standard handbook of petroleum and natural gas engineering

Standard Handbook of
Petroleum & Natural Gas
Engineering
Second Edition
This page intentionally left blank
Standard Handbook of
Petroleum & Natural Gas
Engineering
Second Edition
Editors
William C. Lyons, Ph.D., P.E.
Gary J. Plisga, B.S.
AMSTERDAM
•
BOSTON
•
HEIDELBERG
•
LONDON
•
NEW YORK
•
OXFORD
PARIS
•
SAN DIEGO
•
SAN FRANCISCO
•
SINGAPORE

•
SYDNEY
•
TOKYO
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Library of Congress Cataloging-in-Publication Data
Standard handbook of petroleum & natural gas engineering.—2nd ed./
editors, William C. Lyons, Gary J. Plisga.
p. cm.
Includes bibliographical references and index.
ISBN 0-7506-7785-6
1. Petroleum engineering. 2. Natural gas. I. Title: Standard handbook of
petroleum and natural gas engineering. II. Lyons, William C. III. Plisga, Gary J.
TN870.S6233 2005
665.5–dc22
2004056285
British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.
ISBN: 0-7506-7785-6
For information on all Gulf Professional Publishing
publications visit our Web site at www.gulfpp.com
10987654321
Printed in the United States of America
Contents
Contributing Authors vii
Preface xi
1 Mathematics 1-1
1.1 General 1-2
1.2 Geometry 1-2
1.3 Algebra 1-6
1.4 Trigonometry 1-8
1.5 Differential and Integral Calculus 1-10
1.6 Analytic Geometry 1-16
1.7 Numerical Methods 1-20
1.8 Applied Statistics 1-31
1.9 Computer Applications 1-38
2 General Engineering and Science 2-1
2.1 Basic Mechanics (Statics and
Dynamics) 2-2
2.2 Fluid Mechanics 2-19
2.3 Strength of Materials 2-27
2.4 Thermodynamics 2-32
2.5 Geological Engineering 2-69
2.6 Electricity 2-91
2.7 Chemistry 2-100
2.8 Engineering Design 2-129
3 Auxiliary Equipment 3-1

3.1 Prime Movers 3-2
3.2 Power Transmission 3-17
3.3 Pumps 3-39
3.4 Compressors 3-48
4 Drilling and Well Completions 4-1
4.1 Drilling and Well Servicing Structures 4-2
4.2 Hoisting System 4-9
4.3 Rotary Equipment 4-82
4.4 Mud Pumps 4-95
4.5 Drilling Muds and Completion
Fluids 4-103
4.6 Drill String: Composition and Design 4-124
4.7 Bits and Downhole Tools 4-192
4.8 Drilling Mud Hydraulics 4-255
4.9 Underbalanced Drilling and
Completions 4-259
4.10 Downhole Motors 4-276
4.11 MWD and LWD 4-300
4.12 Directional Drilling 4-356
4.13 Selection of Drilling Practices 4-363
4.14 Well Pressure Control 4-371
4.15 Fishing and Abandonment 4-378
4.16 Casing and Casing String Design 4-406
4.17 Well Cementing 4-438
4.18 Tubing and Tubing String Design 4-467
4.19 Corrosion in Drilling and Well
Completions 4-501
4.20 Environmental Considerations for Drilling
Operations 4-545
4.21 Offshore Drilling Operations 4-558

5 Reservoir Engineering 5-1
5.1 Basic Principles, Definitions, and Data 5-2
5.2 Formation Evaluation 5-53
5.3 Pressure Transient Testing of Oil and
Gas Wells 5-151
5.4 Mechanisms & Recovery of Hydrocarbons
by Natural Means 5-158
5.5 Material Balance and Volumetric
Analysis 5-161
5.6 Decline Curve Analysis 5-168
5.7 Reserve Estimates 5-172
5.8 Secondary Recovery 5-177
5.9 Fluid Movement in Waterflooded
Reservoirs 5-183
5.10 Estimation of Waterflood Residual
Oil Saturation 5-201
5.11 Enhanced Oil Recovery Methods 5-211
6 Production Engineering 6-1
6.1 Properties of Hydrocarbon Mixtures 6-2
6.2 Flow of Fluids 6-40
6.3 Natural Flow Performance 6-89
6.4 Sucker Rod Pumping 6-120
6.5 Stimulation and Remedial Operations 6-218
6.6 Oil and Gas Production Processing
Systems 6-242
6.7 Gas Production Engineering 6-274
6.8 Corrosion in Production Operations 6-371
6.9 Environmental Considerations
in Oil and Gas Operations 6-406
6.10 Offshore Operations 6-424

6.11 Industry Standards for Production
Facilities 6-443
7 Petroleum Economic Evaluation 7-1
7.1 Estimating Producible Volumes and
Future of Production 7-2
7.2 Estimating the Value of Future
Production 7-15
Appendix: Units, Dimensions and
Conversion Factors 1
Index 1
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Contributing Authors
Egill Abrahamsen
Weatherford International Limited
Houston, Texas
Chip Abrant
Weatherford International Limited
Houston, Texas
Bo Anderson
Weatherford International Limited
Houston, Texas
Robert P. Badrak
Weatherford International Limited
Houston, Texas
Frederick Beck
Consultant
Denver, Colorado
Susan Beck
Weatherford International Limited
Houston, Texas

Joe Berry
Varco Incorporated
Houston, Texas
Daniel Boone
Consultant in Petroleum Engineering
Houston, Texas
Gordon Bopp
Environmental Technology and Educational
Services Company
Richland, Washington
Ronald Brimhall
Consultant
College Station, Texas
Ernie Brown
Schlumberger
Sugarland, Texas
Tom Carlson
Halliburton Energy Services Group
Houston, Texas
William X. Chavez, Jr.
New Mexico Institute of Mining and Technology
Socorro, New Mexico
Francesco Ciulla
Weatherford International Limited
Houston, Texas
Vern Cobb
Consultant
Robert Colpitts
Consultant in Geology and Geophysics
Las Vegas, Nevada

Robert B. Coolidge
Weatherford International Limited
Houston, Texas
Heru Danardatu
Schlumberger
Balikpapan, Indonesia
Tracy Darr van Reet
Chevron—retired
El Paso, Texas
Robert Desbrandes
Louisiana State University
Baton Rouge, Louisiana
Aimee Dobbs
Global Santa Fe
Houston, Texas
Patricia Duettra
Consultant in Applied Mathematics and
Computer Analysis
Albuquerque, New Mexico
Ernie Dunn
Weatherford International Limited
Houston, Texas
Michael Economides
University of Houston
Houston, Texas
Jason Fasnacht
Boart Longyear
Salt Lake City, Utah
Joel Ferguson
Weatherford International Limited

Houston, Texas
Jerry W. Fisher
Weatherford International Limited
Houston, Texas
Robert Ford
Smith Bits International
Houston, Texas
Kazimierz Glowacki
Consultant in Energy and Environmental Engineering
Krakow, Poland
Bill Grubb
Weatherford International Limited
Houston, Texas
Mark Heironimus
El Paso Production
El Paso, Texas
Matthew Hill
Unocal Indonesia Company
Jakarta, Indonesia
John Hosford
Chevron Texaco
El Paso, Texas
Phillip Johnson
University of Alabama
Tuscaloosa, Alabama
Harald Jordan
BP America, Inc.
Farmington, New Mexico
Mike Juenke
Weatherford International Limited

Houston, Texas
Reza Kashmiri
International Lubrication and Fuel, Incorporated
Rio Rancho, New Mexico
William Kersting, MS
New Mexico State University
Las Cruces, New Mexico
Murty Kuntamukkla
Westinghouse Savannah River Company
Aiken, South Carolina
Doug LaBombard
Weatherford International Limited
Houston, Texas
Julius Langlinais
Louisiana State University
Baton Rouge, Louisiana
William Lyons
New Mexico Institute of Mining and Technology
Socorro, New Mexico
James Martens
Weatherford International Limited
Houston, Texas
F. David Martin
Consultant
Albuquerque, New Mexico
George McKown
Smith Services
Houston, Texas
David Mildren
Dril Tech Mission

Fort Worth, Texas
Mark Miller
Pathfinder
Texas
Richard J. Miller
Richard J. Miller and Associates, Incorporated
Huntington Beach, California
Stefan Miska
University of Tulsa
Tulsa, Oklahoma
Tom Morrow
Global Santa Fe
Houston, Texas
Abdul Mujeeb
Henkels & McCoy, Incorporated
Blue Bell, Pennsylvania
Bob Murphy
Weatherford International Limited
Houston, Texas
Tim Parker
Weatherford International Limited
Houston, Texas
Pudji Permadi
Institut Teknologi Bandung
Bandung, Indonesia
Jim Pipes
Weatherford International Limited
Houston, Texas
Gary J. Plisga
Consultant in Hydrocarbon Properties

Albuquerque, New Mexico
Floyd Preston
University of Kansas
Lawrence, Kansas
Toby Pugh
Weatherford International Limited
Houston, Texas
Carroll Rambin
Weatherford International Limited
Houston, Texas
Bharath N. Rao
President, Bhavya Technologies, Inc.
Richard S. Reilly
New Mexico Institute of Mining and Technology
Socorro, New Mexico
Cheryl Rofer
Tammoak Enterprises, LLC
Los Alamos, New Mexico
Chris Russell
Consultant in Environmental Engineering
Grand Junction, Colorado
Jorge H.B. Sampaio, Jr.
New Mexico Institute of Mining and Technology
Socorro, New Mexico
Eddie Scales
National Oil Well
Houston, Texas
Ron Schmidt
Weatherford International Limited
Houston, Texas

Ardeshir Shahraki
Dwight’s Energy Data, Inc.
Richardson, Texas
Paul Singer
New Mexico Institute of Mining
and Technology
Socorro, New Mexico
Jack Smith
Weatherford International Limited
Houston, Texas
Mark Trevithick
T&T Engineering Services, Inc.
Houston, Texas
Adrian Vuyk, Jr.
Weatherford International Limited
Houston, Texas
Bill Wamsley
Smith Bits International
Houston, Texas
Sue Weber
Consultant in Computer and Mathematics
Jack Wise
Sandia National Labs
Albuquerque, New Mexico
Andrzej Wojtanowicz
Louisiana State University
Baton Rouge, Louisiana
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Preface
Several objectives guided the preparation of this second

edition of the Standard Handbook of Petroleum and Natu-
ral Gas Engineering. As in thefirst edition, the first objective
in this edition was to continue the effort to create for the
worldwide petroleum and natural gas exploration and pro-
duction industries an engineering handbook written in the
spirit of the classic handbooks of the other important engi-
neering disciplines. Thisnew edition reflects the importance
of these industries to the modern world economies and the
importance of theengineers and technicians that servethese
industries.
The second objective of this edition was to utilize, nearly
exclusively, practicing engineers in industry to carry out the
reviews, revisions, and any re-writes of first edition mate-
rial for the new second edition. The third objective was, of
course, to update the information of the old edition and to
make the new edition more SI friendly. The fourth objective
was to unite the previous two volumes of the first edition
into a single volume that could be available in both book and
CD form. The fifth and final objective of the handbook was
to maintain and enhance the first edition objective of hav-
ing a publication that could be read and understood by any
up-to-date engineer or technician, regardless of discipline.
Theinitialchaptersofthehandbooksetthetonebyinform-
ing the reader of the common language and notation all
engineering disciplines utilize. This common language and
notation is used throughout the handbook (in nearly all
cases consistent with Society of Petroleum Engineers publi-
cation practices). The 75 contributing authors have tried to
avoid the jargon that has crept into petroleum engineering
literature over the past few decades.

The specific petroleum engineering discipline chapters
cover drilling and well completions, reservoir engineering,
production engineering, and economics (with valuation and
risk analysis). These chapters contain information, data,
and example calculations directed toward practical situa-
tions that petroleum engineers often encounter. Also, these
chapters reflect the growing role of natural gas in the world
economies by integrating natural gas topics and related
subjects throughout the volume.
The preparation of this new edition has taken approxi-
mately two years. Throughout the entire effort the authors
have been steadfastly cooperative and supportive of the
editors. In the preparation of the handbook the authors
have used published information from both the American
Petroleum Institute and the Society of Petroleum Engineers.
The authors and editors thank these two institutions for
their cooperation. The authors and editors would also like
to thank all the petroleum production and service company
employees thathave assistedin this project. Specifically, edi-
tors would like to express their great appreciation to the
management and employees of Weatherford International
Limited for providing direct support of this revision. The
editors would also like to specifically thank management
and employees of Burlington Resources Incorporated for
their long term support of the students and faculty at the
New Mexico Institute of Mining and Technology, and for
their assistance in this book. These two companies have
exhibited throughout the longpreparationperiod exemplary
vision regarding the potential value of this new edition to the
industry.

In the detailed preparation of this new edition, the authors
and editors would like to specifically thank Raven Gary. She
started asan undergraduatestudent at New MexicoInstitute
ofMiningandTechnologyinthefallof2000.Sheis now a new
BS graduate in petroleum engineering and is happily work-
ing in the industry. Raven Gary spent her last two years in
college reviewing theincoming materialfrom allthe authors,
checking outlineorganization, figure and table organization,
and references, and communicating with the authors and
Elsevier editors. Our deepest thanks go to Raven Gary. The
authors and editors would also like to thank Phil Carmical
and Andrea Sherman at Elsevier for their very competent
preparation of the final manuscript of this new edition. We
alsothankall those at Elsevier for their support of this project
over the past three years.
All the authors and editors know that this work is not per-
fect. But we also know that this handbook has to be written.
Our greatest hope is that we have given those that will follow
us in future editions of this handbook sound basic material
to work with.
William C. Lyons, Ph.D., P.E.
Socorro, New Mexico
and
Gary J. Plisga, B.S.
Albuquerque, New Mexico
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1
Mathematics
Contents
1.1 GENERAL 1-2

1.2 GEOMETRY 1-2
1.3 ALGEBRA 1-6
1.4 TRIGONOMETRY 1-8
1.5 DIFFERENTIAL AND INTEGRAL
CALCULUS 1-10
1.6 ANALYTIC GEOMETRY 1-16
1.7 NUMERICAL METHODS 1-20
1.8 APPLIED STATISTICS 1-31
1.9 COMPUTER APPLICATIONS 1-38
1-2 MATHEMATICS
1.1 GENERAL
See Reference 1 for additional information.
1.1.1 Sets and Functions
A set is a collection of distinct objects or elements. The inter-
sectionoftwosets S and T is the set of elementswhich belong
to S and which also belong to T. The union (or inclusive) of
S and T is the set of all elements that belong to S or to T (or
to both).
A function canbedefined as aset of orderedpairs, denoted
as (x, y) such that no two such pairs have the same first
element. The element x is referred to as the independent
variable, and the element y is referred to as the dependent
variable. A function is established when a condition exists
that determines y for each x, the condition usually being
defined by an equation such as y = f(x) [2].
References
1. Mark’s Standard Handbook for Mechanical Engineers, 8th
Edition, Baumeister, T., Avallone, E. A., and Baumeister
III, T. (Eds.), McGraw-Hill, New York, 1978.
1.2 GEOMETRY

See References 1 and 2 for additional information.
1.2.1 Angles
Angles can be measured using degrees or with radian mea-
sure. Using the degree system of measurement, a circle has
360
◦
, a straight line has 180
◦
, and a right angle has 90
◦
. The
radian system of measurement uses the arc length of a unit
circle cut off by the angle as the measurement of the angle.
In this system, a circle is measured as 2p radians, a straight
line is p radians and a right angle is p/2 radians. An angle
A is defined as acute if 0
◦
< A < 90
◦
, right if A = 90
◦
,
and obtuse if 90
◦
< A < 180
◦
. Two angles are complemen-
tary if their sum is 90
◦
or are supplementary if their sum is

180
◦
. Angles are congruent if they have the same measure-
ment in degrees and line segments are congruent if they
have the same length. A dihedral angle is 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.
1.2.2 Polygons
A polygon is a closed figure with at least three line segments
that lies within a plane. A regular polygon is a polygon in
which all sides and angles are congruent. Two polygons are
similar if their corresponding angles are congruent and cor-
responding sides are proportional. A segment whose end
points are two nonconsecutive vertices of a polygon is a
diagonal. The perimeteris the sumofthe lengths ofthe sides.
1.2.3 Triangles
A triangle is a three-sided polygon. The sum of the angles of
a triangle is equal to 180
◦
.Anequilateral triangle has three
sides that are the same length, an isosceles triangle has two
sides that are the same length, and a scalene triangle has
three sides of different lengths.
A median of a triangle is a line segment whose end points
are a vertex and the midpoint of the opposite side. An angle
bisector of a triangle is a median that lies on the ray bisect-
ing an angle of the triangle. The altitude of a triangle is a
perpendicular segment from a vertex to the opposite side.
Two triangles are congruent ifoneof the followingis given

(where S = side length and A = angle measurement): SSS,
SAS, AAS, or ASA.
1.2.4 Quadrilaterals
A quadrilateral is a four-sided polygon.
A trapezoid has one pair of opposite parallel sides. A par-
allelogram has both pairs of opposite sides congruent and
parallel. The opposite angles are then congruent, and adja-
cent angles are supplementary. The diagonals bisect each
other and are congruent. A rhombus is a parallelogram
whose four sides are congruent and whose diagonals are
perpendicular to each other.
A rectangle is a parallelogram having four right angles;
therefore, both pairs of opposite sides are congruent.
A rectangle whose sides are all congruent is a square.
1.2.5 Circles and Spheres
If P is a point on a given plane and r is a positive number,
the circle with center P and radius r is the set of all points
of the plane whose distance from P is equal to r. The sphere
with center P and radius r is the set of all points in space
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) isa line segment whose end
points lie on the circle (or sphere). A line which intersects
the circle (or sphere) in two points is a secant of the circle
(or sphere). A diameter of a circle (or sphere) is a chord
containing the center, and a radius is a line segment from
the center to a point on the circle (or sphere).
Theintersection of a spherewitha plane through itscenter
is called a great circle.

A line that intersects a circle at only one point is a tangent
to the circle at that point. Every tangent is perpendicular to
the radius drawn to the point of intersection. Spheres may
have tangent lines or tangent planes.
Pi (p) is the universal ratio of the circumference of any
circle to its diameter and is approximately equal to 3.14159.
Therefore, the circumference of a circle is pdor2pr.
1.2.6 Arcs of Circles
A 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 lie on C (the circle), the
minor arc AB is the union of A, B, and all points on C in the
interior of <APB. The major arc is the union of A, B, and all
points on C on the exterior of <APB. A and B are the end
points of the arc and P is the center. If A and B are the end
points of adiameter, the arc is asemicircle. Asector of acircle
is a region bounded by two radii and an arc of the circle.
1.2.7 Concurrency
Two or morelines are concurrentif there is asingle pointthat
lies on all ofthem.The three altitudesof atriangle (iftakenas
lines, not segments) are always concurrent, and their point
of concurrency is called the orthocenter. The angle bisectors
of a triangle are concurrent at a point equidistant from their
sides, and the medians are concurrent two thirds of the way
along each median from the vertex to the opposite side. The
point of concurrency of the medians is the centroid.
1.2.8 Similarity
Two figures with straight sides are similar if corresponding
angles are congruent andthelengths of corresponding sides
are in the same ratio. A line parallel to one side of a triangle

dividestheother two sides in proportion, producing asecond
triangle similar to the original one.
1.2.9 Prisms and Pyramids
A prism is a three-dimensional figure whose bases are any
congruent and parallel polygons and whose sides are paral-
lelograms. A pyramid is a solid with one base consisting of
GEOMETRY 1-3
any polygon and with triangular sides meeting at a point in
a plane parallel to the base.
Prisms andpyramids are described by their bases: a trian-
gular prism has a triangular base, a parallelpiped is a prism
whosebase is a parallelogram and arectangular parallelpiped
is a right rectangular prism. A cube is a rectangular par-
allelpiped all of whose edges are congruent. A triangular
pyramid has a triangular base, etc. A circular cylinder is a
prism whose base is a circle and a circular cone is a pyramid
whose base is a circle.
1.2.10 Coordinate Systems
Each point on a plane may be defined by a pair of numbers.
The coordinatesystem isrepresented bya lineX in the plane
(the x-axis)and bya line Y (the y-axis) perpendicular to line X
inthe plane, constructedsothattheir intersection, the origin,
is denoted byzero. Any pointP on theplanecan be described
by its two coordinates, which form an ordered pair, so that
P(x
1
,y
1
) is a point whose location corresponds to the real
numbers x and y on the x-axis and the y-axis.

If the coordinate system is extended into space, a third
axis, the z-axis, perpendicular to the plane of the x
1
and y
1
axes, is needed to represent the third dimension coordinate
defining a point P(x
1
,y
1
,z
1
). The z-axis intersects the x and
y axes at their origin, zero. More than three dimensions
are frequently dealt with mathematically but are difficult to
visualize.
The slope m of a line segment in a plane with end points
P
1
(x
1
,y
1
) and P
2
(x
2
,y
2
) is determined by the ratio of the

change in the vertical (y) coordinates to the change in the
horizontal (x) coordinates or
m = (y
2
−y
1
)/(x
2
−x
1
)
except that a vertical line segment (the change in x coor-
dinates 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 are
negative reciprocals are perpendicular to each other.
Because the distance between two points P
1
(x
1
,y
1
) and
P
2
(x
2
,y
2
) is the hypotenuse of a right triangle, the length

(L) of the line segment P
1
P
2
is equal to
L =

(x
2
−x
1
)
2
+(y
2
−y
1
)
2
1.2.11 Graphs
A graph is a set of points lying in a coordinate system and
a graph of a condition (such as x = y + 2) is the set of
all points that satisfy the condition. The graph of the slope-
intercept equation,y= mx+b, isa straight line which passes
through the point (0, b), where b is the y-intercept (x = 0)
and m is the slope. The graph of the equation
(x −a)
2
+(y −b)
2

= r
2
is a circle with center (a, b) and radius r.
1.2.12 Vectors
A vector is described on a coordinate plane by a directed seg-
ment from its initial point to its terminal point. The directed
segment represents the fact that every vector determines a
magnitude and a direction. A vector v is not changed when
moved around the plane, if its magnitude and angular ori-
entation with respect to the x-axis is kept constant. The
initial point of v may therefore be placed at the origin of
the coordinate system and

v may be denoted by

v =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 theorem
v =

a
2
+b
2
Foreverypairof vectors (x
1
,y
1
)and(x

2
,y
2
),thevector sum
is given by (x
1
+x
2
,y
1
+y
2
). The scalar product of the vector
P = (x, y) and a real number (a scalar)risrP= (rx, ry).
Also see the discussion of polar coordinates in the Section
“Trigonometry” and Chapter 2, “Basic Mechanics.”
1.2.13 Lengths and Areas of Plane Figures
For definitions of trigonometric functions, see “Trigonome-
try.”
●
Right triangle (Figure 1.2.1)
c
A
B
c
b
a
c
2
= a

2
+b
2
(Pythagorean theorem)
area = 1/2•ab= 1/2•a
2
cot A
= 1/2•b
2
tan A =1/4•c
2
sin 2A
●
Any triangle (Figure 1.2.2)
A
BC
a
bc
h
area = 1/2base • altitude = 1/2 • ah = 1/2 • absin C
=±1/2 • {(x
1
y
2
−x
2
y
1
)
+ (x

2
y
3
− x
3
y
2
)
+ (x
3
y
1
− x
1
y
3
)}
where (x
1
,y
1
),(x
2
,y
2
),(x
3
,y
3
)are coordinatesofvertices.

●
Rectangle (Figure 1.2.3)
DD
b
a
u
area = ab = 1/2•D
2
sin u
where u = angle between diagonals D, D.
●
Parallelogram (Figure 1.2.4)
D
1
D
2
b
h
C
a
u
area = bh = ab sin c = 1/2•D
1
D
2
sin u
where u = angle between diagonals D
1
and D
2

.
1-4 MATHEMATICS
●
Trapezoid (Figure 1.2.5)
a
u
b
h
D
1
D
2
area = 1/2•(a+b)h = 1/2•D
1
D
2
sin u
where u = angle between diagonals D
1
and D
2
and where bases a and b are parallel.
●
Any quadrilateral (Figure 1.2.6)
b
c
u
d
a
D

1
D
2
area = 1/2•D
1
D
2
sin u
Note: a
2
+b
2
+c
2
+d
2
= D
2
1
+D
2
2
+4m
2
where m = distance between midpoints of D
1
and D
2
.
●

Circles
area = pr
2
= 1/2 • Cr = 1/4 • Cd = 1/4 • pd
2
=
0.785398 d
2
where r = radius
d = diameter
C = circumference = 2pr = pd.
●
Annulus (Figure 1.2.7)
R
r
db
D
area = p(R
2
−r
2
) = p(D
2
−d
2
)/4 = 2pR

b
where R


= mean radius = 1/2•(R+r)
b = R −r
●
Sector (Figure 1.2.8)
r
As
area = 1/2•rs= pr
2
A/360
◦
= 1/2•r
2
rad A
where rad A = radian measure of angle A
s = length of arc = r rad A
●
Ellipse (Figure 1.2.9)
x
b
y
y
b
aa
area of ellipse = pab
area of shaded segment = xy +ab sin
−1
(x/a)
length of perimeter of ellipse = p(a +b)K,
where K = (1 + 1/4 • m
2

+ 1/64 • m
4
+ 1/256 •
m
6
+ )
m = (a −b)/(a +b)
●
Hyperbola (Figure 1.2.10)
a
b
b
A
P
y
y
a
x
For any hyperbola,
shaded area A = ab • ln[(x/a) + (y/b)]
For an equilateral hyperbola (a = b),
area A = a
2
sinh
−1
(y/a) = a
2
cosh
−1
(x/a)

where x and y are coordinates of point P.
●
Parabola (Figure 1.2.11)
A
h
c
GEOMETRY 1-5
shaded area A = 2/3•ch
P
MTOF
P
u
y
In Figure 1.2.12,
length of arc OP = s = 1/2•PT+1/2•p•
ln [cot(1/2 • u) ]
Here c = any chord
p = semilatus rectum
PT = tangent at P
Note: OT =OM = x
1.2.14 Surfaces and Volumes of Solids
●
Regular prism (Figure 1.2.13)
h
a
a
a
r
volume = 1/2 • nrah = Bh
lateral area = nah = Ph

where n = number of sides
B = area of base
P = perimeter of base
●
Right circular cylinder (Figure 1.2.14)
h
r
volume = pr
2
h = Bh
lateral area = 2prh = Ph
where B = area of base
P = perimeter of base
●
Any prism or cylinder (Figure 1.2.15)
h
volume = Bh = Nl
lateral area = Ql
where l = length of an element or lateral edge
B = area of base
N = area of normal section
Q = perimeter of normal section
●
Hollow cylinder (right and circular)
volume = ph(R
2
− r
2
) = phb(D − b) = phb(d + b) =
phbD


= phb(R +r)
where h = altitude
r, R (d, D) = inner and outer radii (diameters)
b = thickness = R −r
D

= mean diam = 1/2•(d+D) = D −b = d +b
●
Sphere
volume = V = 4/3 • pr
3
= 4.188790r
3
= 1/6 • pd
3
=
0.523599d
3
area = A = 4pr
2
= pd
2
where r = radius
d = 2r = diameter =
3

6V/p = 1.24070
3
√

V
=
3

A/p = 0.56419
√
A
●
Hollow sphere, or spherical shell
volume = 4/3 •p(R
3
−r
3
) = 1/6 •p(D
3
−d
3
) = 4pR
2
1
t +
1/3 •pt
3
where R, r = outer and inner radii
D, d = outer and inner diameters
t = thickness = R −r
R
1
= mean radius = 1/2•(R+r)
●

Torus,oranchor ring (Figure 1.2.16)
r
c
volume = 2p
2
cr
2
area = 4pr
2
cr (proof by theorems of Pappus)
1-6 MATHEMATICS
References
1. Moise, E. E., and Downs, Jr., F. L., Geometry, Addison
Wesley, Melano Park, 1982.
2. Graening, J., Geometry, Charles E. Merrill, Columbus,
1980.
1.3 ALGEBRA
See Reference 1.3 for additional information.
1.3.1 Operator Precedence and Notation
Operations in an equation are performed in the following
order of precedence:
1. Parenthesis and grouping symbols
2. Exponents
3. Multiplication or division (left to right)
4. Addition or subtraction (left to right)
For example:
a + b •c −d
3
/e
will be operated upon (calculated) as if it were written

a + (b •c) −[(d
3
)/e]
The symbol |a| means “the absolute value of a,” or the
numerical value of a regardless of sign, so that
|−2|=|2|=2
The n! means “n factorial” (where n is a whole number)
and is the product of the whole numbers 1 to n inclusive, so
that
4!=1 •2 •3 •4 = 24
0!=1 by definition
The notation for the sum of any real numbers a
1
,a
2
, ,a
n
is
n

i=1
a
i
and for their product
n

i=1
a
i
The notation “x ∞ y” is read “x varies directly with y” or

“x is directly proportional to y,” meaning x = ky where k is
some constant. If x ∞ 1/y, then x is inversely proportional
to y and x = k/y.
1.3.2 Rules of Addition
a + b = b +a (commutative property)
(a + b) +c = a + (b +c) (associative property)
a − (−b) = a + b and
a − (x − y + z) = a − x + y − z
(i.e., a minus sign preceding a pair of parentheses operates
to reverse the signs of each term within if the parentheses
are removed)
1.3.3 Rules of Multiplication and Simple Factoring
a • b = b •a (commutative property)
(ab)c = a(bc) (associative property)
a(b +c) = ab +ac (distributive property)
a(−b) =−ab and −a(−b) = ab
(a + b)(a − b) = a
2
−b
2
(a + b)
2
= a
2
+2ab +b
2
and
(a − b)
2
= a

2
−2ab +b
2
(a + b)
3
= a
3
+3a
2
+3ab
2
+b
3
and
(a − b)
3
= a
3
−3a
2
+3ab
2
−b
3
(For higher-order polynomials, see the “Binomial Theo-
rem.”) a
n
+b
n
is factorable by (a +b) if n is odd, and

a
3
+b
3
= (a + b)(a
2
−ab +b
2
)
and a
n
−b
n
is factorable by (a −b), thus
a
n
−b
n
= (a − b)(a
n−1
+a
n−2
b + +ab
n−2
+b
n−1
)
1.3.4 Fractions
The numerator and denominator of a fraction may be mul-
tiplied or divided by any quantity (other than zero) without

altering the value of the fraction, so that, if m = 0,
ma + mb +mc
mx +my
=
a + b +c
x +y
To add fractions, transform each to a common denomina-
tor and add the numerators (b, y = 0):
a
b
+
x
y
=
ay
by
+
bx
by
=
ay +bx
by
To multiply fractions (denominators = 0):
a
b
•
x
y
=
ax

by
a
b
•x =
ax
b
a
b
•
x
y
•
c
z
=
axc
byz
To divide one fraction by another, invert the divisor and
multiply:
a
b
÷
x
y
=
a
b
•
y
x

=
ay
bx
1.3.5 Exponents
a
m
•a
n
= a
m+n
and a
m
÷a
n
= a
m−n
a
0
= 1 (a = 0) and a
1
= a
a
−m
= 1/a
m
(a
m
)
n
= a

mn
a
1/n
=
n
√
a and a
m/n
=
n
√
a
m
(ab)
n
= a
n
b
n
(a/b)
n
= a
n
/b
n
Except in simple cases (square and cube roots), radical
signs are replaced by fractional exponents. If n is odd,
n
√
−a =−

n
√
a
but if n is even, the nth root of −a is imaginary.
1.3.6 Logarithms
The logarithm of a positive number N is the power to
which the base must be raised to produce N. So, x = log
b
N
means b
x
= N. Logarithms to the base 10, frequently used
in numerical computation, are called common or denary log-
arithms, and those to base e, used in theoretical work, are
called natural logarithms and frequently notated as ln.In
any case,
log(ab) = loga + log b
log(a/b) = loga − log b
log(1/n) =−logn
ALGEBRA 1-7
log(a
n
) = n loga
log
b
(b) = 1, where b is either 10 or e
log 0 =−∞
log 1 = 0
log
10

e = M = 0.4342944819 , so for conversion
log
10
x = 0.4343 log
e
x
and since 1/M = 2.302585, for conversion (ln = log
e
)
ln x = 2.3026 log
10
x
1.3.7 Binomial Theorem
Let
n
1
= n
n
2
=
n(n −1)
2!
n
3
=
n(n −1)(n −2)
3!
and so on. Then for any n, |x| < 1,
(1 +x)
n

= 1 +n
1
x +n
2
x
2
+n
3
x
3
+
Ifn is a positive integer,the system is validwithout restriction
on x and completes with the term n
n
x
n
.
Some of the more useful special cases follow [1]:

1 +x = (1 +x)
1/2
= 1 +
1
2
x −
1
8
x
2
+

1
16
x
3
−
5
128
x
4
+ (|x| < 1)
3

1 +x = (1 +x)
1/3
= 1 +
1
3
x −
1
9
x
2
−
5
81
x
3
−
10
243

x
4
+ (|x| < 1)
1
1 +x
= (1 +x)
−1
= 1 −x +x
2
−x
3
+x
4
− (|x| < 1)
1
√
1 +x
= (1 +x)
−1/2
= 1 −
1
2
x +
3
8
x
2
−
5
16

x
3
+
35
128
x
4
− (|x| < 1)
1
3
√
1 +x
= (1 +x)
−1/3
= 1 −
1
3
x +
2
9
x
2
−
14
81
x
3
+
35
243

x
4
− (|x| < 1)

(1 +x)
3
= (1 +x)
3/2
= 1 −
3
2
x +
3
8
x
2
−
1
16
x
3
+
3
128
x
4
− (|x| < 1)
1

(1 +x)

3
= (1 +x)
−3/2
= 1 −
3
2
x +
15
8
x
2
−
35
16
x
3
+
315
128
x
4
− (|x| < 1)
with corresponding formulas for (1 − x)
1/2
, etc., obtained
by reversing the signs of the odd powers of x. Provided
|b| < |a|:
(a + b)
n
= a

n

1 +
b
a

n
= a
n
+n
1
a
n−1
b +n
2
a
n−2
b
2
+n
3
a
n−3
b
3
+
where n
1
,n
2

, etc., have the values given earlier.
1.3.8 Progressions
In an arithmetic progression, (a, a + d, a + 2d, a +3d, ),
each term is obtained from the preceding term by adding a
constant difference, d. If n is the number of terms, the last
termisp= a + (n − 1)d, the “average” term is 1/2(a + p)
and the sum of the terms is n times the average term or
s = n/2(a + p). The arithmetic mean between a and b is
(a +b)/2.
In a geometric progression, (a, ar, ar
2
,ar
3
, ), each term
is obtained from the preceding term by multiplying by acon-
stant ratio, r. The nth term is ar
n−1
, and the sum of the first
n terms is s = a(r
n
−1)/(r −1) = a(1 −r
n
)/(1 −r).Ifrisa
fraction, r
n
will approach zero as n increases and the sum of
n terms will approach a/(1 −r) as a limit.
The geometric mean, also called the “mean proportional,”
between a and b is
√

ab. The harmonic mean between a and
b is 2ab/(a +b).
1.3.9 Sums of the First n Natural Numbers
●
To the first power:
1 +2 +3 + +(n −1) +n = n(n +1)/2
●
To the second power (squared):
1
2
+2
2
+ +(n −1)
2
+n
2
= n(n +1)(2n +1)/6
●
To the third power (cubed):
1
3
+2
3
+ +(n −1)
3
+n
3
=[n(n +1)/2]
2
1.3.10 Solution of Equations in One Unknown

Legitimate operations on equations include addition of any
quantity to both sides, multiplication by any quantity of both
sides (unless this would result in division by zero), raising
both sides to any positive power (if ±is used for even roots)
and taking the logarithm or the trigonometric functions of
both sides.
Any algebraic equation may be written as a polynomial of
nth degree in x of the form
a
0
x
n
+a
1
x
n−1
+a
2
x
n−2
+ +a
n−1
x +a
n
= 0
with, in general, n roots, some of which may be imaginary
and some equal.If the polynomialcan be factoredin the form
(x −p)(x − q)(x − r) = 0
then p, q, r, are the roots of the equation. If |x| is very
large, the terms containing the lower powers of x are least

important, while if |x| is very small, the higher-order terms
are least significant.
First-degree equations (linear equations) have the form
ax +b = c
with the solution x = b − a and the root b − a.
Second-degree equations (quadratic equations) have the
form
ax
2
+bx +c = 0
with the solution
x =
−b ±
√
b
2
−4ac
2a
1-8 MATHEMATICS
and the roots
−b +
√
b
2
−4ac
2a
and
−b −
√
b

2
−4ac
2a
The sum of the roots is −b/a and their product is c/a.
Third-degree equations (cubic equations) have the form,
after division by the coefficient of the highest-order term,
x
3
+ax
2
+bx +c = 0
with the solution
x
3
1
= Ax
1
+B
where x
1
= x − a/3
A = 3(a/3)
2
− b
B =−2(a/3)
3
+ b(a/3) − c
Exponential equations are of the form
a
x

= b
with the solution x = (log b)/(log a) and the root (log b)/
(log a). The complete logarithm must be taken, not just the
mantissa.
1.3.11 Solution of Systems of Simultaneous Equations
A set of simultaneous equations is a system of n equations in
n unknowns. The solutions (if any) are the sets of values for
the unknowns that satisfy all the equations in the system.
First-degree equations in 2 unknowns are of the form
a
1
x
1
+b
1
x
2
= c
1
a
2
x
1
+b
2
x
2
= c
2
The solution is found by multiplication of Equations 1.3.1

and 1.3.2 by some factors that will produce one term in each
that will, upon addition of Equations 1.3.1 and 1.3.2, become
zero. The resulting equation may then be rearranged to
solve for the remaining unknown. For example, by multiply-
ing Equation 1.3.1 by a
2
and Equation 1.3.2 by −a
1
, adding
Equation 1.3.1and Equation1.3.2 and rearranging their sum
x
2
=
a
2
c
1
−a
1
c
2
a
2
b
1
−a
1
b
2
and by substitution in Equation 1.3.1:

x
1
=
b
1
c
2
−b
2
c
1
a
2
b
1
−a
1
b
2
A set of n first-degree equations in n unknowns is solved in
a similar fashion by multiplication and addition to eliminate
n − 1 unknowns and then back substitution. Second-degree
equations in 2unknownsmaybesolvedinthesamewaywhen
two of the following are given: the product of the unknowns,
their sum ordifference, thesum of theirsquares. For further
solutions, see “Numerical Methods.”
1.3.12 Determinants
Determinants of the second order are of the following form
and are evaluated as





a
1
b
1
a
2
b
2




= a
1
b
2
−a
2
b
1
and of the third order as







a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3






= a
1





b
2
c
2
b
3
c
3




−a
2




b
1
c
1
b
3
c
3





+a
3




b
1
c
1
b
2
c
2




and of higher orders, by the general rules as follows. To
evaluate a determinant of the nth order, take the elements
of the first column with alternate plus and minus signs and
form the sum of the products obtained by multiplying each
of these elements by its corresponding minor. The minor
corresponding to any element e
n
is the determinant (of the
next lowest order) obtained by striking out from the given
determinant the row and column containing e
n
.

Some of the general properties of determinants are
1. Columns may be changed to rows and rows to columns.
2. Interchanging two adjacent columns changes the sign of
the result.
3. If two columnsare equalor if oneis a multipleof the other,
the determinant is zero.
4. To multiply a determinant by any number m, multiply all
elements of any one column by m.
Systems of simultaneous equations may be solved by the
useofdeterminantsusingCramer’srule.Althoughtheexam-
ple is a third-order system, larger systems may be solved by
this method. If
a
1
x +b
1
y +c
1
z = p
1
a
2
x +b
2
y +c
2
z +p
2
a
3

x +b
3
y +c
3
z = p
3
and if
D =






a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3

c
3






= 0
then
x = D
1
/D
y = D
2
/D
z = D
3
/D
where
D
1
=






p

1
b
1
c
1
p
2
b
2
c
2
p
3
b
3
c
3






D
2
=







a
1
p
1
c
1
a
2
p
2
c
2
a
3
p
3
c
3






D
3
=







a
1
b
1
p
1
a
2
b
2
p
2
a
3
b
3
p
3






References

1. Benice, D. D., Precalculus Mathematics, 2nd Edition,
Prentice Hall, Englewood Cliffs, 1982.
1.4 TRIGONOMETRY
1.4.1 Directed Angles
If AB and AB

are any two rays with the same end point A,
the directed angle <BAB

is the ordered pair (
−→
AB,
−→
AB

).
−→
AB
is the initial side of <BAB

and
−→
AB

the terminal side. <BAB

= <B

AB and any directed angle may be ≤0
◦

or ≥180
◦
.
A directed angle may be thought of as an amount of rota-
tion rather than a figure. If
−→
AB is considered the initial
position of the ray, which is then rotated about its end point
A to form <BAB

,
−→
AB

is its terminal position.
1.4.2 Basic Trigonometric Functions
A trigonometric function can be defined for an angle q
between 0
◦
and 90
◦
by using Figure 1.4.1.
1.4.3 Trigonometric Properties
sin q = opposite side/hypotenuse = s
1
/h
cos q = adjacent side/hypotenuse = s
2
/h
tan q = opposite side/adjacent side = s

1
/s
2
= sin q/ cos q
TRIGONOMETRY 1-9
S
2
S
1
h
θ
Figure 1.4.1 Trigonometric functions of angles.
andthe reciprocalsofthe basic functions (wherethe function
= 0)
cotangent q = cot q = 1/ tanq = s
2
/s
1
secant q = secq = 1/ cosq = h/s
2
cosecant q = cscq = 1/ sinq = h/s
1
To reduce an angle to the first quadrant of the unit
circle, that is, to a degree measure between 0
◦
and 90
◦
,
see Table 1.4.1. Forfunction valuesat majorangle values,see
Tables 1.4.2 and 1.4.3. Relations between functions and the

sum or difference of two functions are given in Table 1.4.4.
Generally, there will be two angles between 0
◦
and 360
◦
that
correspond to the value of a function.
The trigonometric functions sine and cosine can be
defined for any real number by using the radian measure
of the angle as described in the section on angles. The tan-
gent function is defined on every real number except for
places where cosine is zero.
1.4.4 Graphs of Trigonometric Functions
Graphs of thesineand cosine functionsare identical inshape
and periodic with a period of 360
◦
. The sine function graph
Table 1.4.1 Angle Reduction to First Quadrant
If 90
◦
< x < 180
◦
180
◦
< x < 270
◦
270
◦
< x < 360
◦

sin x =+cos(x −90
◦
) −sin(x −180
◦
) −cos(x −270
◦
)
cos x =−sin(x −90
◦
) −cos(x −180
◦
) +sin(x −270
◦
)
tan x =−cot(x −90
◦
) +tan(x −180
◦
) −cot(x −270
◦
)
csc x =+sec(x −90
◦
) −csc(x − 180
◦
) −sec(x − 270
◦
)
sec x =−csc(x −90
◦

) −sec(x − 180
◦
) +csc(x − 270
◦
)
cot x =−tan(x −90
◦
) +cot(x −180
◦
) −tan(x −270
◦
)
translated ± 90
◦
along the x-axis produces the graph of the
cosine function. The graph of the tangent function is discon-
tinuous when the value of tan q is undefined, that is, at odd
multiples of90
◦
( ,90
◦
, 270
◦
, ).For abbreviated graphs
of the sine, cosine, and tangent functions, see Figure 1.4.2.
1.4.5 Inverse Trigonometric Functions
The inverse sine of x (also referred to as the arc sine of x),
denoted by sin
−1
x, is the principal angle whose sine is x,

that is,
y = sin
−1
x means sin y = x
Inverse functions cos
−1
x and tan
−1
x also exist for the
cosine of y and the tangent of y. The principal angle for
sin
−1
x and tan
−1
x is an angle a, where −90
◦
< a < 90
◦
,
and for cos
−1
x, 0
◦
< a < 180
◦
.
1.4.6 Solution of Plane Triangles
The solution of any part of a plane triangle is determined in
general by any other threeparts given by oneof the following
groups, where S is the length of a side and A is the degree

measure of an angle:
●
AAS
●
SAS
●
SSS
The fourth group, two sides and the angle opposite one
of them, is ambiguous since it may give zero, one, or two
solutions. Given an example triangle with sides a, b, and c
and angles A, B, and C (A being opposite a, etc., and A + B
+ C = 180
◦
), the fundamental laws relating to the solution
of triangles are
1. Law of sines: a/(sin A) = b/(sin B) = c/(sin C)
2. Law of cosines: c
2
= a
2
+ b
2
− 2ab cos C
1.4.7 Hyperbolic Functions
The hyperbolic sine, hyperbolic cosine, etc., of any number x
arefunctionsrelatedtotheexponentialfunctione
x
.Theirdef-
initions and properties are very similar to the trigonometric
functions and are given in Table 1.4.5.

Table 1.4.3 Trigonometric Function Values at Major
Angle Values
Values at 30
◦
45
◦
60
◦
sin x 1/2 1/2
√
2 1/2
√
3
cos x 1/2
√
3 1/2
√
2 1/2
tan x 1/3
√
31
√
3
csc x 2
√
2 2/3
√
3
sec x 2/3
√

3
√
22
cot x
√
3 1 1/3
√
3
Table 1.4.2 Trigonometric Function Values by Quadrant
If 0
◦
< x < 90
◦
90
◦
< x < 180
◦
180
◦
< x < 270
◦
270
◦
< x < 360
◦
sin x +0to+1 +1to+0 −0to−1 −1to−0
cos x +1to+0 −0to−1 −1to−0 +0to+1
tan x +0to+∞ −∞to −0 +0to+∞ −∞ to −0
csc x +∞to +1 +1to+∞ −∞ to −1 −1to−∞
sec x +1to+∞ −∞to −1 −1to−∞ +∞ to +1

cot x +∞ to +0 −0to−∞ +∞ to +0 −0to−∞
1-10 MATHEMATICS
Table 1.4.4 Relations Between Trigonometric Functions
of Angles
Single Angle
sin
2
x +cos
2
x = 1
tan x = (sin x)/(cos x)
cot x = 1/(tan x)
1 +tan
2
x = sec
2
x
1 +cot
2
x = csc
2
x
sin(−x) = −sin x, cos(−x) = cos x, tan(−x)= −tan x
Two Angles
sin(x +y) = sin x cos y +cos x sin y
sin(x −y) = sin x cos y −cos x sin y
cos(x +y) = cos x cos y −sin x sin y
cos(x −y) = cos x cos y +sin x sin y
tan(x +y) = (tan x +tan y)/(1 −tan x tan y)
tan(x −y) = (tan x −tan y)/(1 +tan x tan y)

cot(x +y) = (cot x cot y −1)/(cot y +cot x)
cot(x −y) = (cot x cot y +1)/(cot y −cot x)
sin x +sin y = 2 sin[1/2(x +y)] cos[1/2(x −y)]
sin x −sin y = 2 cos[1/2(x +y)] sin[1/2(x −y)]
cos x +cos y = 2 cos[1/2(x +y)] cos[1/2(x −y)]
cos x −cos y= −2 sin[1/2(x +y)] sin [1/2(x −y)]
tan x +tan y = [sin(x +y)]/[cos x cos y]
tan x −tan y = [sin(x −y)]/[cos x cos y]
cot x +cot y = [sin(x +y)]/[sin x sin y]
cot x −cot y = [sin(y −x)]/[sin x sin y]
sin
2
x −sin
2
y = cos
2
y −cos
2
x
= sin(x +y) sin (x −y)
cos
2
x −sin
2
y = cos
2
y −sin
2
x
= cos(x +y) cos(x −y)

sin(45
◦
+x) = cos(45
◦
−x), tan(45
◦
+x) = cot(45
◦
−x)
sin(45
◦
−x) = cos(45
◦
+x), tan(45
◦
−x) = cot(45
◦
+x)
Multiple and Half Angles
tan 2x = (2 tan x)/(1 −tan
2
x)
cot 2x = (cot
2
x −1)/(2 cot x)
sin(nx) = n sin x cos
n−1
x −(n)
3
sin

3
x cos
n−3
x
+(n)
5
sin
5
x cos
n−5
x −
cos(nx) = cos
n
x −(n)
2
sin
2
x cos
n−2
x
+(n)
4
sin
4
x cos
n−4
x −
(Note: (n)
2
, are the binomial coefficients)

sin(x/2) =±

1/2(1 −cosx)
cos(x/2) =±

1/2(1 +cosx)
tan(x/2) = (sin x)/(1 +cos x) =±

(1 −cosx)/(1 +cos x)
Three Angles Whose Sum = 180
◦
sin A +sin B +sin C = 4 cos(A/2) cos(B/2) cos(C/2)
cos A +cos B +cos C = 4 sin(A/2) sin(B/2) sin(C/2) +1
sin A +sin B −sin C = 4 sin(A/2) sin(B/2) cos(C/2)
cos A +cos B −cos C = 4 cos(A/2) cos(B/2) sin(C/2) −1
sin
2
A +sin
2
B +sin
2
C = 2 cos A cos B cos C +2
sin
2
A +sin
2
B −sin
2
C = 2 sin A sin B cos C
tan A +tan B +tan C = tan A tan B tan C

cot(A/2) +cot(B/2) +cot(C/2)
= cot(A/2) cot(B/2) cot(C/2)
sin 2A +sin 2B +sin 2C = 4 sin A sin B sin C
sin 2A +sin 2B −sin 2C = 4 cos A cos B sin C
The inverse hyperbolic functions, sinh
−1
x, etc., are related
to the logarithmic functions and are particularly useful in
integral calculus. These relationshipsmay be defined forreal
numbers x and y as
sinh
−1
(x/y) = ln(x +

x
2
+y
2
) −ln y
cosh
−1
(x/y) = ln(x +

x
2
−y
2
) −ln y
tanh
−1

(x/y) = 1/2•ln[(y + x)/(y − x)]
coth
−1
(x/y) = 1/2•ln[(x + y)/(x − y)]
1.4.8 Polar Coordinate System
The polar coordinate system describes the location of a point
(denoted as [r, q]) in a plane by specifying a distance r and
an angle q from the origin of the system. There are several
relationships between polar and rectangular coordinates,
diagrammed in Figure1.4.3. From the Pythagorean theorem
r =±

x
2
+y
2
Also
sin q = y/rory= r sinq
cos q = x/rorx= r cos q
tan q = y/xorq = tan
−1
(y/x)
To convert rectangular coordinates to polar coordinates,
given the point (x, y), using the Pythagorean theorem and
the preceding equations.
[r, q]=


x
2

+y
2
, tan
−1
(y/x)

To convert polar to rectangular coordinates, given the
point [r, q]:
(x, y) =[r cos q, r sin q]
For graphic purposes, the polar plane is usually drawn as
a seriesof concentriccircles withthe centerat theorigin and
radii 1, 2, 3, Rays from the center are drawn at 0
◦
,15
◦
,
30
◦
, ,360
◦
or 0, p/12,p/6, p/4, ,2p radians.The origin
is called the pole, and points [r, q] are plotted by moving a
positive or negative distance r horizontally from the pole,
and through an angle q from the horizontal. See Figure 1.4.4
with q given in radians as used in calculus. Also note that
[r, q]=[−r, q + p]
1.5 DIFFERENTIAL AND INTEGRAL CALCULUS
See References 1–4 for additional information.
1.5.1 Derivatives
Geometrically, the derivative of y = f(x) at any value x

n
is
the slope of a tangent line T intersecting the curve at the
point P(x, y).Two conditions applying to differentiation (the
process of determining the derivatives of a function) are
1. The primary (necessary and sufficient) condition is that
lim
Dx→0
Dy
Dx
exists and is independent of the way in which Dx → 0
2. A secondary (necessary, not sufficient) condition is that
lim
Dx→0
f(x +Dx) = f(x)
A short table of derivatives will be found in Table 1.5.1.
1.5.2 Higher-Order Derivatives
The second derivative of a function y = f(x), denoted f

(x)
or d
2
y/dx
2
is the derivative of f

(x) and the third derivative,
f

(x) is thederivative off


(x). Geometrically, interms of f(x):
if f

(x) > 0 then f(x) is concave upwardly, if f

(x) < 0 then
f(x) is concave downwardly.
1.5.3 Partial Derivatives
If u = f(x, y, ) is a function of two or more variables,
the partial derivative of u with respect to x, f
x
(x, y, )or
∂u/∂x, may be formed by assuming x to be the indepen-
dent variable and holding (y, ) as constants. In a similar
manner, f
y
(x, y, )or∂u/∂y may be formed by holding
(x, ) as constants. Second-order partial derivatives of
f(x, y) are denoted by the manner of their formation as f
xx
,
DIFFERENTIAL AND INTEGRAL CALCULUS 1-11
sin X
1
0
90
The sine graph
period 360˚
The cosine graph

period 360˚
The tangent graph
period 360˚
180 270 360
−1
cos X tan X
12
00
90 90180 180270 270360
−1 −2
−90
Figure 1.4.2 Graphs of the trigonometric functions.
Table 1.4.5 Hyperbolic Functions
sinh x = 1/2(e
x
−e
−x
)
cosh x = 1/2(e
x
+e
−x
)
tanh x = sinh x/cosh x
csch x = 1/sinh x
sech x = 1/cosh x
coth x = 1/tanh x
sinh(−x) =−sinh x
cosh(−x) = cosh x
tanh(−x) =−tanh x

cosh
2
x −sinh
2
x = 1
1 −tanh
2
x = sech
2
x
1 −coth
2
x =−csch
2
x
sinh(x ±y) = sinh x cosh y ±cosh x sinh y
cosh(x ±y) = cosh x cosh y ±sinh x sinh y
tanh(x ±y) = (tanh x ±tanh y)/(1 ±tanh x tanh y)
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh
2
x +sinh
2
x
tanh 2x = (2 tanh x)/(1 +tanh
2
x)
sinh(x/2) =

1/2(cosh x − 1)

cosh(x/2) =

1/2(cosh x + 1)
tanh(x/2) = (cosh x −1)/(sinh x) = (sinh x)/(cosh x +1)
f
xy
(equal to f
yx
), f
yy
or as ∂
2
u/∂x
2
, ∂
2
u/∂x∂y, ∂
2
u/∂y
2
, and
the higher-order partial derivatives are likewise formed.
Implicit functions (i.e., f(x, y) = 0) may be solved by the
formula
dy
dx
=−
f
x
f

y
at the point in question.
1.5.4 Maxima and Minima
A critical point on a curve y = f(x) is a point where y

= 0,
that is, where the tangent to thecurve is horizontal.A critical
value of x is therefore a value such that f

(x) =0. All roots of
the equation f

(x) = 0 are critical values of x, and the corre-
sponding values of y are the critical values of the function.
A function f(x) has a relative maximum at x = a if f(x)
< f(a) for all values of x (except a) in some open interval
containing a and a relative minimum at x = b if f(x) > f(b)
for allx (except b) in the interval containingb. Atthe relative
maximum a of f(x), f

(a) = 0, i.e., slope = 0, and f

(a) < 0,
i.e., the curve is downwardly concave at this point, and at
the relative minimum b, f

(b) = 0 and f

(b) > 0 (upward
concavity). In Figure 1.5.1 A, B, C, and D are critical points

and x
1
,x
2
,x
3
, and x
4
are critical values of x. A and C are
maxima, B is a minimum, and D is neither. D, F, G, and
Harepoints of inflection where the slope is minimum or
maximum. In special cases, such as E, maxima or minima
may occur where f

(x) is undefined or infinite.
Y
Y
X
X
r
θ
Figure 1.4.3 Polar coordinates.
π
/2
π
/3
π
/4
π
/6

11π
/6
7π
/4
5π
/3
3π
/2
4π
/3
5π
/4
7π
/5
5π
/6
3π
/4
2π
/3
π
0
Figure 1.4.4 The polar plane.
The absolute maximum(or minimum)of f(x)at x=aexists
if f(x) ≤ f(a) (or f(x) ≥ f(a)) for all x in the domain of the
function and need not be a relative maximum or minimum.
If a function is defined and continuous on a closed interval, it
1-12 MATHEMATICS
Table 1.5.1 Table of Derivatives
a

d
dx
(x) = 1
d
dx
sin
−1
u =
1
√
1−u
2
du
dx
d
dx
(a) = 0
d
dx
cos
−1
u =−
1
√
1−u
2
du
dx
d
dx

(u ±v ± ) =
du
dx
±
dv
dx
±
d
dx
tan
−1
u =
1
1+u
2
du
dx
d
dx
(au) = a
du
dx
d
dx
cot
−1
u =−
1
1+u
2

du
dx
d
dx
(uv) = u
dv
dx
+v
du
dx
d
dx
sec
−1
u =
1
u
√
u
2
−1
du
dx
d
dx
u
v
=
v
du

dx
−u
dv
dx
v
2
d
dx
csc
−1
u =−
1
u
√
u
2
−1
du
dx
d
dx
(u
a
) = nu
a−1
du
dx
d
dx
vers

−1
u =
1
√
2u−u
2
du
dx
d
dx
log
a
u =
log
a
e
u
du
dx
d
dx
sinh u = cosh u
du
dx
d
dx
log u =
1
u
du

dx
d
dx
cosh u = sinh u
du
dx
d
dx
a
u
= a
u
• log
e
a•
du
dx
d
dx
tanh u = sec h
2
u
du
dx
d
dx
e
u
= e
u

du
dx
d
dx
coth u =−csc h
2
u
du
dx
d
dx
u
v
= vu
v−1
du
dx
+u
v
log
e
u
dv
dx
d
dx
sechu =−sechu tanh u
du
dx
d

dx
sin u = cos u
du
dx
d
dx
cschu =−cschu coth u
du
dx
d
dx
cos u =−sinu
du
dx
d
dx
sinh
−1
u =
1
√
u
2
−1
du
dx
d
dx
tan u = sec
2

u
du
dx
d
dx
cosh
−1
u =
1
√
u
2
−1
du
dx
d
dx
cot u =−csc
2
u
du
dx
d
dx
tanh
−1
u =
1
1−u
2

du
dx
d
dx
sec u = sec u tan u
du
dx
d
dx
coth
−1
u =−
1
u
2
−1
du
dx
d
dx
csc u =−csc u cot u
du
dx
d
dx
sech
−1
u =−
1
u

√
1−u
2
du
dx
d
dx
vers u = sin u
du
dx
d
dx
csc h
−1
u =−
1
u
√
u
2
−1
du
dx
a
The u and v represent functions of x. All angles are in radians.
will always have an absolute minimum and an absolute max-
imum, and they will be found either at a relative minimum
and a relative maximum or at the endpoints of the interval.
1.5.5 Differentials
If y = f(x) and Dx and Dy are the increments of x and y,

respectively, because y + Dy = f(x + Dx), then
Dy = f(x + Dx) −f(x)
AsDx approachesitslimit0and(sincexistheindependent
variable) dx = Dx
dy
dx
∼
=
f(x +Dx) −f(x)
Dx
and
dy
∼
=
Dy
By defining dy and dx separately, it is now possible to write
dy
dx
= f

(x)
as
dy = f

= (x)dx
In functions of two or more variables, where f(x, y, ) =
0, if dx, dy, are assigned to the independent variables x,
y, , the differential du is given by differentiating term by
term or by taking
du = f

x
•dx +f
y
•dy +
If x, y, are functions of t, then
du
dt
= (f
x
)
dx
dt
+(f
y
)
dy
dt
+
expresses the rate of change of u with respect to t, in terms
of the separate rates of change of x, y, with respect to t.
Y
B
F
C
G
D
H
X
E
x

4
x
3
x
2
x
1
A
Figure 1.5.1 Maxima and minima.
Y
P
s
X
∆y
∆x
∆s
z
Figure 1.5.2 Radius of curvature in rectangular
coordinates.
1.5.6 Radius of Curvature
The radius of curvature R ofa plane curve at anypoint Pis the
distance along the normal (the perpendicular to the tangent
to the curve at point P) on the concave side of the curve to
the center of curvature (Figure 1.5.2). If the equation of the
curve is y = f(x)
R =
ds
du
=
[1 +f


(x)
2
]
3/2
f

(x)
where the rate of change (ds/dx) and the differential of the
arc (ds), s being the length of the arc, are defined as
ds
dx
=

1 +

dy
dx

2
and
ds =

dx
2
+dy
2
and dx = ds cos u
dy = ds sin u
u = tan

−1
[f

(x)]
with u being the angle of the tangent at P with respect to the
x-axis. (Essentially, ds, dx, and y correspond to the sides of
a right triangle.) The curvature K is the rate at which <uis
changing with respect to s, and
K =
1
R
=
du
ds
If f

(x) is small, K
∼
=
f

(x).