Handbook of mathematics for engineers and scienteists part 108 pps
Handbook of mathematics for engineers and scienteists part 108 pps
... above formulas for the coordinates of the first and second prolongation, (15.8.1.9)
and (15.8.1.14), will be required later for the analysis of differential equations.
15.8.2. Symmetries of Nonlinear ... CLASSICAL METHOD OF STUDYING SYMMETRIES OF DIFFERENTIAL EQUATIONS 717
Remark. In the special case of transformations in the plane, the functions ϕ
1
and ϕ
2
in (15.8.1.1)...
Handbook of mathematics for engineers and scienteists part 1 ppsx
... 123
HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
C5025_C000a.indd 1 10/16/06 2:53:21 PM
CONTENTS
Authors xxv
Preface xxvii
Main Notation xxix
Part I. Definitions, Formulas, Methods, and ... a not for profit organization that provides licenses and registration for a variety of users. For orga‑
nizations that have been granted a photocopy license by the CCC, a se...
Handbook of mathematics for engineers and scienteists part 3 ppsx
... 989
ReferencesforChapter18 990
xviii CONTENTS
19. Calculus of Variations and Optimization 991
19.1. CalculusofVariationsandOptimalControl 991
19.1.1. Some DefinitionsandFormulas 991
19.1.2. SimplestProblemofCalculusofVariations ... 748
15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method
Description 748
15.11.2. Examples of Performing the Painlev
´
e Test a...
Handbook of mathematics for engineers and scienteists part 6 ppsx
... A and |b| ≤ B it follows that |a + b| ≤ A + B and |ab| ≤ AB.
1.3. Powers and Logarithms
1.3.1. Powers and Roots
1.3.1-1. Powers and roots: the main definitions.
Given a positive real number a and ... 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 re...
Handbook of mathematics for engineers and scienteists part 8 pps
... 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 the functions y =log
2
x and y ... multiplication, and division) and the
operation of composition (composite functions).
The graphs and the main properties of the basic as well as some other frequently
occurr...
Handbook of mathematics for engineers and scienteists part 11 ppsx
... 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...
Handbook of mathematics for engineers and scienteists part 25 ppsx
... 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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