Handbook of mathematics for engineers and scienteists part 50 pps
Handbook of mathematics for engineers and scienteists part 1 ppsx
... 8
1. 3. PowersandLogarithms 8
1. 3 .1. PowersandRoots 8
1. 3.2. Logarithms 9
1. 4. BinomialTheoremandRelatedFormulas 10
1. 4 .1. Factorials. Binomial Coefficients.BinomialTheorem 10
1. 4.2. RelatedFormulas ... 1 10 /16 /06 2:53: 21 PM
CONTENTS
Authors xxv
Preface xxvii
Main Notation xxix
Part I. Definitions, Formulas, Methods, and Theorems 1
1. Arithmetic and Elementary Algeb...
Handbook of mathematics for engineers and scienteists part 3 ppsx
... 930
ReferencesforChapter17 935
18. Special Functions and Their Properties 937
18.1. Some Coefficients,Symbols,andNumbers 937
18.1.1. Binomial Coefficients 937
18.1.2. PochhammerSymbol 938
18.1 .3. ... Equation and a Nonlinear Wave Equation 733
15.10. DifferentialConstraintsMethod 737
15.10.1. DescriptionoftheMethod 737
15.10.2. First-OrderDifferentialConstraints 739
15.10 .3. Seco...
Handbook of mathematics for engineers and scienteists part 6 ppsx
... integer
p and q (q ≠ 0). It is only the rational numbers that can be written in the form of finite
(terminating) or periodic (recurring) decimals (e.g., 1/8 = 0.125 and 1 /6 = 0. 166 66 ).
Any integer ... m/n,wherem and n are natural numbers:
a
p
= a
m/n
=
n
√
a
m
, a ≥ 0.
6 ARITHMETIC AND ELEMENTARY ALGEBRA
1.2.2. Addition and Multiplication of Numbers
1.2.2-1. Addition of...
Handbook of mathematics for engineers and scienteists part 8 pps
... (Napierian) logarithms, and the function e
x
:
e = lim
n→∞
1 +
1
n
n
= 2.7 182 81 , e
x
= lim
n→∞
1 +
x
n
n
.
The formula for passing from an arbitrary base a to the base e of natural logarithms:
a
x
= ... modulus of the logarithmic function tends to infinity slower than any
power of x as x → +∞; and it tends to zero slower than any power of 1/x as x → +0.The
graphs of...
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... 1).
References for Chapter 2
Abramowitz,M.andStegun,I.A.(Editors) ,Handbook of Mathematical Functions with Formulas, Graphs
and Mathematical Tables, National Bureau of Standards Applied Mathematics, ... 1998.
Korn,G.A.andKorn,T.M.,Mathematical Handbook for Scientists and Engineers, 2nd Edition, Dover
Publications, New York, 2000.
Prudnikov, A. P., Brychkov, Yu. A., and Mari...
Handbook of mathematics for engineers and scienteists part 17 pps
... (b), and
under translation and rotation (c)ofaxes.
Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi-
nates (ˆx,ˆy) in the system O
X
Y . The transformation of rectangular ... system OXY and coordi-
nates (ˆx,ˆy) in the system
O
X
Y . The transformation of rectangular Cartesian coordinates
by the parallel translation and rotation of axes i...
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... R
.(4.6.3.23)
Remark. To obtain formulas (4.6.3.22) and (4.6.3.23), one should rewrite the equation of the straight line
in parametric form and replace x, y,andz in equation (4.6.3.16) of the plane by their ... length of the
perpendicular drawn from this point to the plane and taken with sign + if the point and the
origin lie on opposite sides of the plane and with sign – if...
Handbook of mathematics for engineers and scienteists part 26 ppsx
... under rotations of the coordinate axes. Here Δ
ij
is the
cofactor of the entry a
ij
in Δ.
The classification of quadrics based on the invariants S, T , δ ,and and the semi-
invariants σ and Σ is given ... the coefficients, has 1 of 17 canonical forms, each of which is associated with a
certain class of quadrics (see Table 4.3).
4.7.2-3. Invariants of quadrics.
The shape of a...
Handbook of mathematics for engineers and scienteists part 28 ppsx
... = 0
by the substitution y = x + 1/x.
5.1.5-2. Equations of general form and their properties.
An algebraic equation of degree n has the form
a
n
x
n
+ a
n–1
x
n–1
+ ···+ a
1
x + a
0
= 0 (a
n
≠ ... form
p/q
,where
p
is a divisor of
a
0
and
q
is a divisor of
a
n
.If
a
n
= 1
, then all rational roots of equation (5.1.5.1) (if they exist) are integer divisors of the
free term.
THE...
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... ALGEBRA
Properties of the set of all solutions of a homogeneous system:
1. All solutions of a homogeneous system (5.5.1.3) form a linear space.
2. The linear space of all solutions of a homogeneous ... unknown
quantities and a basic matrix of rank r is isomorphic to the space A
n–r
of all ordered
systems of (n – r) numbers. The dimension of the space of solutions is eq...
Handbook of mathematics for engineers and scienteists part 36 pps
... basic idea of the method consists of consecutive transformations
of the quadratic form: on every step, one should single out the perfect square of some linear
form.
Consider a quadratic form
B(x, ... on the reduction of a matrix to canonical Jordan form.
5.7. Bilinear and Quadratic Forms
5.7.1. Linear and Sesquilinear Forms
5.7.1-1. Linear forms in a unitary space.
A linear form...
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... graph of the function y = |f (x)| is obtained from that of y = f (x) by preserving
the parts of the latter for which f (x) ≥ 0 and symmetric reflection, with respect to the axis
Ox, of the parts for ... term.
The remainders in the form of Lagrange and Cauchy can be obtained as special cases of the
Schl
¨
omilch formula with p = n + 1 and p = 1, respectively.
For a = 0,...
Handbook of mathematics for engineers and scienteists part 44 pps
... boundary points of the domain, and then choose the
largest and the smallest values.
6.3.4-2. Extremal points of functions of three variables.
For functions of three variables, points of extremum ... Points of Functions of Several Variables
6.3.4-1. Conditions of extremum of a function of two variables.
1
◦
. Points of minimum, maximum, or extremum. A point (x
0
, y...
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... λ ≤ λ
2
and let it has a continuous partial derivative
∂
∂λ
f(x, λ)inthe
284 INTEGRALS
7.1.6-2. Integration of the product of a polynomial by a trigonometric function.
1
◦
. General formulas:
P
n
(x)cos(ax) ... ···
–cos(ax)
P
n
(x)
a
–
P
n
(x)
a
3
+ ···
+ C.
These formulas are obtained by repeated integration by parts; see formula 4 from Para-
graph 7.1.2-2 with f (x)=P
n
(x)...
Handbook of mathematics for engineers and scienteists part 50 pps
... of
integral.
7.3. Double and Triple Integrals
7.3.1. Definition and Properties of the Double Integral
7.3.1-1. Definition and properties of the double integral.
Suppose there is a bounded set of ... is uniformly distributed on the interval
[0, 1]
; the symbol
[z]
stands for the
integer part of
z
(the maximum integer not exceeding
z
).
2
◦
. Uniformly distributed sequences ca...