Handbook of mathematics for engineers and scienteists part 59 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...

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... 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...

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... 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)...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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....

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... 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 |...

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... = 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...

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... infinity as x → 0, and hence formula (12.4.2.6) cannot be applied to it; this formula is only valid for finite initial values of the function and its derivatives. 12.4.2-7. Solution of equations using ... conditions f (x) ≠ 0 and D(x) ≠ 0 are satisfied, the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.1) are described b...

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