Handbook of mathematics for engineers and scienteists part 74 doc
Handbook of mathematics for engineers and scienteists part 2 doc
... GeneralFormofIntegralTransforms.SomeFormulas 435
11.1.1. IntegralTransformsandInversionFormulas 435
11.1 .2. Residues.JordanLemma 435
11 .2. LaplaceTransform 436
11 .2. 1. LaplaceTransformandtheInverseLaplaceTransform ... LinearSystemsofOrdinaryDifferentialEquations 528
12. 6.1. Systems of Linear Constant-CoefficientEquations 528
12. 6 .2. Systems of Linear Variable-CoefficientEquations...
Handbook of mathematics for engineers and scienteists part 7 docx
... right-hand
sides of the above formulas is called the index of summation (for series) or the index of
multiplication (for products). The 1 and n (or the m and n)aretheupper and lower limits
of summation ... b
2
).
1.6. MEAN VALUES AND INEQUALITIES OF GENERAL FORM 13
1.6. Mean Values and Inequalities of General Form
1.6.1. Arithmetic Mean, Geometric Mean, and Other Mea...
Handbook of mathematics for engineers and scienteists part 13 doc
... area of the triangle with vertices at the center of the disk and at the endpoints
of the radii bounding the corresponding sector. One takes the minus sign for α < 180 and
the plus sign for ... 3.21b).
13. For secants, AC ⋅ AD = AB ⋅ AE = m
2
– R
2
(Fig. 3.21c).
14. For a tangent and a secant, AF ⋅ AF = AC ⋅ AD (Fig. 3.21c).
3.1.3-2. Segment and sector.
Aplanefigure bo...
Handbook of mathematics for engineers and scienteists part 31 docx
... rank of the matrix A.
Properties of the rank of a matrix:
1. For any matrices A and B of the same size the following inequality holds:
rank (A + B) ≤ rank (A)+rank(B).
2. For a matrix A of size ... x
n–1
n
=
1≤j<i≤n
(x
i
– x
j
).
5.2.2-6. Determinant of a sum and a product of matrices.
The determinant of the product of two matrices A and B...
Handbook of mathematics for engineers and scienteists part 35 docx
... system of
eigenvectors of A and the subspace consisting of vectors orthogonal to all eigenvectors
of A.Inthefinite-dimensional case, an orthonormal system of eigenvectors of A is a
basis of V.
5. ... =detA.
The trace ofthe matrix of a linear operator, Tr(A), isalso independent of the basis. Therefore,
one can correctly define the trace Tr(A) of a linear operator as the tra...
Handbook of mathematics for engineers and scienteists part 49 doc
... notation for the Laplace transform:
f(p)=L
f(x)
.
Section 11.2 presents properties and methods for determining the Laplace transform,
and Section T3.1 gives tables of the Laplace transforms of ... method for the calculation of improper integrals using the
theory of functions of a complex variable.
7.2.8-5. Calculation of improper integrals using the Laplace transfo...
Handbook of mathematics for engineers and scienteists part 51 doc
... r
v
.
Remark. The formulas from Items 3 and 4 are equivalent—they define one and the same surface in two
forms, scalar and vector, respectively.
5. Calculation of volumes. If a domain U of the three-dimensional ... sphere of radius R and a right circular cylinder of radius a < R whose axis passes
through the sphere center, find the volume of the figure the cylinder cuts o...
Handbook of mathematics for engineers and scienteists part 65 doc
... property) and transforms any pair of points symmetric about a circle
C
into a pair of points symmetric about the image of the circle
C
(preservation of symmetric
points).
420 FUNCTIONS OF COMPLEX ... zeros and
poles of
f(z)
. Moreover, the logarithmic residue of
f(z)
at a zero or a pole of
f(z)
is equal
to the order of the zero or minus the order of the pole, respectiv...
Handbook of mathematics for engineers and scienteists part 74 doc
... the
change of variable z =A
2
x
2
+A
1
x+A
0
+y (the constants A
2
, A
1
,and A
0
are selected using
the method of undetermined coefficients). In particular, the nonhomogeneous boundary
conditions of ... continuous in x for fixed s, with x
1
≤ x ≤ x
2
and x
1
≤ s ≤ x
2
.
2
◦
. G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0,forall
x
1
< x < x
2
exclu...
Handbook of mathematics for engineers and scienteists part 82 docx
... stated
Discriminant D,
formula (12.6.1.29)
Roots of quadratic
equation (12.6.1.27), λ
1
and λ
2
Conditions for coefficients of
differential equations (12.6.1.25)
Type of equilibrium points or
shape of phase ... trajectories of a system of differential equations near an equilibrium point of the center
type.
12.6. LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 535
By conve...
Handbook of mathematics for engineers and scienteists part 84 docx
... Theorems of stability and instability.
In the cases where the theorems of stability and instability by first approximation fail to
resolve the issue of stability for a specific system of nonlinear ... solution in the form
u
k
= Ψ(u
3–k
),
where k = 1, 2 and Ψ(u) is an arbitrary function of one variable.
2
◦
. For linear equations (13.1.1.1) with the functions f , g,andh inde...
Handbook of mathematics for engineers and scienteists part 85 doc
... f, g,andh of equation (13.1.1.1) be continuously differentiable functions
of x, y,andw in G and let x = h
1
(ξ), y = h
2
(ξ), and w = h
3
(ξ) be continuously differentiable
functions of ξ for ... slightly).
y
x
line of discontinuity
characteristics
characteristics
Figure 13.7. Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shock...
Handbook of mathematics for engineers and scienteists part 86 docx
... η
x
.(13.1.3.13)
This is a continuous function of η for fixed x and y. It can be shown that for fixed x and with
the exception of a countable set of values of y, function (13.1.3.13) has a unique minimum
with ... At the lines of discontinuity, y = y(x), the left and right limit values of w exist: w(x, y–0)
and w(x, y + 0).
13.1. LINEAR AND QUASILINEAR EQUATIONS 565
1...
Handbook of mathematics for engineers and scienteists part 87 doc
... Jacobian of F , Φ ,and with respect to w, p,andq ,and[ Φ, Ψ]isthe
Jacobi–Mayer bracket. In this case, relations (13.2.1.1) and (13.2.1.10) form a parametric
representation of the complete integral of ... Method of separation of variables. Equations of special form.
The method of separation of variables implies searching for a complete integral as the sum or
product of fu...
Handbook of mathematics for engineers and scienteists part 92 doc
... of auxiliary problems for equation (14.4.4.1) and problems for associated
functions ϕ(x )and (y) that determine particular solutions of the form (14.4.4.3).
The abbreviation HBC below stands for ... detail).
14.5.1. Laplace Transform and Its Application in Mathematical
Physics
14.5.1-1. Laplace and inverse Laplace transforms. Laplace transforms for derivatives.
The Laplace t...
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