Có 60+ tài liệu thuộc chủ đề "toán học trong nghiên cứu"
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Then f (x) has bounded variation and. Then, f (x) is of bounded variation and V a b f (x. Properties of functions of bounded variation.. Any function of bounded variation is bounded.. The sum, difference, or product of finitely many functions of bounded variation is a function of bounded variation.. Let f(x) and g(x) be two functions of bounded...
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Expressions of the form 0. Indeterminate expressions of the form 1. 0 , 0 0 can be reduced to expressions of the form 0. The derivative of the second derivative of a function y = f (x) is called the third-order derivative, y = (y. The nth-order derivative of the function y = f (x) is defined as the derivative...
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Suppose that the derivatives f (x) and f (x) exist on the interval [a, b] and the inequalities f (x. The straight line (secant) passing through the points (a, f (a)) and (b, f (b)) of the curve y = f (x) meets the abscissa axis at the point x 1 . the value x n +1 is the abscissa...
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Extremal Points of Functions of Several Variables 6.3.4-1. Conditions of extremum of a function of two variables.. Points of minimum, maximum, or extremum. A point (x 0 , y 0 ) is called a point of local minimum (resp., maximum) of a function z = f(x, y) if there is a neighborhood of (x 0 , y 0 ) in...
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The integration of a polynomial multiplied by an exponential function can be accom- plished by using the formula of integration by parts (or repeated integration by parts) given in Paragraph 7.1.2-2.. Compute the integral. (3x + 1) e 2x dx.. e 2x dx = 1. More complex examples of the application of integration by parts or repeated integration by parts...
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sin α cos β = 1 2 [sin(α + β. sin α sin β = 1 2 [cos(α – β. Integrals of the form. sin m x cos n x dx, where m and n are integers, are evaluated as follows:. (a) if m is odd, one uses the change of variable cos x = z, with sin x dx...
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Let f (x) be a continuous function on [0, π] and let it have everywhere on [0, π], except maybe at finitely many points, a square integrable deriva- tive f (x). If either of the conditions. 0 f (x) dx = 0 is satisfied, then the following inequality holds:. The arithmetic mean, geometric mean, harmonic mean, and quadratic mean of...
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x λ dx is convergent for a >. a A sin x dx. |cos a – cos A. If an improper integral is convergent and the integrand function tends to a limit as x. 0 is not a necessary condition for convergence of the integral (7.2.7.1).. 0 sin(x 2 ) dx. Furthermore, it can be shown that the integral x....
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Principal value of a singular integral.. dx x – c = ln b – c c – a + lim. (7.2.11.2) The limit of the last expression obviously depends on the way in which ε 1 and ε 2 tend to zero. This integral is called a singular integral. we arrive at the notion of the Cauchy principal value of...
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If f (x, y) is continuous in D, then the double integral. D f (x, y) dx dy exists.. If f (x, y) is bounded and the set of points of discontinuity of f (x, y) has a zero area (e.g., the points of discontinuity lie on finitely many continuous curves in the x, y plane), then the double integral...
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If a domain U is split into two subdomains, U 1 and U 2 , that do not have common internal points and if a function f (x, y, z) is integrable in either subdomain, then U f (x, y, z) dx dy dz. f (x, y, z) dx dy dz. f (x, y, z) dx dy dz.. M in...
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(1) the vector field a is potential;. AB a ⋅ dr is independent of the shape of. Surface Integral of the First Kind. Definition of the surface integral of the first kind.. of a partition D n is the largest of the diameters of the cells (see Paragraph 7.3.4-1). f (x i , y i , z i ) ΔS...
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The Kummer transformation is used for accelerating convergence of series, since the generic term a (1) n. a n of the transformed series (8.1.5.7) tends to zero faster that the generic term {a n } of the original series lim. The auxiliary sequence {b n } is chosen in such a way that the sum (8.1.5.6) is known beforehand.. The...
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A special case of the Taylor series (for x 0 = 0) is the Maclaurin series:. 2) convergent in a neighborhood of x 0 to a function different from f (x), 3) convergent in a neighborhood of x 0 to the function f (x).. A necessary and sufficient condition for a function f (x) to be represented by its Taylor...
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Fourier expansion of odd and even functions.. Then the Fourier expansion of f (x) on the interval (–l, l) has the form of the cosine Fourier series:. a n cos nπx l , where the Fourier coefficients have the form. Then the Fourier expansion of f (x) on the interval (–l, l) has the form of the sine Fourier series:....
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In a polar coordinate system, the curve is usually given by the equation r = r(ϕ),. If a curve is given parametrically then the positive sense is defined on this curve, i.e., the direction in which the point M(x(t), y(t)) of the curve moves as the parameter t increases. If the curve is given explicitly by then the positive sense...
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A straight line is called an asymptote of a curve Γ if the distance from a point M (x, y) of the curve to this straight line tends to zero as x 2 + y 2. The limit position of the tangent to a regular point of the curve is an asymptote. Let us find the asymptotes of the curve...
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At a regular point M 0 , the equation of the normal plane has the form. and for the curve Γ obtained as the intersection (9.1.2.2) of two planes, we have. Thus the first derivative with respect to the natural parameter s of the position vector of a point on a curve is the unit vector tangent to the curve.....
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Suppose that a surface is given in vector form (9.2.1.1) and a curve lying on it is parametrized by the parameter t. Then to each parameter value there corresponds a point of the curve, and the position of this point on the surface is specified by some values of the curvilinear coordinates u and v. Thus the curvilinear coordinates of...
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Suppose that two surfaces U and U ∗ are given and there is one-to-one correspondence between their points such that the length of each curve on U is equal to the length of the corresponding curve on U. Such a one-to-one mapping of U into U ∗ is called a bending of the surface U into the surface U. and...