Handbook of mathematics for engineers and sciente119 pptx

... 15.13 Systems of equations of the form (15.14.5.1) admitting invariant solutions that are found using combinations of translation and scaling transformations (C is an arbitrary constant) Systems of equations ... lists some systems of the form (15.14.5.1) admitting other invariant solu- tions that are found using combinations of translation and scaling transformations in the in...

Ngày tải lên: 02/07/2014, 13:20

... right-hand sides of the above formulas is called the index of summation (for series) or the index of multiplication (for products). The 1 and n (or the m and n)aretheupper and lower limits of summation ... compendium of mathematical definitions, formulas, and theorems intended for researchers, university teachers, engineers, and students of various backgrounds in ma...

Ngày tải lên: 08/03/2014, 12:20

... 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

... 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

... 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

... = 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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... the above holds for α = 1, 2, and β = 0,–1,–2, 14.4. Method of Separation of Variables (Fourier Method) 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution 14.4.1-1. ... 1. The method of separation of variables is also used to solve linear boundary value problems for elliptic equations of the form (14.4.1.1), provided that α(t)<0...

Ngày tải lên: 02/07/2014, 13:20

... constant, for some k and m. Transformation (15.3.4.2) is a combination of a translation transformation in t and scaling transformations in x and w. It should be emphasized that these transformations ... differential equations. 15.5. Method of Generalized Separation of Variables 15.5.1. Structure of Generalized Separable Solutions 15.5.1-1. General form of solutions. The clas...

Ngày tải lên: 02/07/2014, 13:20

... condition of compatibility of equations (15.10.1.4) and (15.10.1.5). The compatibility condition is a differential equation for ϕ(y)andisa consequence of equations (15.10.1.4), (15.10.1.5), and those ... C 1 and C 2 are arbitrary constants. For convenience, we introduce the new constant C 1 =exp  3 2 C 1  andlookforasolutionintheformC 2 =– 3 √ 2 C 1 F (C 1 ). Inserting (15.9...

Ngày tải lên: 02/07/2014, 13:20