APPENDIX Mathematics, Symbols, and Physical Constants - The Civil Engineering Handbook.pdf

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Appendix: Mathematics, Symbols, and Physical Constants
Mathematics, Symbols,
A
PPENDIX
and Physical Constants
Greek Alphabet
D
efinitions of SI Base Units
•
N
ames and Symbols for the SI Base Units
•
SI D
erived Units
with Special Names and Symbols
U
nits in Use Together with the SI
R
ecommended Decimal Multiples and Submultiples
•
Conversion Factors — Metric to
English
•
C
onversion Factors — English to Metric
•
C
onversion Factors — General
•
T
emperature Factors
•
C
onversion of Temperatures
Ge
neral
•
C
onstants
C
onstants Involving
e
N
umerical Constants
Elementary Algebra and Geometry
F
undamental Properties (Real Numbers)
E
xponents
•
F
ractional Exponents
•
I
rrational
Exponents
L
ogarithms
•
F
actorials
B
inomial Theorem
•
F
actors and Expansion
•
P
rogression
C
omplex Numbers
P
olar Form
•
P
ermutations
•
C
ombinations
•
Algebraic
Equations
•
Ge
ometry
D
eterminants
•
E
valuation by Cofactors
•
P
roperties of Determinants
•
M
atrices
O
perations
•
P
roperties
•
T
ranspose
•
I
dentity Matrix
•
Adjoint
I
nverse Matrix
•
S
ystems of Linear
Equations
•
M
atrix Solution
Trigonometry
T
riangles
•
T
rigonometric Functions of an Angle
I
nverse Trigonometric Functions
R
ectangular Coordinates
D
istance between Two Points; Slope
•
E
quations of Straight Lines
•
D
istance from a Point to a Line
•
C
ircle
•
P
arabola
•
El
lipse
H
yperbola (
e
> 1)
C
hange of Axes
Series
B
ernoulli and Euler Numbers
•
S
eries of Functions
Er
ror Function
•
S
eries Expansion
N
otation
•
S
lope of a Curve
•
Angle of Intersection of Two Curves
•
R
adius of Curvature
•
R
elative Maxima and Minima
•
oints of Inflection of a Curve
•
T
aylor’s Formula
•
I
ndeterminant Forms
• Numerical Methods • Functions of Two Variables • Partial Derivatives
Indefinite Integral • Definite Integral • Properties • Common Applications of the Definite
Vo lume by Double Integration • Centroid
Vectors • Ve c t o r D ifferentiation • Divergence Theorem (Gauss) • Stokes’ Theorem • Planar
Motion in Polar Coordinates
© 2003 by CRC Press LLC
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
P
Integral • Cylindrical and Spherical Coordinates • Double Integration • Surface Area and
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Hyperbolic Functions • Laplace Tr ansforms • z -Transform • Trigonometric Identities • Fourier
• Bessel Functions • Legendre Polynomials •
Laguerre Polynomials • Hermite Polynomials • Orthogonality
Statistics
Arithmetic Mean • Median • Mode • Geometric Mean • Harmonic Mean • Va r iance • Standard
Deviation • Coefficient of Variation • Probability • Binomial Distribution • Mean of Binomially
Distributed Variable • Normal Distribution • Poisson Distribution
Areas under the Standard Normal Curve • Poisson Distribution • t -Distribution •
2 Distribution • Va r iance Ratio
Elementary Forms • Forms Containing ( a + bx )
Fourier Transforms • Finite Sine Transforms • Finite Cosine Transforms • Fourier Sine
Tr ansforms • Fourier Cosine Transforms • Fourier Transforms
Solution of Equations by Iteration • Finite Differences • Interpolation
Definitions • Definition of Probability • Marginal and Conditional Probability • Probability
Theorems • Random Variable • Probability Function (Discrete Case) • Cumulative
Distribution Function (Discrete Case) • Probability Density (Continuous Case) • Cumulative
Distribution Function (Continuous Case) • Mathematical Expectation
Change of Base • Examples
Credits
Ethics
Greek Alphabet
Greek
Letter
Greek
Name
EnglishEquivalent
Greek
Letter
Greek
Name
English
Equivalent
Alpha
a
Nu
n
Beta
b
Xi
x
Gamma
g
Omicron
o
Delta
d
Pi
p
Epsilon
e
P
Rho
r
Zeta
z
s
Sigma
s
Eta
e
Tau
t
Theta
th
Y
υ
Upsilon
u
Ι
ι
Iota
i
Φ
φ ϕ
Phi
ph
Κ
κ
Kappa
k
X
Chi
ch
Λ
λ
Lambda
l
Ψ
ψ
Psi
ps
Μ
µ
Mu
m
Ω
ω
Omega
–
© 2003 by CRC Press LLC
Series • Functions with Period Other Than 2
8450414.003.png
International System of Units (SI)
Measures (CGPM) in 1960. It is a coherent system of units built from seven SI base units, one for each
of the seven dimensionally independent base quantities: the meter, kilogram, second, ampere, kelvin,
mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature,
amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given
below. The SI derived units are expressed as products of powers of the base units, analogous to the
corresponding relations between physical quantities but with numerical factors equal to unity.
In the International System there is only one SI unit for each physical quantity. This is either the
appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal
prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units.
It is recommended that only SI units be used in science and technology (with SI prefixes where
appropriate). Where there are special reasons for making an exception to this rule, it is recommended
always to define the units used in terms of SI units. This section is based on information supplied by
IUPAC.
792 458 of a second (17th CGPM, 1983).
Kilogram — The kilogram is the unit of mass; it is equal to the mass of the international prototype
of the kilogram (3rd CGPM, 1901).
Second — The second is the duration of 9 192 631 770 periods of the radiation corresponding to the
transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM,
1967).
Ampere — The ampere is that constant current which, if maintained in two straight parallel conduc-
tors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum,
×
10 –7 newton per meter of length
(9th CGPM, 1948).
Kelvin — The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermody-
namic temperature of the triple point of water (13th CGPM, 1967).
Mole — The mole is the amount of substance of a system that contains as many elementary entities
must be specified and may be atoms, molecules, ions, electrons, or other particles, or specified
groups of such particles (14th CGPM, 1971).
Examples of the use of the mole:
1 mol of H 2 contains about 6.022
×
10 23 H 2 molecules, or 12.044
×
10 23 H atoms
1 mol of HgCl has a mass of 236.04 g
1 mol of Hg 2 Cl 2 has a mass of 472.08 g
1 mol of Hg 2 2
has a mass of 401.18 g and a charge of 192.97 kC
1 mol of Fe 0.91 S has a mass of 82.88 g
1 mol of e – has a mass of 548.60
96.49 kC
1 mol of photons whose frequency is 10 14 Hz has energy of about 39.90 kJ
µ
g and a charge of –
Candela — The candela is the luminous intensity, in a given direction, of a source that emits
direction of (1/683) watt per steradian (16th CGPM, 1979).
×
10 12 hertz and that has a radiant intensity in that
© 2003 by CRC Press LLC
The International System of Units (SI) was adopted by the 11th General Conference on Weights and
Definitions of SI Base Units
Meter — The meter is the length of path traveled by light in vacuum during a time interval of 1/299
would produce between these conductors a force equal to 2
as there are atoms in 0.012 kilogram of carbon-12. When the mole is used, the elementary entities
monochromatic radiation of frequency 540
8450414.004.png
Names and Symbols for the SI Base Units
Physical Quantity
Name of SI Unit
Symbol for SI Unit
Length
Meter
m
Mass
Kilogram
kg
Time
Second
s
Electric current
Ampere
A
Thermodynamic temperature
Kelvin
K
Amount of substance
Mole
mol
Luminous intensity
Candela
cd
SI Derived Units with Special Names and Symbols
Physical Quantity
Name of
SI Unit
Symbol for
SI Unit
Expression in
Terms of SI Base Units
Frequency 1
Hertz
Hz
s –1
Force
Newton
N
m kg s –2
Pressure, stress
Pascal
Pa
N m –2 = m –1 kg s –2
Energy, work, heat
Joule
J
N m = m 2 kg s –2
Power, radiant flux
Watt
W
J s –1 = m 2 kg s –3
Electric charge
Coulomb
C
A s
Electric potential,
electromotive force
Volt
V
J C –1 = m 2 kg s –3 A –1
Electric resistance
Ohm
Ω
V A –1 = m 2 kg s –3 A –2
Electric conductance
Siemens
S
Ω
–1 = m –2 kg –1 s 3 A 2
Electric capacitance
Farad
F
C V –1 = m –2 kg –1 s 4 A 2
Magnetic flux density
Tesla
T
V s m –2 = kg s –2 A –1
Magnetic flux
Weber
Wb
V s = m 2 kg s –2 A –1
Inductance
Henry
H
V A –1 s = m 2 kg s –2 A –2
Celsius temperature 2
Degree Celsius
°C
K
Luminous flux
Lumen
lm
cd sr
Illuminance
Lux
lx
cd sr m –2
Activity (radioactive)
Becquerel
Bq
s –1
Absorbed dose (of radiation)
Gray
Gy
J kg –1 = m 2 s –2
Dose equivalent
(dose equivalent index)
Sievert
Sv
J kg –1 = m 2 s –2
Plane angle
Radian
rad
I = m m –1
Solid angle
Steradian
sr
I = m 2 m –2
1 For radial (circular) frequency and for angular velocity, the unit rad s –1 , or simply s –1 ,
should be used, and this may not be simplified to Hz. The unit Hz should be used only
for frequency in the sense of cycles per second.
2
is defined by the equation:
θ °C
=
T K 273.15
–
The SI unit of Celsius temperature interval is the degree Celsius, °C, which is equal to the
kelvin, K. °C should be treated as a single symbol, with no space between the ° sign and
the letter C. (The symbol °K, and the symbol °, should no longer be used.)
© 2003 by CRC Press LLC
The Celsius temperature
8450414.005.png
Units in Use Together with the SI
These units are not part of the SI, but it is recognized that they will continue to be used in appropriate
megaelectronvolt, MeV; and kilotonne, ktonne.
Physical
Quantity
Name of Unit
Symbol for Unit
Value in SI Units
Time
Minute
min
60 s
Time
Hour
h
3600 s
Time
Day
d
86 400 s
Planeangle
Degree
°
(
/180) rad
Planeangle
Minute
′
(
/10 800) rad
Planeangle
Second
″
(
/648 000) rad
Length
Ångstrom 1
Å
10 –10 m
Area
Barn
b
10 –28 m 2
Volume
Liter
l, L
dm 3 = 10 –3 m 3
Mass
Tonne
t
Mg = 10 3 kg
Pressure
Bar 1
bar
10 5 Pa = 10 5 N m –2
Energy
Electronvolt 2
eV (= e
×
V)
≈
1.60218
×
10 –19 J
Mass
Unified atomic
mass unit 2,3
u (= m a ( 12 C)/12)
≈
1.66054
×
10 –27 kg
1 The ångstrom and the bar are approved by CIPM for “temporary use with
SI units,” until CIPM makes a further recommendation. However, they
should not be introduced where they are not used at present.
2 The values of these units in terms of the corresponding SI units are not
exact, since they depend on the values of the physical constants e (for the
electronvolt) and N A (for the unified atomic mass unit), which are deter-
mined by experiment.
3 The unified atomic mass unit is also sometimes called the dalton, with
symbol Da, although the name and symbol have not been approved by
CGPM.
Conversion Constants and Multipliers
Recommended Decimal Multiples and Submultiples
Multiples and
Submultiples
Prefixes
Symbols
Multiples and
Submultiples
Prefixes
Symbols
10 18
exa
E
10 –1
deci
d
10 15
peta
P
10 –2
centi
c
10 12
tera
T
10 –3
milli
m
10 9
giga
G
10 –6
micro
µ (Greek mu)
10 6
mega
M
10 –9
nano
n
10 3
kilo
k
10 –12
pico
p
10 2
hecto
h
10 –15
femto
f
10
deca
da
10 –18
atto
a
© 2003 by CRC Press LLC
contexts. SI prefixes may be attached to some of these units, such as milliliter, ml; millibar, mbar;
8450414.001.png

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