Handbook of mathematics for engineers and scienteists part 21 pptx
Handbook of mathematics for engineers and scienteists part 18 pptx
... equations of a curve.
Parametric equations of a curve on the plane have the form
x = ϕ(t), y = ψ(t), (4.2.1.4)
where x and y are treated as the coordinates of some point A for each value of the ... coordinate
system OXY has the form
x
a
+
y
b
= 1,(4.3.1.13)
where a and b are the x-andy-intercepts of the line, i.e., the signed distances from the
points of intersection of th...
Handbook of mathematics for engineers and scienteists part 21 pptx
... directions of principal axes and of their conjugate chords are called the principal
directions of a second-order curve. The angle between the positive direction of the axis OX
and each of the two ... I and δ of second-order curves are expressed as follows in
terms of the roots λ
1
and λ
2
of the characteristic equation (4.4.5.4):
I = λ
1
+ λ
2
, δ = λ
1
λ
2
.(4.4.5.5)...
Handbook of mathematics for engineers and scienteists part 23 pptx
... the surface that is formed by points of both surfaces in (4.5.2.18) and does not contain any other
points.
4.6. Line and Plane in Space
4.6.1. Plane in Space
4.6.1-1. Equation of plane passing through ... z
0
) and perpendicularly to
a vector N =(A, B, C)hastheform
A(x – x
0
)+B(y – y
0
)+C(z – z
0
)=0,or(r – r
0
) ⋅ N = 0,(4.6.1.1)
where r and r
0
are the position vectors of th...
Handbook of mathematics for engineers and scienteists part 56 pptx
... series
∞
n=0
kx
k–1
can be obtained
from the left-hand side of the second formula in (8.3.3.7) for m = 0 and n = 1. Substituting S(x)=1/(1 – x)
into the right-hand side of that formula, we get
∞
k=0
kx
k–1
=
d
dx
1
1 ... subtraction, multiplication, and division of power series.
1. Addition and subtraction of power series.Twoseries
∞
n=0
a
n
x
n
and
∞
n=0
b
n
x
n
wit...
Handbook of mathematics for engineers and scienteists part 76 pptx
... =
1; z
p
and C are arbitrary constants; and the a
j
(j ≥ 3) are uniquely defined
in terms of α, β, z
p
,andC.
2
◦
. Two solutions of equation (12.3.4.8) corresponding to different values of the ... z
p
and C are arbitrary constants; the coefficients a
j
(j ≥ 7) are uniquely defined
in terms of z
p
and C.
2
◦
. In a neighborhood of a fixed point z = z
0
, the solution of the Ca...
Handbook of mathematics for engineers and scienteists part 91 pptx
... the above holds for
α = 1, 2, and β = 0,–1,–2,
14.4. Method of Separation of Variables (Fourier Method)
14.4.1. Description of the Method of Separation of Variables.
General Stage of Solution
14.4.1-1. ... Derivation of equations and boundary conditions for particular solutions.
The approach is based on searching for particular solutions of equation (14.4.1.1) in th...
Handbook of mathematics for engineers and scienteists part 102 pptx
... constant, for some k and m. Transformation (15.3.4.2) is a
combination of a translation transformation in t and scaling transformations in x and w.
It should be emphasized that these transformations ... differential equations.
15.5. Method of Generalized Separation of Variables
15.5.1. Structure of Generalized Separable Solutions
15.5.1-1. General form of solutions. The clas...
Handbook of mathematics for engineers and scienteists part 110 pptx
... 15.5. Algorithm for the construction of exact solutions by a nonclassical method for second-order
evolution equations. Here, ODE stands for ordinary differential equation and PDE for partial differential
equation.
which ... condition of compatibility of
equations (15.10.1.4) and (15.10.1.5). The compatibility condition is a differential equation for ϕ(y)andisa
consequence...
Handbook of mathematics for engineers and scienteists part 111 pptx
... operators of (15.10.2.4) in which the partial
derivatives w
t
, w
xx
, w
xt
,andw
tt
must be expressed in terms of x, t, w,andw
x
with
the help of relations (15.10.2.12) and (15.10.2.2) and those ... to t and (15.10.2.2) with respect to t and x, and then equating the resulting
expressions of the third derivatives w
xtt
and w
ttx
to one another:
D
t
F =D
x
[D
t
G]. (15.10.2...
Handbook of mathematics for engineers and scienteists part 124 pptx
... Neumann series converges absolutely and uniformly on
[a, b].
A solution of a Fredholm equation of the second kind of the form (16.4.3.1) is expressed
by the formula
y(x)=f(x)+λ
b
a
R(x, t; λ)f ... value.
For any n, the set of characteristic values of the nth iterated kernel coincides with the
set of nth powers of the characteristic values of the first kernel.
The eigenfun...
Handbook of mathematics for engineers and scienteists part 132 pptx
... general solution of the corresponding homogeneous equation
(with g ≡ 0), and the second term y(x) is a particular solution of equation (17.2.1.15).
A formula for the general solution of equation ... 18.18.1.
2
◦
. For the equation with polynomial right-hand side
y(x + 1)–ay(x)=
n
k=0
b
k
x
k
, a ≠ 1,(17.2.1.19)
a particular solution is sought by the method of indefinite coeffic...
Handbook of mathematics for engineers and scienteists part 148 pptx
... calculus of variations.
19.1. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 999
L
EMMA 1(FUNDAMENTAL LEMMA OR LAGRANGE’S LEMMA).
Let
f(t) C[t
0
, t
1
]
and
assume that
t
1
t
0
f(t)g(t) dt = 0
for each ... 0.
One of the best-known isoperimetric problems, after which the entire class of problems
was named, is the Dido problem.
Example 1. Find a smooth curve of given length fixed a...
Handbook of mathematics for engineers and scienteists part 164 pptx
... expectations
introduced by the first and second factors under the action at levels i and j.
References for Chapter 21
Arora, P. N. and Anand, S. K., Mathematical and Statistical Tables and Formulae, Anmol Publications ... Computational Handbook of Statistics, 4th Edition, Allyn & Bacon, Boston,
1997.
Bulmer,M.G.,Principles of Statistics, Dover Publishers, New York, 1979....
Handbook of mathematics for engineers and scienteists part 165 pptx
... π.
15.
∞
k=1
1
k
2n+1
sin(kx)=
(–1)
n–1
(2π)
2n+1
2(2n + 1)!
B
2n+1
x
2π
,
where 0 ≤ x ≤ 2π for n = 1, 2, ; 0 < x < 2π for n = 0;andtheB
n
(x) are Bernoulli
polynomials.
1120 FINITE SUMS AND INFINITE SERIES
16.
∞
k=1
1
k2
k
=ln2.
17.
∞
k=0
(–1)
k
n
2k
=
n
2
n
2
+ ... a)=2
n
cosh
n
x
2
sinh
nx
2
+ a
.
10.
n
k=0
C
k
n
cosh(kx + a)=2
n
cosh
n
x
2
cosh
nx
2
+ a
.
1...
Handbook of mathematics for engineers and scienteists part 174 pptx
... 4b, ϕ =
1
2
arctan(b/a)
1182 INTEGRAL TRANSFORMS
T3.4. Tables of Fourier Sine Transforms
T3.4.1. General Formulas
No. Original function, f(x) Sine transform,
ˇ
f
s
(u)=
∞
0
f(x)sin(ux) dx
1 af
1
(x)+bf
2
(x)
a
ˇ
f
1s
(u)+b
ˇ
f
2s
(u)
2 ... n
(–1)
n+m
π
2n!
∂
n
∂a
n
a
m
e
–u
√
a
11
1
√
x
π
2u
12
1
x
√
x
√
2πu
T3.4. TABLES OF FOURIER SINE TRANSFORMS 1185
T3.4.6. Expressions with Tri...
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