Handbook of mathematics for engineers and scienteists part 14 ppt

... = 2R and d = 2r are the outer and inner diameters, ρ =12(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 angleis less that 2π (or 360◦).4◦.Asolid angle is a part of space bounded ... volume of the frustum isV =13h(S1+ S2+S1S2)=13hS21 +aA+a2A2,(3.2.2.7)where S1 and S2are the areas of the bases, a and A are two respective sides of the bases,and...

Handbook of mathematics for engineers and scienteists part 18 pptx

... equations of a curve.Parametric equations of a curve on the plane have the formx = 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 ... coordinatesystem OXY has the formxa+yb= 1,(4.3.1.13)where a and b are the x-andy-intercepts of the line, i.e., the signed distances from thepoints of intersection of the line with the coordinate ... Equation of a pencil of straight lines.The set of all straight lines passing through a point A on the plane is called a pencil of straightlines, and the point A itself is called the center of the...

Handbook of mathematics for engineers and scienteists part 21 pptx

... directions of principal axes and of their conjugate chords are called the principaldirections 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 interms of the roots λ1 and λ2 of the characteristic equation (4.4.5.4):I = λ1+ λ2, δ = λ1λ2.(4.4.5.5)4.4.5-5. Centers and ... δ ,and and the roots λ1 and λ2(λ1≥ λ2)ofthecharacteristic equation (4.4.5.4) by the formulasp =1I–ΔI=1λ1–Δλ1> 0, λ2= 0.(4.4.5.20)4.4.5-9. Geometric deﬁnition of...

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 equationsF ... e33z.(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XYZ are given and the ﬁrst system ... Z and coordinates(ˆx,ˆy,ˆz) in the systemOXYZ. The transformation of rectangular Cartesian coordinates bythe parallel translation and the rotation of axes is determined by the formulasx=e11ˆx...

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 otherpoints.4.6. Line and Plane in Space4.6.1. Plane in Space4.6.1-1. Equation of plane passing through ... z0) and perpendicularly toa vector N =(A, B, C)hastheformA(x – x0)+B(y – y0)+C(z – z0)=0,or(r – r0) ⋅ N = 0,(4.6.1.1)where r and r0are the position vectors of the point M(x, y, z)andM0(x0, ... theequation of a plane is zero, then the equation is said to be incomplete:122 ANALYTIC GEOMETRY3. Vectors a and b are collinear if and only if a × b = 0. In particular, a × a = 0 and a(a × b)=b(a...

Handbook of mathematics for engineers and scienteists part 39 ppt

... inﬁnite) of a subsequence of a given sequence xnis called a partial limit of xn. In the set of all partial limits of any sequence of real numbers, there always exists thelargest 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 deﬁnition of a function consists of ... limit if and only if for anyε > 0,thereisNsuch that the inequality|xn– xm| < εholds for alln > N and m > N.240 LIMITS AND DERIVATIVES6.1.2-7. Upper and lower limits of a...

Handbook of mathematics for engineers and scienteists part 43 ppt

... o(ρ)forsmallΔx and Δy is widely used for approximatecalculations, 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, allstatements can be easily extended to the case of n variables.6.3.2-1. Total and partial increments ... limits exist). Partial derivatives are also denoted by zx and zy, ∂xz and ∂yz,or fx(x, y)andfy(x, y).266 LIMITS AND DERIVATIVES6.3.2-4. Second partial derivatives and second differentials.The...

Handbook of mathematics for engineers and scienteists part 56 pptx

... series∞n=0kxk–1can be obtainedfrom 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=0kxk–1=ddx11 ... subtraction, multiplication, and division of power series.1. Addition and subtraction of power series.Twoseries∞n=0anxn and ∞n=0bnxnwithconvergence radii Ra and Rb, respectively, ... dependent on all values of the functions.* What is meant here is the fact of convergence or divergence of the Fourier series at x0,andalsothenumerical value of its sum in the case of convergence.8.3....

Handbook of mathematics for engineers and scienteists part 59 ppt

... treat xc and ycin the equations for the coordinates of the center of curvature as the current coordinates of the evolute.380 DIFFERENTIAL GEOMETRYThe 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 (theyare constant at all points of the circle); for a straight ... tangency of the envelopeto a curve of the family is called a characteristic point of the curve of the family (Fig. 9.13).Figure 9.13. The envelope of the family of curves.9.1. THEORY OF CURVES...

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., Handbook of Differential Geometry, ... formulaΓk,ij=12∂gki∂uj+∂gkj∂ui–∂gij∂uk.This is one of the basic formulas in the theory of surfaces; this formula means that thescalar products of the second partial derivatives of the position vector r(ui, uj) by its partialderivatives...

Handbook of mathematics for engineers and scienteists part 67 ppt

... 11Integral Transforms11.1. General Form of Integral Transforms. SomeFormulas11.1.1. Integral Transforms and Inversion FormulasNormally an integral transform has the formf(λ)=baϕ(x, λ)f(x) ... the transform of the function f(x )and (x, λ) is called thekernel of the integral transform. The function f (x) is called the inverse transform of f(λ).The limits of integration a and b are ... see Paragraphs 10.1.2-7 and 10.1.2-8.11.2. Laplace Transform11.2.1. Laplace Transform and the Inverse Laplace Transform11.2.1-1. Laplace transform.The Laplace transform of an arbitrary (complex-valued)...

Handbook of mathematics for engineers and scienteists part 73 ppt

... canonical form; see Paragraph 12.2.1-3 for the reduction of equationsto this form. Let y1(x) be any nontrivial partial solution of this equation. The generalsolution 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 eitherformula (12.2.3.2).12.2.3-3. Equations of special ... each of the three cases, the following procedure shouldbe performed: substitute the above expressions of y1 and y2into the original equa-tion (12.2.2.1) and equate the coefﬁcients of (x...

Handbook of mathematics for engineers and scienteists part 75 ppt

... transformations.1◦. The equations of the formF (eλxy, eλxyx, eλxyxx)=0 (12.3.2.8)remain unchanged under simultaneous translation and scaling of variables, x → x + α and y ... ﬁnite.If f (x) → k2> 0 for x → and f≥ 0, then the solutions of the equation for large xbehave similarly to those of the equation y+ k2y = 0.3◦.Letf (x) → –∞ for |x| →∞. Then every ... (12.3.2.10)is invariant under the simultaneous scaling and translation of variables, x → αx and y → y + β,where α =e–βλ/n and β is an arbitrary number. The transformation z = xneλy,w = xyxbrings...

Handbook of mathematics for engineers and scienteists part 76 pptx

... =1; zp and C are arbitrary constants; and the aj(j ≥ 3) are uniquely deﬁnedin terms of α, β, zp,andC.2◦. Two solutions of equation (12.3.4.8) corresponding to different values of the ... zp and C are arbitrary constants; the coefﬁcients aj(j ≥ 7) are uniquely deﬁnedin terms of zp and C.2◦. In a neighborhood of a ﬁxed point z = z0, the solution of the Cauchy problem for the ... –z0)+12(6A2+z0)(z–z0)2+16(12AB+1)(z–z0)3+12(6A3+B2+Az0)(z–z0)4+···,where A and B are initial data of the Cauchy problem, so that y|z=z0= A and yz|z=z0= B.Remark. The solutions of the Cauchy problems for the second and fourth Painlev´e...

Handbook of mathematics for engineers and scienteists part 80 ppt

... 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 by the formulasyk=[λk(x)]–1/2[D(x)]–1/4exp1ελk(x) ... variables,x → x + α and y → βy,whereβ =exp(–αλ/n )and α is an arbitrary constant. Thetransformation z = eλxyn, w = yx/y leads to an equation of order n – 1.12.5.2-8. Equations of the form Fxneλy,...

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