Handbook of mathematics for engineers and scienteists part 149 ppt

... of stationarity with respect to t k (only for movable endpoints of the integration interval), Λ t k ( ˆ t k )=0 (k = 0, 1). 1008 CALCULUS OF VARIATIONS AND OPTIMIZATION Necessary conditions for ... (19.1.3.6) and the regularity condition is satisfied (i.e., the functions g i (t)are linearly independent on any of the intervals [t 0 , τ ]and[ τ, t 1 ]foranyτ), then x(t)isan extre...

Ngày tải lên: 02/07/2014, 13:20

... = 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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 ... roots λ 1 and λ 2 of the characteristic equation (4.4.5.4) are the eigenvalues of the real symmetric matrix [a ij ] and, as a consequence, are real. Obviously, the invariants...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20

... 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...

Ngày tải lên: 02/07/2014, 13:20