Handbook of mathematics for engineers and scienteists part 9 pot
Handbook of mathematics for engineers and scienteists part 4 potx
... Press, 1995 and
2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press,
1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers
and Scientists, ... V. F. Zaitsev, and
A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis,
2002; and A. D. Polyanin and V. F. Zaitsev, Hand...
Handbook of mathematics for engineers and scienteists part 5 potx
... stands for the union of sets A and B
∩ intersection (Boolean multiplication); A ∩B stands for the intersection (com-
mon part) of sets A and B
⊂ inclusion; A ⊂ B means that the set A is part of ... of numbers a and b is denoted by a + b and has the property
a + b = b + a
– minus sign; the difference of numbers a and b is denoted by a – b
⋅ multiplication sign; the p...
Handbook of mathematics for engineers and scienteists part 9 pot
... 2πn]. For x =
π
2
+ 2πn, it attains its maximal value (y = 1), and for
x =–
π
2
+ 2πn it attains its minimal value (y =–1). The graph of the function y =sinx is
called the sinusoid or sine curve and ... The graph of the function y =cotx isgiveninFig.2 .9.
O
1
x
y
π
π
1
yx= tan
π
2
π
2
Figure 2.8. The graph of the function y =tanx.
O
1
x
y
π
π
1
yx= cot
π
2
π
2
3π
2
Figure 2 .9....
Handbook of mathematics for engineers and scienteists part 15 pot
... r is the radius of the upper base and h is the altitude of the frustum of a cone.
3.2.3-3. Sphere. Spherical parts. Torus.
1
◦
.Thesphere of radius R centered at O is the set of points in space ... h),
(3.2.3.10)
where R and h are the radius and the height of the spherical cap.
3
◦
. A portion of a ball bounded by the curved surface of a spherical cap and the conical
s...
Handbook of mathematics for engineers and scienteists part 19 potx
... r
1
(M
0
)andr
2
(M
0
) are the lengths of the focal radii of M
0
.
The tangent at an arbitrary point M
0
(x
0
, y
0
) of an ellipse forms acute angles ϕ
1
and ϕ
2
with the focal radii of the point of ... y) of an ellipse with the foci F
1
(–c, 0)andF
2
(c, 0)
are called the left and right focal radii of this point. We denote the lengths of the left and
right focal radii...
Handbook of mathematics for engineers and scienteists part 20 pot
... is the locus of points for which the
ratio of distances to the focus and the directrix is equal to 1:
r
|x + p/2|
= 1.(4.4.4.6)
4.4.4-4. Equation of tangent and optical property of parabola.
The ... ϕ
,(4.4.3.16)
where upper and lower signs correspond to right and left parts of a hyperbola, respectively.
4.4.4. Parabola
4.4.4-1. Definition and canonical equation of para...
Handbook of mathematics for engineers and scienteists part 24 pot
... 0.
4.6.1-12. Equation of plane passing through line of intersection of planes.
The planes passing through the line of intersection of the planes A
1
x +B
1
y + C
1
z + D
1
= 0
and A
2
x + B
2
y + ... cosines of the angles α, β ,and formed by this straight line (the direction of
the vector R
0
) with the coordinate axes OX, OY ,andOZ. These cosines can be expressed
via the coor...
Handbook of mathematics for engineers and scienteists part 30 pot
... a
nn
=
n
k=1
a
ik
A
i
k
=
n
k=1
a
kj
A
k
j
.
This formula is also called the ith row expansion of the determinant of A andalsothejth
column expansion of the determinant of A.
Example 1. Let us find the third-order determinant of the matrix
A ... [a
i
a
j
a
B].
Note that if A and B are square matrices and the number of rows in C is equal to the
number of ro...
Handbook of mathematics for engineers and scienteists part 32 potx
... to both spaces L
1
and L
2
.Such
elements form a subspace of V.
The sum of subspaces L
1
and L
2
of one and the same linear space V is, by definition,
the set of all elements of V that can be represented ... of matrices A and B consist of eigenvalues λ
j
and μ
k
, respec-
tively. Then the spectrum of the Kronecker product A ⊗ B is the set of all products λ
j
μ
k
.
T...
Handbook of mathematics for engineers and scienteists part 37 pot
... called addition and c is
called the sum of a and b.
2. Multiplicative form: c = ab; the corresponding composition law is called multiplication
and c is called the product of a and b.
A composition ... group and let
G be a set with a composition law. A mapping f : G →
G is
called a homomorphism if
f(ab)=f(a)f(b)foralla, b
G;
and the subset of
G consisting of all eleme...
Handbook of mathematics for engineers and scienteists part 38 potx
... Baltimore, Maryland, 1996.
Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 1, North Holland, Amsterdam, 1996.
Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 2, North Holland, Amsterdam, ... transformations,forwhichdetP =+1.
2. Improper orthogonal transformations,forwhichdetP =–1.
The set of proper orthogonal transformations forms a group called the special orthogonal
group o...
Handbook of mathematics for engineers and scienteists part 40 pot
... point of its domain.
6.1.5-3. Points of discontinuity of a function.
A point a is called a point of discontinuity of the first kind for a function f(x)ifthereexist
finite one-sided limits f(a+0)andf(a–0), ... small for x → a.
Functions f (x)andg(x) are said to be of the same order for x → a, and one writes
f(x)=O
g(x)
if lim
x→a
f(x)
g(x)
= K, 0 < |K| < ∞.*
A funct...
Handbook of mathematics for engineers and scienteists part 41 pot
... expressions of the form 0/0 and ∞ /∞.
THEOREM 1.
Let
f(x)
and
g(x)
be two functions defined in a neighborhood of a point
a
, vanishing at this point,
f(a)=g(a)=0
, and having the derivatives
f
(a)
and
g
(a)
, ... Calculus for Functions of a Single
Variable
6.2.1. Derivative and Differential, Their Geometrical and Physical
Meaning
6.2.1-1. Definition of derivative and...
Handbook of mathematics for engineers and scienteists part 45 pot
... integrals of the form 3
◦
from Paragraph 7.1.4-1.
7.1.5. Integration of Exponential and Trigonometric Functions
7.1.5-1. Integration of exponential and hyperbolic functions.
1. Integrals of the form
R(e
px
, ... of a proper fraction.
1
◦
. To integrate a proper fraction, one should first rewrite the integrand (7.1.3.1) in the
form of a sum of partial fractions. Below are th...
Handbook of mathematics for engineers and scienteists part 52 potx
... S(t) is the boundary of the domain U(t), n is the unit normal to S(t), and v is the
velocity of motion of the points of S(t).
7.3.5-4. Some geometric and physical applications of the triple integral.
1. ... neither the
partition L
n
nor the selection of the points (x
i
, y
i
, z
i
), then it is called the line integral of
the first kind of the function f(x, y, z) over the...
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