsthat determines y for each x, the condition usually being defined by an equationsuch as y f(x) [2].AnglesAn angle A may be acute, 0" A go", right, A go", or obtuse, 90" A 180".Directed angles, A 2 0" or 2 180", are discussed in the section "Trigonometry."Two angles are complementary if their sum is 90" or are supplementary if theirsum is 180". Angles are congruent if they have the same measurement in degreesand line segments are congruent if they have the same length. A dihedral angleis formed by two half-planes having the same edge, but not lying in the same plane.A plane angle is the intersection of a perpendicular plane with a dihedral angle.PolygonsA polygon is the union of a finite number of triangular regions in a plane,such that if two regions intersect, their intersection is either a point o r a linesegment. Two polygons are similar if corresponding angles are congruent andcorresponding sides are proportional with some constant k of proportionality.A segment whose end points are two nonconsecutive vertices of a polygon is adiagonal. The perimeter is the sum of the lengths of the sides.TrianglesA median of a triangle is a line segment whose end points are a vertex andthe midpoint of the opposite side. An angle bisector of a triangle is a medianthat lies on the ray bisecting an angle of the triangle. The altitude of a triangleis a perpendicular segment from a vertex to the opposite side. The sum of theangles of a triangle equals 180". An isosceles triangle has two congruent sidesand the angles opposite them are also congruent. If a triangle has threecongruent sides (and, therefore, angles), it is equilateral and equiangular. A scalene3

4Mathematicstriangle has no congruent sides. A set of congruent triangles can be drawn ifone set of the following is given (where S side length and A angle measurement): SSS, SAS, AAS or ASA.QuadrilateralsA quadrilateral is a four-sided polygon determined by four coplanar points(three of which are noncollinear), if the line segments thus formed intersect eachother only at their end points, forming four angles.A trapezoid has one pair of opposite parallel sides and therefore the otherpair of opposite sides is congruent. A parallelogram has both pairs of oppositesides congruent and parallel. The opposite angles are then congruent andadjacent angles are supplementary. The diagonals bisect each other and arecongruent. A rhombus is a parallelogram whose four sides are congruent andwhose diagonals are perpendicular to each other.A rectangle is a parallelogram having four right angles, therefore both pairsof opposite sides are congruent. A rectangle whose sides are all congruent isa square.Circles and SpheresIf P is a point on a given plane and r is a positive number, the circle withcenter P and radius r is the set of all points of the plane whose distance fromP is equal to r. The sphere with center P and radius r is the set of all points inspace whose distance from P is equal to r. Two or more circles (or spheres)with the same P, but different values of r are concentric.A chord of a circle (or sphere) is a line segment whose end points lie on thecircle (or sphere). A line which intersects the circle (or sphere) in two points isa secant of the circle (or sphere). A diameter of a circle (or sphere) is a chordcontaining the center and a radius is a line segment from the center to a pointon the circle (or sphere).The intersection of a sphere with a plane through its center is called agreat circle.A line which intersects a circle at only one point is a tangent to the circle atthat point. Every tangent is perpendicular to the radius drawn to the point ofintersection. Spheres may have tangent lines or tangent planes.Pi (x) is the universal ratio of the circumference of any circle to its diameterand is equivalent to 3.1415927. Therefore the circumference of a circle is ndor 2nr.Arcs of CirclesA central angle of a circle is an angle whose vertex is the center of the circle.If P is the center and A and B are points, not on the same diameter, which lieon C (the circle), the minor arc AB is the union of A, B, and all points on C inthe interior of APB. The major arc is the union of A, B, and all points on Con the exterior of APB. A and B are the end points of the arc and P is thecenter. If A and B are the end points of a diameter, the arc is a semicircle. Asector of a circle is a region bounded by two radii and an arc of the circle.The degree measure (m) of a minor arc is the measure of the correspondingcentral angle (m of a semicircle is 180") and of a major arc 360" minus the mof the corresponding minor arc. If an arc has a measure q and a radius r, thenits length is

GeometryL 5q/l80*nrSome of the properties of arcs are defined by the following theorems:1. In congruent circles, if two chords are congruent, so are the correspondingminor arcs.2. Tangent-Secant Theorem-If given an angle with its vertex on a circle, formedby a secant ray and a tangent ray, then the measure of the angle is halfthe measure of the intercepted arc.3. Two-Tangent Power Theorem-The two tangent segments to a circle from anexterior point are congruent and determine congruent angles with thesegment from the exterior point to the center of the circle.4. Two-Secant Power Theorem-If given a circle C and an exterior point Q, letL, be a secant line through Q, intersecting C at points R and S, and let L,be another secant line through Q, intersecting C at U and T, thenQRQS QUQT5. Tangent-Secant Power Theorem-If given a tangent segment QT to a circleand a secant line through Q, intersecting the circle at R and S, thenQRQS QT2-6. Two-Chord Power Theorem-Ifintersecting at Q, thenQRQS QU-RS and TU are chords of the same circle,QTConcurrencyTwo or more lines are concurrent if there is a single point which lies on allof them. The three altitudes of a triangle (if taken as lines, not segments) arealways concurrent, and their point of concurrency is called the orthocenter. Theangle bisectors of a triangle are concurrent at a point equidistant from theirsides, and the medians are concurrent two thirds of the way along each medianfrom the vertex to the opposite side. The point of concurrency of the mediansis the centroid.SimilarityTwo figures with straight sides are similar if corresponding angles are congruent and the lengths of corresponding sides are in the same ratio. A lineparallel to one side of a triangle divides the other two sides in proportion,producing a second triangle similar to the original one.Prisms and PyramidsA prism is a three dimensional figure whose bases are any congruent andparallel polygons and whose sides are parallelograms. A pyramid is a solid withone base consisting of any polygon and with triangular sides meeting at a pointin a plane parallel to the base.Prisms and pyramids are described by their bases: a triangular prism has atriangular base, a parallelpiped is a prism whose base is a parallelogram and a

6Mathematicsrectangular parallelpiped is a right rectangular prism. A cube is a rectangularparallelpiped all of whose edges are congruent. A triangular pyramid has atriangular base, etc. A circular cylinder is a prism whose base is a circle and acircular cone is a pyramid whose base is a circle.Coordinate SystemsEach point on a plane may be defined by a pair of numbers. The coordinatesystem is represented by a line X in the plane (the x-axis) and by a line Y (they-axis) perpendicular to line X in the plane, constructed so that their intersection,the origin, is denoted by zero. Any point P on the plane can now be describedby its two coordinates which form an ordered pair, so that P(x,,y,) is a pointwhose location corresponds to the real numbers x and y on the x-axis andthe y-axis.If the coordinate system is extended into space, a third axis, the z-axis,perpendicular to the plane of the xI and y, axes, is needed to represent thethird dimension coordinate defining a point P(x,,y,,z,).The z-axis intersects thex and y axes at their origin, zero. More than three dimensions are frequentlydealt with mathematically, but are difficult to visualize.The slope m of a line segment in a plane with end points P,(x,,y,) and P,(x,,y,)is determined by the ratio of the change in the vertical (y) coordinates to thechange in the horizontal (x) coordinates orm (Y' -YI)/(X2- XI)except that a vertical line segment (the change in x coordinates equal to zero)has no slope, i.e., m is undefined. A horizontal segment has a slope of zero.Two lines with the same slope are parallel and two lines whose slopes arenegative reciprocals are perpendicular to each other.Since the distance between two points P,(x,,y,) and P,(x,,y,) is the hypotenuseof a right triangle, the length of the line segment PIP, is equal toGraphsA graph is a figure, i.e., a set of points, lying in a coordinate system and agraph of a condition (such as x y 2) is the set of all points that satisfy thecondition. The graph of the slope-intercept equation, y mx b, is the line whichpasses through the point (O,b), where b is the y-intercept (x 0) and m is theslope. The graph of the equation(x-a)' (y-b)' r2is a circle with center (a,b) and radius r.VectorsA vector is described on a coordinate plane by a directed segment from its initialpoint to its terminal point. The directed segment represents the fact that everyvector determines not only a magnitude, but also a direction. A vector v is not

Geometry7changed when moved around the plane, if its magnitude and angular orientationwith respect to the x-axis is kept constant. The initial point of v may thereforebe placed at the origin of the coordinate system a n d ? ’ m a y be denoted byd a,b where a is the x-component and b is the y-component of the terminal point.The magnitude may then be determined by the Pythagorean theoremFor every pair of vectors (xI,yI)and (x2,y2),the vector sum is given by (x, xp, (x,y) and a real number (a scalar) ris rP (rx,ry). Also see the discussion of polar coordinates in the section“Trigonometry” and Chapter 2, “Basic Mechanics.”y1 y2). The scalar product of a vector PLengths and Areas of Plane Figures [ l ](For definitions of trigonometric functions, see “Trigonometry.”)Right triangle (Figure 1-1)A a2 area 1/2c2b2ab 1/2a2 cot AEquilateral triangle (Figure 1-2)area 114 a2& 0. 43301a2 1/2b2 tan A 1/4c2 sin 2A

8MathematicsAny triangle (Figure 1-3.)Aarea 1/2 base altitude 1/21/2 {(XIY2 - yI)ah 1/2ab sin C (XZYS(XSYI-X.SY2)- XIY,))where (x,,yl),(xz,yz),(x,,y,) are coordinates of vertices.Rectangle (Figure 1-4)area ab 1/2D2 sin uwhere u angle between diagonals D, DRhombus (Figure 1-5)Area a'sin C 1/2DID,where C angle between two adjacent sidesD,, D, diagonalsParallelogram (Figure 1-6)c\

Geometryarea bhwhere u ab sin c D,D, sin u1/2angle between diagonals D, and D,Trapezoid (Figure 1-7)area 1/2where u (a b ) h ' 1/2D,D, sin uangle between diagonals D, and D,and where bases a and b are parallel.Any quadrilateral (Figure 1-8)area 1/2D,D, sin uNote: a2 b2 where m c2 d* D21 D*2 4m*distance between midpoints of D, and D,Circlesarea nr2 1/2Cr where r radiusd diameterC circumference1/4 2nrCd nd1/4nd2 0.785398d29

10MathematicsAnnulus (Figure 1-9)area n(R2 - r2) n(D2 - d2)/4where R' mean radiusb R - r 1/2 2nR'b(R r)Sector (Figure 1-10)area 1/20rswhere rad A s xr2A/36O0 1/2r2 rad Aradian measure of angle Alength of arc r rad ASegment (Figure 1-11)area 1/2r2 (rad A- sin A) 1/2[r(s-c) ch]

Geometrywhere rad A 11radian measure of angle AFor small arcs,s ( 8 -‘ C )1/3where c’ chord of half of the arc (Huygen’s approximation)Note: c 2 2 / h oc’ Jdh or d d 2 / hwhere dhs diameter of circler[l - cos (1/2 A)]2r rad (1/2 A)Ellipse (Figure 1-12)area of ellipse nabarea of shaded segment xy ab sin-’ (x/a)length of perimeter of ellipse( 1 1/4m2 1/64(a - b)/(a b)where Km For mKFor mK0.11.0020.61.092 Hyperbola (Figure 1-13)0.21.0100.71.127x(a b)K,m4 1/2560.31.0230.81.168m6 . . .)0.41.0400.91.2160.51.0641 .01.273

12MathematicsIn any hyperbola,shaded area A abIn [(x/a)In an equilateral hyperbola (aarea A a2 sinh-’ (y/a) (y/b)]b),a2 cosh-’ (x/a)Here x and y are coordinates of point P.Parabola (Figure 1-14)shaded area A 2/3chIn Figure 1-15,length of arc OPHere c p PT s 1/2PT 1/2pIn [cot(1/2any chordsemilatus rectumtangent at PNote: OT OM xSurfaces and Volumes of Solids [I]Regular prism (Figure 1-16)u)]

Geometryvolume1/2 lateral areawhere nBP nrahnah BhPh number of sidesarea of baseperimeter of baseRight circular cylinder (Figure 1-17)hvolume nreh Bhlateral areawhere BP 2nrh Pharea of baseperimeter of base13

14.MathematicsAny prism or cylinder (Figure 1-18),volume Bhlateral area N1Q1where 1 length of a n element o r lateral edgeB area of baseN area of normal sectionQ perimeter of normal sectionHollow cylinder (right and circular)volume where hr,R, (d,D)bD'rch(R2 - r2) nhb(D - b) nhb(d b)altitudeinner and outer radii (diameters)thickness R - rmean diam 1/2 (d D) D - bRegular pyramid (Figure 1-19) rchbD'd b rchb(R r)

Geometryvolume1/3 altitude lateral area area of base1/2 slant height radius of inscribed circleside (of regular polygon)number of sidess JTTi7 1/6hranperimeter of basewhere ran 15 1/2san(vertex of pyramid directly above center of base)Right circular conevolume1/3 lateral area zr2hzrswhere rh radius of basealtitudes slant height dmAny pyramid or conevolume1/3 where BhBharea of baseperpendicular distance from vertex to plane in which base lies Wedge (Figure 1-20)a.(Rectangular base; a, parallel to a,a and at distance h above base)volume 1/6 Vhb(2a a,)Spherevolumearea A 4/3zrJ 4.188790r5 1/6zd3 0.523599d’4zr2 nd2where r radiusd 2r diameter 1.240706 0.56419fi

169MathematicsHollow sphere, or spherical shellvolume 4/3where R,rD,d t R, x(D3 - d3) 4xR;t 1/3z(R3 - r3) 1/6outer and inner radiiouter and inner diametersthickness R - rmean radius 1/2 (R r)Any spherical segment (Zone) (Figure 1-21)volume 1/6sch(3a2 3a:lateral area (zone)where r h2)2nrhradius of sphereSpherical sector (Figure 1-22)volume total area1/3 rarea of cap 2/3area of cap area of coneNote: as h(2r- h)m2h 2xrh m aztq

Geometry17Spherical wedge bounded by two plane semicircles and a lune (Figure 1-23)volume of wedge/volume of spherearea of lune/area of spherewhere u u/36Oou/360 dihedral angle of the wedgeRegular polyhedraName of SolidBounded ByA/ dron4 triangles6 squares8 triangles12 pentagons20 000.47147.66312.1817where AVa area of surfacevolumeedgeEllipsoid (Figure 1-24)volume 4/3where a, b, cxabc*semiaxes

18MathematicsSpheroid (or ellipsoid of revolution)By the prismoidal formula:volume1/6 where hA and BM h(A B 4M)altitudeareas of basesarea of a plane section midway between the basesParaboloid of revolution (Figure 1-25)0 # - -#7--,/volume 1/2\nr2h 1/2 volume of circumscribed cylinderTorus, or anchor ring (Figure 1-26)volumearea 2n2cr24nr2cr (proof by theorems of Pappus)ALGEBRASee References 1 and 4 for additional information.Operator Precedence and NotationOperations in a given equation are performed in decreasing order of precedence as follows:

Algebra191. Exponentiation2. Multiplication or division3. Addition or subtractionfrom left to right unless this order is changed by the insertion of parentheses.For example:c - d3/ea bwill be operated upon (calculated) as if it were writtenc) - [(ds)/e]a (bThe symbol I a I means “the absolute value of a,” or the numerical value of aregardless of sign, so that1-21 121 2n! means “n factorial” (where n is a whole number) and is the product ofthe whole numbers 1 to n inclusive, so that24! 14 243The notation for the sum of any real numbers a,, a2, . . . , an isand for their productThe notation ‘‘x y” is read “x varies directly with y” or ‘‘x is directlyproportional to y,” meaning x ky where k is some constant. If x l/y, thenx is inversely proportional to y and x k/y.Rules of Additiona b b a(a b) a - (-b)a-c a (b c)a b and(x - y z) a - x y - z(i.e., a minus sign preceding a pair of parentheses operates to reverse the signsof each term within, if the parentheses are removed)

20MathematicsRules of Multiplication and Simple Factoringa * b b * a(ab)c a(bc)a(b c)a(-b) ab ac-ab and -a(- b) (a b)(a-b) aba2 - b2(a b)2 a2 2ab b2and(a-b)2 a2 - 2ab b2(a b)s a3 3a2 3ab2 bSand(a-b)s a3-3a2 3ab2-bs(For higher order polynomials see the section “Binomial Theorem”)a” b” is factorable by (a b) if n is odd, thusa3 b” (a b)(a2 - ab b2)and a” - b” is factorable by (aa”- b), thus- b” (a - b)(a”-’ a”-2b . . . abn-2 bn-11FractionsThe numerator and denominator of a fraction may be multiplied or dividedby any quantity (other than zero) without altering the value of the fraction, sothat, if m # 0,ma mb mcmx my-a b cX YTo add fractions, transform each to a common denominator and add thenumerators (b,y # 0):To multiply fractions (denominators#0):

Algebra21-ao x

The specific petroleum engineering discipline chapters cover drilling and well completions, reservoir engineering, production, and economics and valuation. These chapters contain information, data, and example calculations related to practical situations that petroleum engineers often encounter. Also, these chapters reflect the growing