1. N C E E S
FUNDAMENTALS OF ENGINEERING
SUPPLIED-REFERENCE HANDBOOK
FIFTH EDITION
N A T I O N A L C O U N C I L O F E X A M I N E R S
F O R E N G I N E E R I N G A N D S U R V E Y I N G
 2001 by the National Council of Examiners for Engineering and Surveying® .

2. FUNDAMENTALS OF ENGINEERING
SUPPLIED-REFERENCE HANDBOOK
FIFTH EDITION
Prepared by
National Council of Examiners for Engineering and Surveying® (NCEES®)
280 Seneca Creek Road
P.O. Box 1686
Clemson, SC 29633-1686
Telephone: (800) 250-3196
Fax: (864) 654-6033
www.ncees.org

3.  2001 by the National Council of Examiners for Engineering and Surveying® .
All rights reserved. First edition 1996
Fifth edition 2001

4. FOREWORD
During its August 1991 Annual Business Meeting, the National Council of Examiners for Engineering and Surveying (NCEES)
voted to make the Fundamentals of Engineering (FE) examination an NCEES supplied-reference examination. Then during its
August 1994 Annual Business Meeting, the NCEES voted to make the FE examination a discipline-specific examination. As a
result of the 1994 vote, the FE examination was developed to test the lower-division subjects of a typical bachelor engineering
degree program during the morning portion of the examination, and to test the upper-division subjects of a typical bachelor
engineering degree program during the afternoon. The lower-division subjects refer to the first 90 semester credit hours (five
semesters at 18 credit hours per semester) of engineering coursework. The upper-division subjects refer to the remainder of the
engineering coursework.
Since engineers rely heavily on reference materials, the FE Supplied-Reference Handbook will be made available prior to the
examination. The examinee may use this handbook while preparing for the examination. The handbook contains only reference
formulas and tables; no example questions are included. Many commercially available books contain worked examples and
sample questions. An examinee can also perform a self-test using one of the NCEES FE Sample Questions and Solutions books
(a partial examination), which may be purchased by calling (800) 250-3196.
The examinee is not allowed to bring reference material into the examination room. Another copy of the FE Supplied-Reference
Handbook will be made available to each examinee in the room. When the examinee departs the examination room, the FE
Supplied-Reference Handbook supplied in the room shall be returned to the examination proctors.
The FE Supplied-Reference Handbook has been prepared to support the FE examination process. The FE Supplied-Reference
Handbook is not designed to assist in all parts of the FE examination. For example, some of the basic theories, conversions,
formulas, and definitions that examinees are expected to know have not been included. The FE Supplied-Reference Handbook
may not include some special material required for the solution of a particular question. In such a situation, the required special
information will be included in the question statement.
DISCLAIMER: The NCEES in no event shall be liable for not providing reference material to support all the
questions in the FE examination. In the interest of constant improvement, the NCEES reserves the right to
revise and update the FE Supplied-Reference Handbook as it deems appropriate without informing interested
parties. Each NCEES FE examination will be administered using the latest version of the FE Supplied-
Reference Handbook.
So that this handbook can be reused, PLEASE, at the examination site,
DO NOT WRITE IN THIS HANDBOOK.
iii

5. TABLE OF CONTENTS
UNITS ........................................................................................................................................ 1
CONVERSION FACTORS ............................................................................................................ 2
MATHEMATICS .......................................................................................................................... 3
STATICS.................................................................................................................................... 22
DYNAMICS ............................................................................................................................... 24
MECHANICS OF MATERIALS ..................................................................................................... 33
FLUID MECHANICS ................................................................................................................... 38
THERMODYNAMICS .................................................................................................................. 47
HEAT TRANSFER ...................................................................................................................... 58
TRANSPORT PHENOMENA ......................................................................................................... 63
CHEMISTRY .............................................................................................................................. 64
MATERIALS SCIENCE/STRUCTURE OF MATTER ........................................................................ 68
ELECTRIC CIRCUITS ................................................................................................................. 72
COMPUTERS, MEASUREMENT, AND CONTROLS........................................................................ 76
ENGINEERING ECONOMICS ....................................................................................................... 79
ETHICS ..................................................................................................................................... 86
CHEMICAL ENGINEERING ......................................................................................................... 88
CIVIL ENGINEERING ................................................................................................................. 92
ENVIRONMENTAL ENGINEERING ............................................................................................ 117
ELECTRICAL AND COMPUTER ENGINEERING .......................................................................... 134
INDUSTRIAL ENGINEERING ..................................................................................................... 143
MECHANICAL ENGINEERING .................................................................................................. 155
INDEX ..................................................................................................................................... 166
v

6. UNITS
This handbook uses the metric system of units. Ultimately, the FE examination will be entirely metric. However, currently some
of the problems use both metric and U.S. Customary System (USCS). In the USCS system of units, both force and mass are
called pounds. Therefore, one must distinguish the pound-force (lbf) from the pound-mass (lbm).
The pound-force is that force which accelerates one pound-mass at 32.174 ft/s2. Thus, 1 lbf = 32.174 lbm-ft/s2. The expression
32.174 lbm-ft/(lbf-s2) is designated as gc and is used to resolve expressions involving both mass and force expressed as pounds.
For instance, in writing Newton's second law, the equation would be written as F = ma/gc, where F is in lbf, m in lbm, and a is
in ft/s2.
Similar expressions exist for other quantities. Kinetic Energy: KE = mv2/2gc, with KE in (ft-lbf); Potential Energy: PE = mgh/gc,
with PE in (ft-lbf); Fluid Pressure: p = ρgh/gc, with p in (lbf/ft2); Specific Weight: SW = ρg/gc, in (lbf/ft3); Shear Stress: τ =
(µ/gc)(dv/dy), with shear stress in (lbf/ft2). In all these examples, gc should be regarded as a unit conversion factor. It is
frequently not written explicitly in engineering equations. However, its use is required to produce a consistent set of units.
Note that the conversion factor gc [lbm-ft/(lbf-s2)] should not be confused with the local acceleration of gravity g, which has
different units (m/s2) and may be either its standard value (9.807 m/s2) or some other local value.
If the problem is presented in USCS units, it may be necessary to use the constant gc in the equation to have a consistent set of
units.
METRIC PREFIXES
Multiple Prefix Symbol COMMONLY USED EQUIVALENTS
10–18 atto a 1 gallon of water weighs 8.34 lbf
10–15 femto f 1 cubic foot of water weighs 62.4 lbf
10–12 pico p
1 cubic inch of mercury weighs 0.491 lbf
10–9 nano n
The mass of one cubic meter of water is 1,000 kilograms
10–6 micro µ
10–3 milli m
10–2 centi c TEMPERATURE CONVERSIONS
10–1 deci d ºF = 1.8 (ºC) + 32
101 deka da ºC = (ºF – 32)/1.8
102 hecto h
ºR = ºF + 459.69
103 kilo k
106 mega K = ºC + 273.15
M
109 giga G
1012 tera T
1015 peta P
1018 exa E
FUNDAMENTAL CONSTANTS
Quantity Symbol Value Units
electron charge e 1.6022 × 10−19 C (coulombs)
Faraday constant 96,485 coulombs/(mol)
gas constant metric R 8,314 J/(kmol·K)
gas constant metric R 8.314 kPa·m3/(kmol·K)
gas constant USCS R 1,545 ft-lbf/(lb mole-ºR)
R 0.08206 L-atm/mole-K
gravitation - newtonian constant G 6.673 × 10–11 m3/(kg·s2)
gravitation - newtonian constant G 6.673 × 10–11 N·m2/kg2
gravity acceleration (standard) metric g 9.807 m/s2
gravity acceleration (standard) USCS g 32.174 ft/s2
molar volume (ideal gas), T = 273.15K, p = 101.3 kPa Vm 22,414 L/kmol
speed of light in vacuum c 299,792,000 m/s
1

7. CONVERSION FACTORS
Multiply By To Obtain Multiply By To Obtain
acre 43,560 square feet (ft2) joule (J) 9.478×10–4 Btu
ampere-hr (A-hr) 3,600 coulomb (C) J 0.7376 ft-lbf
ångström (Å) 1×10–10 meter (m) J 1 newton·m (N·m)
atmosphere (atm) 76.0 cm, mercury (Hg) J/s 1 watt (W)
atm, std 29.92 in, mercury (Hg)
atm, std 14.70 lbf/in2 abs (psia) kilogram (kg) 2.205 pound (lbm)
atm, std 33.90 ft, water kgf 9.8066 newton (N)
atm, std 1.013×105 pascal (Pa) kilometer (km) 3,281 feet (ft)
km/hr 0.621 mph
bar 1×105 Pa kilopascal (kPa) 0.145 lbf/in2 (psi)
barrels–oil 42 gallons–oil kilowatt (kW) 1.341 horsepower (hp)
Btu 1,055 joule (J) kW 3,413 Btu/hr
Btu 2.928×10–4 kilowatt-hr (kWh) kW 737.6 (ft-lbf )/sec
Btu 778 ft-lbf kW-hour (kWh) 3,413 Btu
Btu/hr 3.930×10–4 horsepower (hp) kWh 1.341 hp-hr
Btu/hr 0.293 watt (W) kWh 3.6×106 joule (J)
Btu/hr 0.216 ft-lbf/sec kip (K) 1,000 lbf
K 4,448 newton (N)
calorie (g-cal) 3.968×10–3 Btu
cal 1.560×10–6 hp-hr liter (L) 61.02 in3
cal 4.186 joule (J) L 0.264 gal (US Liq)
cal/sec 4.186 watt (W) L 10–3 m3
centimeter (cm) 3.281×10–2 foot (ft) L/second (L/s) 2.119 ft3/min (cfm)
cm 0.394 inch (in) L/s 15.85 gal (US)/min (gpm)
centipoise (cP) 0.001 pascal·sec (Pa·s)
centistokes (cSt) 1×10–6 m2/sec (m2/s) meter (m) 3.281 feet (ft)
cubic feet/second (cfs) 0.646317 million gallons/day m 1.094 yard
(mgd)
cubic foot (ft3) 7.481 gallon m/second (m/s) 196.8 feet/min (ft/min)
cubic meters (m3) 1,000 Liters mile (statute) 5,280 feet (ft)
electronvolt (eV) 1.602×10–19 joule (J) mile (statute) 1.609 kilometer (km)
mile/hour (mph) 88.0 ft/min (fpm)
foot (ft) 30.48 cm mph 1.609 km/h
ft 0.3048 meter (m) mm of Hg 1.316×10–3 atm
–3
ft-pound (ft-lbf) 1.285×10 Btu mm of H2O 9.678×10–5 atm
ft-lbf 3.766×10–7 kilowatt-hr (kWh)
ft-lbf 0.324 calorie (g-cal) newton (N) 0.225 lbf
ft-lbf 1.356 joule (J) N·m 0.7376 ft-lbf
ft-lbf/sec 1.818×10–3 horsepower (hp) N·m 1 joule (J)
gallon (US Liq) 3.785 liter (L) pascal (Pa) 9.869×10–6 atmosphere (atm)
gallon (US Liq) 0.134 ft3 Pa 1 newton/m2 (N/m2)
gallons of water 8.3453 pounds of water Pa·sec (Pa·s) 10 poise (P)
gamma (γ, Γ) 1×10–9 tesla (T) pound (lbm,avdp) 0.454 kilogram (kg)
gauss 1×10–4 T lbf 4.448 N
gram (g) 2.205×10–3 pound (lbm) lbf-ft 1.356 N·m
lbf/in2 (psi) 0.068 atm
hectare 1×104 square meters (m2) psi 2.307 ft of H2O
hectare 2.47104 acres psi 2.036 in of Hg
horsepower (hp) 42.4 Btu/min psi 6,895 Pa
hp 745.7 watt (W)
hp 33,000 (ft-lbf)/min radian 180/π degree
hp 550 (ft-lbf)/sec
hp-hr 2,544 Btu stokes 1×10–4 m2/s
6
hp-hr 1.98×10 ft-lbf
hp-hr 2.68×106 joule (J) therm 1×105 Btu
hp-hr 0.746 kWh
watt (W) 3.413 Btu/hr
inch (in) 2.540 centimeter (cm) W 1.341×10–3 horsepower (hp)
in of Hg 0.0334 atm W 1 joule/sec (J/s)
in of Hg 13.60 in of H2O weber/m2 (Wb/m2) 10,000 gauss
in of H2O 0.0361 lbf/in2 (psi)
in of H2O 0.002458 atm
2

8. MATHEMATICS
STRAIGHT LINE Case 2. Ellipse e < 1:
The general form of the equation is •
Ax + By + C = 0
The standard form of the equation is
y = mx + b,
which is also known as the slope-intercept form.
The point-slope form is y – y1 = m(x – x1)
Given two points: slope, m = (y2 – y1)/(x2 – x1)
The angle between lines with slopes m1 and m2 is
(x − h)2 + ( y − k )2 = 1; Center at (h , k )
α = arctan [(m2 – m1)/(1 + m2·m1)] a2 b2
Two lines are perpendicular if m1 = –1/m2 is the standard form of the equation. When h = k = 0,
The distance between two points is
Eccentricity: (
e = 1 − b2 a2 = c / a )
d= ( y2 − y1 )
2
+ (x2 − x1 )
2
b = a 1 − e2 ;
QUADRATIC EQUATION Focus: (± ae ,0); Directrix: x = ± a / e
ax2 + bx + c = 0
Case 3. Hyperbola e > 1:
2
− b ± b − 4ac •
Roots =
2a
CONIC SECTIONS
(x − h )2 − ( y − k )2 = 1; Center at (h , k )
a2 b2
e = eccentricity = cos θ/(cos φ) is the standard form of the equation. When h = k = 0,
[Note: X′ and Y′, in the following cases, are translated
axes.]
Eccentricity: (
e = 1 + b2 a2 = c / a )
Case 1. Parabola e = 1: b = a e 2 − 1;
• Focus: (± ae ,0 ); Directrix: x = ± a / e
• Brink, R.W., A First Year of College Mathematics, Copyright © 1937 by D. Appleton-Century
Co., Inc. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.
(y – k)2 = 2p(x – h); Center at (h, k)
is the standard form of the equation. When h = k = 0,
Focus: (p/2,0); Directrix: x = –p/2
3

9. MATHEMATICS (continued)
Case 4. Circle e = 0: h = –a; k = –b
(x – h)2 + (y – k)2 = r2; Center at (h, k) r = a 2 + b2 − c
is the general form of the equation with radius If a2 + b2 – c is positive, a circle, center (–a, –b).
r= (x − h )2 + ( y − k )2 If a2 + b2 – c equals zero, a point at (–a, –b).
If a2 + b2 – c is negative, locus is imaginary.
•
QUADRIC SURFACE (SPHERE)
The general form of the equation is
(x – h)2 + (y – k)2 + (z – m)2 = r2
with center at (h, k, m).
In a three-dimensional space, the distance between two
points is
d= (x2 − x1 )2 + ( y 2 − y1 )2 + (z 2 − z1 )2
Length of the tangent from a point. Using the general form
of the equation of a circle, the length of the tangent is found LOGARITHMS
from The logarithm of x to the Base b is defined by
t2 = (x′ – h)2 + (y′ – k)2 – r2 logb (x) = c, where bc = x
by substituting the coordinates of a point P(x′,y′) and the Special definitions for b = e or b = 10 are:
coordinates of the center of the circle into the equation and ln x, Base = e
computing.
log x, Base = 10
To change from one Base to another:
•
logb x = (loga x)/(loga b)
e.g., ln x = (log10 x)/(log10 e) = 2.302585 (log10 x)
Identities
logb bn = n
log xc = c log x; xc = antilog (c log x)
log xy = log x + log y
logb b = 1; log 1 = 0
log x/y = log x – log y
Conic Section Equation
The general form of the conic section equation is
• Brink, R.W., A First Year of College Mathematics, Copyright  1937 by D.
Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0 Appleton-Century Co., Inc. Reprinted by permission of Prentice-Hall, Inc.,
Englewood Cliffs, NJ.
where not both A and C are zero.
2
If B – AC < 0, an ellipse is defined.
If B2 – AC > 0, a hyperbola is defined.
If B2 – AC = 0, the conic is a parabola.
If A = C and B = 0, a circle is defined.
If A = B = C = 0, a straight line is defined.
x2 + y2 + 2ax + 2by + c = 0
is the normal form of the conic section equation, if that
conic section has a principal axis parallel to a coordinate
axis.
4

10. MATHEMATICS (continued)
TRIGONOMETRY sin α sin β = (1/2)[cos (α – β) – cos (α + β)]
Trigonometric functions are defined using a right triangle. cos α cos β = (1/2)[cos (α – β) + cos (α + β)]
sin θ = y/r, cos θ = x/r sin α cos β = (1/2)[sin (α + β) + sin (α – β)]
tan θ = y/x, cot θ = x/y sin α + sin β = 2 sin (1/2)(α + β) cos (1/2)(α – β)
csc θ = r/y, sec θ = r/x sin α – sin β = 2 cos (1/2)(α + β) sin (1/2)(α – β)
cos α + cos β = 2 cos (1/2)(α + β) cos (1/2)(α – β)
cos α – cos β = – 2 sin (1/2)(α + β) sin (1/2)(α – β)
a b c COMPLEX NUMBERS
Law of Sines = =
sin A sin B sin C Definition i = −1
Law of Cosines (a + ib) + (c + id) = (a + c) + i (b + d)
2 2 2
a = b + c – 2bc cos A (a + ib) – (c + id) = (a – c) + i (b – d)
2 2 2
b = a + c – 2ac cos B (a + ib)(c + id) = (ac – bd) + i (ad + bc)
a + ib (a + ib )(c − id ) (ac + bd ) + i(bc − ad )
2 2 2
c = a + b – 2ab cos C
= =
Identities c + id (c + id )(c − id ) c2 + d 2
csc θ = 1/sin θ (a + ib) + (a – ib) = 2a
sec θ = 1/cos θ (a + ib) – (a – ib) = 2ib
tan θ = sin θ/cos θ (a + ib)(a – ib) = a2 + b2
cot θ = 1/tan θ Polar Coordinates
2 2
sin θ + cos θ = 1 x = r cos θ; y = r sin θ; θ = arctan (y/x)
tan2θ + 1 = sec2θ r = x + iy = x2 + y2
cot2θ + 1 = csc2θ
x + iy = r (cos θ + i sin θ) = reiθ
sin (α + β) = sin α cos β + cos α sin β
[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =
cos (α + β) = cos α cos β – sin α sin β
r1r2[cos (θ1 + θ2) + i sin (θ1 + θ2)]
sin 2α = 2 sin α cos α n
2 2 2 2
(x + iy) = [r (cos θ + i sin θ)]n
cos 2α = cos α – sin α = 1 – 2 sin α = 2 cos α – 1
2
= r n(cos nθ + i sin nθ)
tan 2α = (2 tan α)/(1 – tan α)
r1 (cos θ + i sin θ1 ) r
cot 2α = (cot2α – 1)/(2 cot α) = 1 [cos(θ1 − θ 2 ) + i sin (θ1 − θ 2 )]
r2 (cos θ 2 + i sin θ 2 ) r2
tan (α + β) = (tan α + tan β)/(1 – tan α tan β)
Euler's Identity
cot (α + β) = (cot α cot β – 1)/(cot α + cot β)
eiθ = cos θ + i sin θ
sin (α – β) = sin α cos β – cos α sin β
e−iθ = cos θ – i sin θ
cos (α – β) = cos α cos β + sin α sin β
eiθ + e −iθ eiθ − e −iθ
tan (α – β) = (tan α – tan β)/(1 + tan α tan β) cos θ = , sin θ =
2 2i
cot (α – β) = (cot α cot β + 1)/(cot β – cot α)
Roots
sin (α/2) = ± (1 − cos α ) 2 If k is any positive integer, any complex number (other than
zero) has k distinct roots. The k roots of r (cos θ + i sin θ)
cos (α/2) = ± (1 + cos α ) 2
can be found by substituting successively n = 0, 1, 2, …,
tan (α/2) = ± (1 − cos α ) (1 + cos α ) (k – 1) in the formula
cot (α/2) = ± (1 + cos α ) (1 − cos α ) é æθ 360o ö æθ 360o öù
w = k r êcosç + n
ç
÷ + i sin ç + n ÷ú
ê èk
ë k ÷ ø
çk
è k ÷úøû
5

11. MATHEMATICS (continued)
MATRICES For a third-order determinant:
A matrix is an ordered rectangular array of numbers with m
rows and n columns. The element aij refers to row i and a1 a2 a3
column j. b1 b2 b3 = a1b2 c3 + a2 b3 c1 + a3b1c2 − a3b2 c1 − a2b1c3 − a1b3c2
Multiplication c1 c2 c3
If A = (aik) is an m × n matrix and B = (bkj) is an n × s
matrix, the matrix product AB is an m × s matrix
VECTORS
( ) æn ö
C = ci j = ç å ail blj ÷
è l =1 ø
j
where n is the common integer representing the number of
columns of A and the number of rows of B (l and k = 1, 2,
…, n). i
k
Addition
If A = (aij) and B = (bij) are two matrices of the same size
m × n, the sum A + B is the m × n matrix C = (cij) where
cij = aij + bij.
Identity
The matrix I = (aij) is a square n × n identity matrix where
aii = 1 for i = 1, 2, …, n and aij = 0 for i ≠ j.
Transpose
The matrix B is the transpose of the matrix A if each entry
bji in B is the same as the entry aij in A and conversely. In
A = axi + ayj + azk
equation form, the transpose is B = AT.
Addition and subtraction:
Inverse
A + B = (ax + bx)i + (ay + by)j + (az + bz)k
The inverse B of a square n × n matrix A is
adj ( A) A – B = (ax – bx)i + (ay – by)j + (az – bz)k
B = A −1 = , where
A The dot product is a scalar product and represents the
projection of B onto A times A. It is given by
adj(A) = adjoint of A (obtained by replacing AT elements
A·B = axbx + ayby + azbz
with their cofactors, see DETERMINANTS) and
= AB cos θ = B·A
A = determinant of A.
The cross product is a vector product of magnitude
DETERMINANTS BA sin θ which is perpendicular to the plane
A determinant of order n consists of n2 numbers, called the containing A and B. The product is
elements of the determinant, arranged in n rows and n
columns and enclosed by two vertical lines. In any i j k
determinant, the minor of a given element is the A × B = a x a y a z = −B × A
determinant that remains after all of the elements are struck
out that lie in the same row and in the same column as the bx b y bz
given element. Consider an element which lies in the hth
column and the kth row. The cofactor of this element is the
value of the minor of the element (if h + k is even), and it is
the negative of the value of the minor of the element (if h +
k is odd).
If n is greater than 1, the value of a determinant of order n
is the sum of the n products formed by multiplying each
element of some specified row (or column) by its cofactor.
This sum is called the expansion of the determinant
[according to the elements of the specified row (or The sense of A × B is determined by the right-hand rule.
column)]. For a second-order determinant:
A × B = AB n sin θ, where
a1 a2
= a1b2 − a2b1 n = unit vector perpendicular to the plane of A and B.
b1 b2
6

12. MATHEMATICS (continued)
Gradient, Divergence, and Curl Geometric Progression
æ ∂ ∂ ∂ ö To determine whether a given finite sequence is a
∇φ = ç i +
ç ∂x j + k ÷φ geometric progression (G.P.), divide each number after the
è ∂y ∂z ÷
ø first by the preceding number. If the quotients are equal, the
series is geometric.
æ ∂
∇⋅V = ç i +
ç ∂x
∂
∂y
∂ ö
(
j + k ÷ ⋅ V1 i + V2 j + V3k
∂z ÷
) 1. The first term is a.
è ø
2. The common ratio is r.
æ ∂
∇×V = ç i +
ç ∂x
∂
∂y
∂ ö
(
j + k ÷ × V1 i + V2 j + V3k
∂z ÷
) 3. The number of terms is n.
è ø
4. The last or nth term is l.
The Laplacian of a scalar function φ is
5. The sum of n terms is S.
2 ∂ 2φ ∂ 2φ ∂ 2φ
∇ φ= 2 + 2 + 2 l = arn−1
∂x ∂y ∂z
S = a (1 – rn)/(1 – r); r ≠ 1
Identities
S = (a – rl)/(1 – r); r ≠ 1
A·B = B·A; A·(B + C) = A·B + A·C
limit S n = a (1 − r ); r < 1
A·A = A2 n →∞
i·i = j·j = k·k = 1 A G.P. converges if r < 1 and it diverges if r ≥ 1.
i·j = j·k = k·i = 0 Properties of Series
If A·B = 0, then either A = 0, B = 0, or A is perpendicular n
to B. å c = nc; c = constant
i =1
A × B = –B × A n n
å cxi = c å xi
A × (B + C) = (A × B) + (A × C) i =1 i =1
(B + C) × A = (B × A) + (C × A) n n n
å (xi + yi − zi ) = å xi + å yi − å zi
n
i =1 i =1 i =1 i =1
i×i =j×j =k×k=0
i × j = k = –j × i; j × k = i = –k × j
n
x =1
(
2
åx= n+n 2 )
k × i = j = –i × k
1. A power series in x, or in x – a, which is convergent in
If A × B = 0, then either A = 0, B = 0, or A is parallel to B. the interval –1 < x < 1 (or –1 < x – a < 1), defines a
function of x which is continuous for all values of x
∇ 2 φ = ∇ ⋅ (∇ φ) = (∇ ⋅ ∇ )φ within the interval and is said to represent the function
∇ × ∇φ = 0 in that interval.
∇ ⋅ (∇ × A ) = 0 2. A power series may be differentiated term by term, and
the resulting series has the same interval of
∇ × (∇ × A) = ∇ (∇ ⋅ A) − ∇ 2 A convergence as the original series (except possibly at
the end points of the interval).
PROGRESSIONS AND SERIES 3. A power series may be integrated term by term
Arithmetic Progression provided the limits of integration are within the interval
To determine whether a given finite sequence of numbers is of convergence of the series.
an arithmetic progression, subtract each number from the 4. Two power series may be added, subtracted, or
following number. If the differences are equal, the series is multiplied, and the resulting series in each case is
arithmetic. convergent, at least, in the interval common to the two
1. The first term is a. series.
2. The common difference is d. 5. Using the process of long division (as for polynomials),
two power series may be divided one by the other.
3. The number of terms is n.
4. The last or nth term is l.
5. The sum of n terms is S.
l = a + (n – 1)d
S = n(a + l)/2 = n [2a + (n – 1) d]/2
7

13. MATHEMATICS (continued)
Taylor's Series Property 3. Law of Compound or Joint Probability
f ′(a )
(x − a ) + f (a ) (x − a )2
′′ If neither P(A) nor P(B) is zero,
f (x ) = f (a ) +
1! 2! P(A, B) = P(A)P(B | A) = P(B)P(A | B), where
f (n ) (a ) P(B | A) = the probability that B occurs given the fact that A
+K+ (x − a )n + K has occurred, and
n!
is called Taylor's series, and the function f (x) is said to be P(A | B) = the probability that A occurs given the fact that B
expanded about the point a in a Taylor's series. has occurred.
If a = 0, the Taylor's series equation becomes a Maclaurin's If either P(A) or P(B) is zero, then P(A, B) = 0.
series.
Probability Functions
PROBABILITY AND STATISTICS A random variable x has a probability associated with each
of its values. The probability is termed a discrete
Permutations and Combinations probability if x can assume only the discrete values
A permutation is a particular sequence of a given set of
x = X1, X2, …, Xi, …, XN
objects. A combination is the set itself without reference to
order. The discrete probability of the event X = xi occurring is
defined as P(Xi).
1. The number of different permutations of n distinct
objects taken r at a time is Probability Density Functions
n! If x is continuous, then the probability density function f (x)
P (n , r ) =
(n − r ) ! is defined so that
òx f ( x )dx = the probability that x lies between x1 and x2.
x 2
2. The number of different combinations of n distinct 1
objects taken r at a time is The probability is determined by defining the equation for
f (x) and integrating between the values of x required.
P(n , r ) n!
C (n , r ) = =
r! [r!(n − r )!] Probability Distribution Functions
The probability distribution function F(Xn) of the discrete
3. The number of different permutations of n objects
probability function P(Xi) is defined by
taken n at a time, given that ni are of type i,
where i = 1, 2,…, k and Σni = n, is n
F ( X n ) = å P ( X k ) = P( X i ≤ X n )
k =1
n!
P (n; n1 , n2 ,K nk ) =
n1!n2 !Kn k ! When x is continuous, the probability distribution function
F(x) is defined by
Laws of Probability
F (x ) = ò−∞ f (t )dt
x
Property 1. General Character of Probability
The probability P(E) of an event E is a real number in the which implies that F(a) is the probability that x ≤ a.
range of 0 to 1. The probability of an impossible event is 0 The expected value g(x) of any function is defined as
and that of an event certain to occur is 1.
E{g (x )} = ò −∞ g (t ) f (t )dt
x
Property 2. Law of Total Probability
P(A + B) = P(A) + P(B) – P(A, B), where Binomial Distribution
P(A + B) = the probability that either A or B occur alone P(x) is the probability that x will occur in n trials. If p =
or that both occur together, probability of success and q = probability of failure = 1 – p,
P(A) = the probability that A occurs, then
n!
P(B) = the probability that B occurs, and P (x ) = C (n , x ) p x q n − x = p x q n − x , where
x! (n − x ) !
P(A, B) = the probability that both A and B occur
simultaneously. x = 0, 1, 2, …, n,
C(n, x) = the number of combinations, and
n, p = parameters.
8

14. MATHEMATICS (continued)
Normal Distribution (Gaussian Distribution) The standard deviation of a population is
This is a unimodal distribution, the mode being x = µ, with σ= (1 N )å ( X i − µ )2
two points of inflection (each located at a distance σ to
either side of the mode). The averages of n observations The sample variance is
tend to become normally distributed as n increases. The 2
( )
n
variate x is said to be normally distributed if its density s 2 = [1 (n − 1)]å X i − X
function f (x) is given by an expression of the form i =1
1 The sample standard deviation is
f (x ) = e − ( x −µ )
2
2σ 2
, where
σ 2π
µ = the population mean,
é 1 ùn
s= ê úå Xi − X ( )2
ë n − 1û i =1
σ = the standard deviation of the population, and
The coefficient of variation = CV = s/ X
–∞ ≤ x ≤ ∞
When µ = 0 and σ2 = σ = 1, the distribution is called a The geometric mean = n X1 X 2 X 3 K X n
standardized or unit normal distribution. Then
The root-mean-square value = (1 n)å X i2
1
f (x ) =
2
e−x 2
, where − ∞ ≤ x ≤ ∞ .
2π The median is defined as the value of the middle item when
the data are rank-ordered and the number of items is odd.
A unit normal distribution table is included at the end of The median is the average of the middle two items when
this section. In the table, the following notations are the rank-ordered data consists of an even number of items.
utilized:
The mode of a set of data is the value that occurs with
F(x) = the area under the curve from –∞ to x, greatest frequency.
R(x) = the area under the curve from x to ∞, and
t-Distribution
W(x) = the area under the curve between –x and x.
The variate t is defined as the quotient of two independent
Dispersion, Mean, Median, and Mode Values variates x and r where x is unit normal and r is the root
If X1, X2, …, Xn represent the values of n items or mean square of n other independent unit normal variates;
observations, the arithmetic mean of these items or that is,
observations, denoted X , is defined as t = x/r. The following is the t-distribution with n degrees of
freedom:
n
X = (1 n )( X 1 + X 2 + K + X n ) = (1 n )å X i Γ [(n + 1)] 2 1
f (t ) =
( )(
i =1 n +1) 2
Γ(n 2) nπ 1 + t 2 n
X → µ for sufficiently large values of n.
where – ∞ ≤ t ≤ ∞.
The weighted arithmetic mean is
A table at the end of this section gives the values of tα, n for
å wi X i
Xw = , where values of α and n. Note that in view of the symmetry of the
å wi t-distribution,
X w = the weighted arithmetic mean, t1−α,n = –tα,n. The function for α follows:
Xi = the values of the observations to be averaged, and
α = òt∞,n f (t )dt
wi = the weight applied to the Xi value. α
The variance of the observations is the arithmetic mean of A table showing probability and density functions is
the squared deviations from the population mean. In included on page 149 in the INDUSTRIAL
symbols, X1, X2, …, Xn represent the values of the n sample ENGINEERING SECTION of this handbook.
observations of a population of size N. If µ is the arithmetic
mean of the population, the population variance is defined
by
σ 2 = (1 / N )[( X 1 − µ ) + ( X 2 − µ ) + K + ( X N − µ ) ]
2 2 2
2
= (1 / N )å ( X i − µ )
N
i =1
9

15. MATHEMATICS (continued)
GAMMA FUNCTION
Γ (n ) = òo t n −1e −t dt , n > 0
∞
CONFIDENCE INTERVALS
Confidence Interval for the Mean µ of a Normal
Distribution
(a) Standard deviation σ is known
σ σ
X − Zα 2 ≤ µ ≤ X + Zα 2
n n
(b) Standard deviation σ is not known
s s
X − tα 2 ≤ µ ≤ X + tα 2
n n
where t α 2 corresponds to n – 1 degrees of freedom.
Confidence Interval for the Difference Between Two
Means µ1 and µ2
(a) Standard deviations σ1 and σ2 known
σ1 σ 2
2
σ2 σ2
X1 − X 2 − Zα 2 + 2 ≤ µ1 − µ 2 ≤ X 1 − X 2 + Z α 2 1 + 2
n1 n2 n1 n2
(b) Standard deviations σ1 and σ2 are not known
æ 1
ç
1 ö
÷[
ç n + n ÷ (n − 1)S 1 + (n 2 − 1)S 2
2 2
] æ 1 1 ö
[
ç n + n ÷ (n − 1)S 1 + (n 2 − 1)S 2
ç ÷
2 2
]
è 1 2 ø è 1 2 ø
X 1 − X 2 − tα 2 ≤ µ1 − µ 2 ≤ X 1 − X 2 − t α 2
n1 + n 2 − 2 n1 + n 2 − 2
where t α 2 corresponds to n1 + n2 – 2 degrees of freedom.
10

16. MATHEMATICS (continued)
UNIT NORMAL DISTRIBUTION TABLE
x f(x) F(x) R(x) 2R(x) W(x)
0.0 0.3989 0.5000 0.5000 1.0000 0.0000
0.1 0.3970 0.5398 0.4602 0.9203 0.0797
0.2 0.3910 0.5793 0.4207 0.8415 0.1585
0.3 0.3814 0.6179 0.3821 0.7642 0.2358
0.4 0.3683 0.6554 0.3446 0.6892 0.3108
0.5 0.3521 0.6915 0.3085 0.6171 0.3829
0.6 0.3332 0.7257 0.2743 0.5485 0.4515
0.7 0.3123 0.7580 0.2420 0.4839 0.5161
0.8 0.2897 0.7881 0.2119 0.4237 0.5763
0.9 0.2661 0.8159 0.1841 0.3681 0.6319
1.0 0.2420 0.8413 0.1587 0.3173 0.6827
1.1 0.2179 0.8643 0.1357 0.2713 0.7287
1.2 0.1942 0.8849 0.1151 0.2301 0.7699
1.3 0.1714 0.9032 0.0968 0.1936 0.8064
1.4 0.1497 0.9192 0.0808 0.1615 0.8385
1.5 0.1295 0.9332 0.0668 0.1336 0.8664
1.6 0.1109 0.9452 0.0548 0.1096 0.8904
1.7 0.0940 0.9554 0.0446 0.0891 0.9109
1.8 0.0790 0.9641 0.0359 0.0719 0.9281
1.9 0.0656 0.9713 0.0287 0.0574 0.9426
2.0 0.0540 0.9772 0.0228 0.0455 0.9545
2.1 0.0440 0.9821 0.0179 0.0357 0.9643
2.2 0.0355 0.9861 0.0139 0.0278 0.9722
2.3 0.0283 0.9893 0.0107 0.0214 0.9786
2.4 0.0224 0.9918 0.0082 0.0164 0.9836
2.5 0.0175 0.9938 0.0062 0.0124 0.9876
2.6 0.0136 0.9953 0.0047 0.0093 0.9907
2.7 0.0104 0.9965 0.0035 0.0069 0.9931
2.8 0.0079 0.9974 0.0026 0.0051 0.9949
2.9 0.0060 0.9981 0.0019 0.0037 0.9963
3.0 0.0044 0.9987 0.0013 0.0027 0.9973
Fractiles
1.2816 0.1755 0.9000 0.1000 0.2000 0.8000
1.6449 0.1031 0.9500 0.0500 0.1000 0.9000
1.9600 0.0584 0.9750 0.0250 0.0500 0.9500
2.0537 0.0484 0.9800 0.0200 0.0400 0.9600
2.3263 0.0267 0.9900 0.0100 0.0200 0.9800
2.5758 0.0145 0.9950 0.0050 0.0100 0.9900
11

17. MATHEMATICS (continued)
t-DISTRIBUTION TABLE
VALUES OF tα,n α
n α = 0.10 α = 0.05 α = 0.025 α = 0.01 α = 0.005 n
1 3.078 6.314 12.706 31.821 63.657 1
2 1.886 2.920 4.303 6.965 9.925 2
3 1.638 2.353 3.182 4.541 5.841 3
4 1.533 2.132 2.776 3.747 4.604 4
5 1.476 2.015 2.571 3.365 4.032 5
6 1.440 1.943 2.447 3.143 3.707 6
7 1.415 1.895 2.365 2.998 3.499 7
8 1.397 1.860 2.306 2.896 3.355 8
9 1.383 1.833 2.262 2.821 3.250 9
10 1.372 1.812 2.228 2.764 3.169 10
11 1.363 1.796 2.201 2.718 3.106 11
12 1.356 1.782 2.179 2.681 3.055 12
13 1.350 1.771 2.160 2.650 3.012 13
14 1.345 1.761 2.145 2.624 2.977 14
15 1.341 1.753 2.131 2.602 2.947 15
16 1.337 1.746 2.120 2.583 2.921 16
17 1.333 1.740 2.110 2.567 2.898 17
18 1.330 1.734 2.101 2.552 2.878 18
19 1.328 1.729 2.093 2.539 2.861 19
20 1.325 1.725 2.086 2.528 2.845 20
21 1.323 1.721 2.080 2.518 2.831 21
22 1.321 1.717 2.074 2.508 2.819 22
23 1.319 1.714 2.069 2.500 2.807 23
24 1.318 1.711 2.064 2.492 2.797 24
25 1.316 1.708 2.060 2.485 2.787 25
26 1.315 1.706 2.056 2.479 2.779 26
27 1.314 1.703 2.052 2.473 2.771 27
28 1.313 1.701 2.048 2.467 2.763 28
29 1.311 1.699 2.045 2.462 2.756 29
inf. 1.282 1.645 1.960 2.326 2.576 inf.
12

18. MATHEMATICS (continued)
CRITICAL VALUES OF THE F DISTRIBUTION – TABLE
For a particular combination of
numerator and denominator degrees
of freedom, entry represents the
critical values of F corresponding
to a specified upper tail area (α).
Denominator Numerator df1
df2 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
13
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67
28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25
∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

19. MATHEMATICS (continued)
DIFFERENTIAL CALCULUS Curvature in Rectangular Coordinates
The Derivative y ′′
K=
For any function y = f (x), [
1 + ( y ′)2 3 2 ]
the derivative = Dx y = dy/dx = y′ When it may be easier to differentiate the function with
y ′ = limit [(∆y ) (∆x )] respect to y rather than x, the notation x′ will be used for the
∆x → 0 derivative.
= limit {[ f (x + ∆x ) − f (x )] (∆x )} x′ = dx/dy
∆x →0
y′ = the slope of the curve f(x). − x′′
K=
Test for a Maximum [1 + (x′) ] 2 3 2
y = f (x) is a maximum for The Radius of Curvature
x = a, if f ′(a) = 0 and f ″(a) < 0. The radius of curvature R at any point on a curve is defined
as the absolute value of the reciprocal of the curvature K at
Test for a Minimum that point.
y = f (x) is a minimum for
1
x = a, if f ′(a) = 0 and f ″(a) > 0. R= (K ≠ 0 )
K
[1 + ( y′) ]
Test for a Point of Inflection 2 32
y = f (x) has a point of inflection at x = a, R= ( y′′ ≠ 0)
y ′′
if f ″(a) = 0, and
if f ″(x) changes sign as x increases through L'Hospital's Rule (L'Hôpital's Rule)
x = a. If the fractional function f(x)/g(x) assumes one of the
indeterminate forms 0/0 or ∞/∞ (where α is finite or
The Partial Derivative infinite), then
In a function of two independent variables x and y, a
derivative with respect to one of the variables may be found limit f (x ) g (x )
x →α
if the other variable is assumed to remain constant. If y is
kept fixed, the function is equal to the first of the expressions
z = f (x, y) f ′(x ) f ′′(x ) f ′′′(x )
limit , limit , limit
becomes a function of the single variable x, and its x →α g ′(x ) x→α g ′′(x ) x →α g ′′′(x )
derivative (if it exists) can be found. This derivative is which is not indeterminate, provided such first indicated
called the partial derivative of z with respect to x. The limit exists.
partial derivative with respect to x is denoted as follows:
∂z ∂f (x , y ) INTEGRAL CALCULUS
=
∂x ∂x The definite integral is defined as:
n
The Curvature of Any Curve limit å f (xi )∆xi = òa f (x )dx
b
n → ∞ i =1
♦
Also, ∆xi → 0 for all i .
A table of derivatives and integrals is available on page 15.
The integral equations can be used along with the following
methods of integration:
A. Integration by Parts (integral equation #6),
B. Integration by Substitution, and
The curvature K of a curve at P is the limit of its average C. Separation of Rational Fractions into Partial Fractions.
curvature for the arc PQ as Q approaches P. This is also
expressed as: the curvature of a curve at a given point is the ♦ Wade, Thomas L., Calculus, Copyright © 1953 by Ginn & Company. Diagram reprinted by permission
rate-of-change of its inclination with respect to its arc of Simon & Schuster Publishers.
length.
∆ α dα
K = limit =
∆s →0 ∆s ds
14

20. MATHEMATICS (continued)
DERIVATIVES AND INDEFINITE INTEGRALS
In these formulas, u, v, and w represent functions of x. Also, a, c, and n represent constants. All arguments of the trigonometric
functions are in radians. A constant of integration should be added to the integrals. To avoid terminology difficulty, the
following definitions are followed: arcsin u = sin–1 u, (sin u) –1 = 1/sin u.
1. dc/dx = 0 1. ò d f (x) = f (x)
2. dx/dx = 1 2. ò dx = x
3. d(cu)/dx = c du/dx 3. ò a f(x) dx = a ò f(x) dx
4. d(u + v – w)/dx = du/dx + dv/dx – dw/dx 4. ò [u(x) ± v(x)] dx = ò u(x) dx ± ò v(x) dx
5. d(uv)/dx = u dv/dx + v du/dx x m+1
6. d(uvw)/dx = uv dw/dx + uw dv/dx + vw du/dx 5. ò x m dx = (m ≠ −1)
m +1
d (u v ) v du dx − u dv dx 6. ò u(x) dv(x) = u(x) v(x) – ò v (x) du(x)
7. =
dx v2 dx 1
n n–1 7. ò = ln ax + b
8. d(u )/dx = nu du/dx ax + b a
9. d[f (u)]/dx = {d[f (u)]/du} du/dx dx
10. du/dx = 1/(dx/du) 8. ò =2 x
x
d (log a u ) 1 du
11. = (log a e ) ax
dx u dx 9. ò ax dx =
ln a
d (lnu ) 1 du
12. = 10. ò sin x dx = – cos x
dx u dx 11. ò cos x dx = sin x
13.
( )
d au
= (lna )a u
du
12. ò sin 2 xdx =
x sin 2 x
−
dx dx 2 4
14. d(eu)/dx = eu du/dx x sin 2 x
15. d(uv)/dx = vuv–1 du/dx + (ln u) uv dv/dx 13. ò cos 2 xdx = +
2 4
16. d(sin u)/dx = cos u du/dx
14. ò x sin x dx = sin x – x cos x
17. d(cos u)/dx = –sin u du/dx
15. ò x cos x dx = cos x + x sin x
18. d(tan u)/dx = sec2u du/dx
16. ò sin x cos x dx = (sin2x)/2
19. d(cot u)/dx = –csc2u du/dx
cos(a − b )x cos(a + b )x
20. d(sec u)/dx = sec u tan u du/dx 17. ò sin ax cos bx dx = −
2(a − b )
−
2(a + b )
(a 2
≠ b2 )
21. d(csc u)/dx = –csc u cot u du/dx
18. ò tan x dx = –lncos x = ln sec x
22.
(
d sin −1u
=
1) du
(− π 2 ≤ sin −1
u≤π 2 ) 19. ò cot x dx = –ln csc x  = ln sin x
dx 1 − u 2 dx 20. ò tan2x dx = tan x – x
21. ò cot2x dx = –cot x – x
23.
(
d cos −1u
=−
) 1 du
(0 ≤ cos −1
u≤π ) 22. ò eax dx = (1/a) eax
dx 2 dx 23. ò xeax dx = (eax/a2)(ax – 1)
1− u
24. ò ln x dx = x [ln (x) – 1] (x > 0)
24.
(
d tan u
=
−1
)
1 du
(− π 2 < tan −1
u<π 2 ) 25. ò 2
dx
2
1
= tan −1
x
(a ≠ 0)
dx 1 + u 2 dx a +x a a
æ aö
25.
(
d cot −1u
=−
)1 du
(0 < cot −1
u<π ) 26. ò
dx
2
ax + c
=
1
ac
tan −1 ç x
ç c÷
÷, (a > 0, c > 0)
è ø
dx 1 + u 2 dx
dx 2 2ax + b
= tan −1
(
d sec −1u
=
)
1 du
ò 2
27a. ax + bx + c 4ac − b 2
4ac − b 2
26. dx u u − 1 dx
2
(4ac − b 2
>0 )
(0 ≤ sec −1
)( −1
u < π 2 − π ≤ sec u < − π 2 ) dx 1 2ax + b − b 2 − 4ac
ò = ln
(
d csc −1u
=−
1 ) du
2
27b. ax + bx + c b 2 − 4ac 2ax + b + b 2 − 4ac
27. dx u u 2 − 1 dx (b 2
− 4ac > 0 )
(0 < csc −1
)(
u ≤ π 2 − π < csc −1u ≤ − π 2 ) 27c. ò
dx
=−
2
, (b 2
− 4ac = 0 )
2
ax + bx + c 2ax + b
15

21. MATHEMATICS (continued)
MENSURATION OF AREAS AND VOLUMES
Nomenclature Circular Segment
A = total surface area ♦
P = perimeter
V = volume
Parabola
A = [r2 (φ – sin φ)]/2
φ = s/r = 2{arccos [(r – d)/r]}
Circular Sector
♦
Ellipse
♦
A = φr2/2 = sr/2
φ = s/r
Sphere
♦
A = πab
(
Papprox = 2π a 2 + b 2 2 )
é1 + ( 1 2 )2 λ2 + ( 1 2 × 1 4 )2 λ4 ù
ê ú
P = π(a + b )ê+ ( 1 × 1 × 3 )2 λ6 + ( 1 × 1 × 3 × 5 )2 λ8 ú ,
2 4 6 2 4 6 8
ê ú
ê+ ( 1 × 1 × 3 × 5 × 7 )2 λ10 + K ú
ê
ë 2 4 6 8 10 ú
û
where V = 4πr3/3 = πd 3/6
λ = (a – b)/(a + b) A = 4πr2 = πd 2
♦ Gieck, K. & Gieck R., Engineering Formulas, 6th Ed., Copyright  1967 by Gieck Publishing.
Diagrams reprinted by permission of Kurt Gieck.
16

22. MATHEMATICS (continued)
MENSURATION OF AREAS AND VOLUMES
Parallelogram Right Circular Cone
♦
P = 2(a + b)
d1 = a 2 + b 2 − 2ab(cosφ)
V = (πr2h)/3
d 2 = a + b + 2ab(cosφ)
2 2
A = side area + base area
(
d12 + d 2 = 2 a 2 + b 2
2
) = π r æ r + r 2 + h2 ö
ç ÷
A = ah = ab(sinφ) è ø
If a = b, the parallelogram is a rhombus. Ax: Ab = x2: h2
Regular Polygon (n equal sides)
Right Circular Cylinder
♦
♦
φ= 2π /n
é π (n − 2) ù æ 2ö
θ=ê ú = π ç1 − n ÷ π d 2h
ë n û è ø V = πr 2 h =
4
P = ns
A = side area + end areas = 2πr (h + r )
s = 2r [tan (φ/2)]
A = (nsr)/2
Paraboloid of Revolution
Prismoid
♦
π d 2h
V=
8
V = (h/6)( A1 + A2 + 4A)
♦ Gieck, K. & R. Gieck, Engineering Formulas, 6th Ed., Copyright 8 1967 by Gieck Publishing.
Diagrams reprinted by permission of Kurt Gieck.
17