Handbook of mathematics for engineers and scienteists part 89 ppt
Handbook of mathematics for engineers and scienteists part 14 ppt
... = 2R and d = 2r are the outer and inner diameters, ρ =
1
2
(R + r) is the midradius,
and δ = R – r is the width of the annulus.
The area of the part of the annulus contained in a sector of central ... on one side of each of its faces. The sum
∠AOB +∠BOC + ···+∠EOA of face angles (Fig. 3.25a) of any convex polyhedral angle
is less that 2π (or 360
◦
).
4
◦
.Asolid angle...
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 22 ppt
... v)(4.5.2.17)
for appropriate values of the parameters u and v.
In spatial analytic geometry, each curve is treated as the intersection of two surfaces and
hence is defined by a system of two equations
F ... e
33
z.
(4.5.2.12)
4.5.2-4. Transformation of coordinates under translation and rotation of axes.
Suppose that two rectangular Cartesian coordinate systems OXY Z and
O...
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 39 ppt
... infinite) of a subsequence of a given sequence x
n
is called a partial limit
of x
n
. In the set of all partial limits of any sequence of real numbers, there always exists the
largest and the least ... definition of a function consists of using an equation of the form F (x, y)=0,
from which one calculates the value y for any fixed value of the argument x.
Parametric defini...
Handbook of mathematics for engineers and scienteists part 43 ppt
... o(ρ)forsmallΔx and Δy is widely used for approximate
calculations, in particular, for fi nding errors in numerical calculations of values of a function.
Example 1. Suppose that the values of the ... Variables
For the sake of brevity, we consider the case of a function of two variables. However, all
statements can be easily extended to the case of n variables.
6.3.2-1. Tot...
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 59 ppt
... treat x
c
and y
c
in the equations
for the coordinates of the center of curvature as the current coordinates of the evolute.
380 DIFFERENTIAL GEOMETRY
The sign of ds is chosen arbitrarily, and it ... larger k is and the smaller ρ is at this point.
For a circle of radius a, the radius of curvature is ρ = a and the curvature is k = 1/a (they
are constant at all points o...
Handbook of mathematics for engineers and scienteists part 62 ppt
... second group of derivation formulas (Weingarten formulas).
The formulas in the second group of derivation formulas express the partial derivatives of
the unit normal vector N in terms of the variables ... 2003.
Dillen, F. J. E. and Verstraelen, L. C. A., Handbook of Differential Geometry, Vol. 1, North Holland,
Amsterdam, 2000.
Dillen, F. J. E. and Verstraelen, L. C. A., Hand...
Handbook of mathematics for engineers and scienteists part 67 ppt
... 11
Integral Transforms
11.1. General Form of Integral Transforms. Some
Formulas
11.1.1. Integral Transforms and Inversion Formulas
Normally an integral transform has the form
f(λ)=
b
a
ϕ(x, λ)f(x) ... the transform of the function f(x )and (x, λ) is called the
kernel of the integral transform. The function f (x) is called the inverse transform of
f(λ).
The limits of integrat...
Handbook of mathematics for engineers and scienteists part 73 ppt
... canonical form; see Paragraph 12.2.1-3 for the reduction of equations
to this form. Let y
1
(x) be any nontrivial partial solution of this equation. The general
solution can be constructed by formula ... expansions of the fundamental system of solutions of equation
(12.2.3.1) with f < 0 are derived by separating the real and imaginary parts in either
formula (12.2.3.2).
12.2....
Handbook of mathematics for engineers and scienteists part 75 ppt
... transformations.
1
◦
. The equations of the form
F (e
λx
y, e
λx
y
x
, e
λx
y
xx
)=0 (12.3.2.8)
remain unchanged under simultaneous translation and scaling of variables, x → x + α
and y ... finite.
If f (x) → k
2
> 0 for x → and f
≥ 0, then the solutions of the equation for large x
behave similarly to those of the equation y
+ k
2
y = 0.
3
◦
.Letf (x) → –∞ for |...
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 89 ppt
... canonical form of a hyperbolic equation.
Apart from notation, the left-hand side of the last equation coincides with that of the wave
equation (14.1.1.2).
In some cases, reduction of an equation ... constants k and k
1
are given by formulas (14.1.1.10)
Type of equation,
conditions on coefficients
Vari ab les ξ and η in
transformation (14.1.1.9)
Coefficients β
1
and β
2
in
transf...