Có 20+ tài liệu thuộc chủ đề "quy luật trong toán học"
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Trigonometric Functions. Definition of Trigonometric Functions. Graphs of Trigonometric Functions. Properties of Trigonometric Functions. Inverse Trigonometric Functions. Graphs of Inverse Trigonometric Functions. Properties of Inverse Trigonometric Functions
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References for Chapter 4. References for Chapter 5. Discontinuities of the First and the Second Kind. Differential Operators of the Field Theory. References for Chapter 6. Properties of the Indefinite Integral. Geometrical Meaning of the Definite Integral. Geometric and Physical Applications of the Definite Integral. Definition and Properties of the Double Integral. Computation of the Double Integral. Geometric and Physical...
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Description of the Method. Examples: The Newell–Whitehead Equation and a Nonlinear Wave Equation 733 15.10. 15.10.1. 15.10.2. 15.10.3. 15.10.4. 15.11. Painlev´e Test for Nonlinear Equations of Mathematical Physics. 15.11.1. 15.11.2. 15.11.3. 15.12. 15.12.1. 15.12.2. 15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 15.13. 15.13.1. 15.13.2. 15.14. 15.14.1. 15.14.2. 15.14.3. 15.14.4. 15.14.5. Systems of Second-Order Equations...
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Equations of the Form y xx = f (x, y. Equations of the Form f (x, y)y xx = g(x, y, y x. Systems of Ordinary Differential Equations. Linear Systems of Two Equations. Systems of First-Order Equations. Systems of Second-Order Equations. Linear Systems of Three and More Equations. Nonlinear Systems of Two Equations. Nonlinear Systems of Three or More Equations....
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We also appreciate the help of Vladimir Nazaikinskii and Grigorii Yosifian for translating several chapters of this book and are thankful to Kirill Kazakov and Mikhail Mikhin for their assistance in preparing the camera-ready copy of the book.. equal to. less than. “a less than b” is written as a <. less than or equal to. a much less than...
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Real Numbers. A prime number is a positive integer that is greater than 1 and has no positive integer divisors other than 1 and itself. A composite number is a positive integer that is greater than 1 and is not prime, i.e., has factors other than 1 and itself. The following numbers are composite . The number 1 is a...
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(2k – 2)(2k. (2k – 1)(2k + 1), where n and k are natural numbers.. Let a, b, and c be real (or complex) numbers. Arithmetic progression.. An arithmetic progression, or arithmetic sequence, is a sequence of real numbers for which each term, starting from the second, is the previous term plus a constant d, called the common difference. In...
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The first step, the formulation of an original hypothesis, is the most difficult part of the method of mathematical induction. Refutation of the proposition: one specifies an element a ∗ (counterexample) that belongs to A but does not belong to B. Proposition: Numbers in the form 2 2 n + 1, where n is a positive integer, were once thought...
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Trigonometric Functions. Definition of Trigonometric Functions 2.2.1-1. Consider rotation of the polar radius issuing from the origin O and ending at a point M of the trigonometric circle. Let α be the angle between the x-axis and the polar radius OM measured from the positive direction of the x-axis. One radian is the angle at the vertex of the sector...
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The principal (single-valued) branches of the inverse trigonometric functions are denoted by. arcsin x ≡ sin –1 x (arcsine is the inverse of sine), arccos x ≡ cos –1 x (arccosine is the inverse of cosine), arctan x ≡ tan –1 x (arctangent is the inverse of tangent), arccot x ≡ cot –1 x (arccotangent is the inverse of cotangent)...
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Hyperbolic functions of multiple argument.. cosh 2x = 2 cosh 2 x – 1,. cosh 3x = –3 cosh x + 4 cosh 3 x, cosh 4x = 1 – 8 cosh 2 x + 8 cosh 4 x,. cosh 5x = 5 cosh x – 20 cosh 3 x + 16 cosh 5 x,. k + 1 C n...
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From the relations given in Tables 3.1 and 3.2, one can derive all missing relations by cyclic permutations of the sides a, b, and c and the angles α, β, and γ. One of the angles is determined by the law of cosines, cos α = b 2 + c 2 – a 2 2bc . Then either the law...
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One can find the area of an arbitrary polygon by dividing it into triangles.. Properties of quadrilaterals.. The diagonals of a convex quadrilateral meet.. The lengths of the sides a, b, c, and d, the diagonals d 1 and d 2 , and the segment m connecting the midpoints of the diagonals satisfy the relation a 2 + b 2...
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Figure 3.22. Then the area of the annulus is given by the formula. 2πρδ where D = 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...
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of edges Total surface area Volume. the generator, then the lateral surface area S lat and the volume V of the cylinder are given by the formulas. In a right cylinder, the bases are perpendicular to the generator. The volume, the lateral surface area, and the total surface area of a right circular cylinder are given by the formulas. where...
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The sum of lengths of two sides is greater than the length of the third side.. The absolute value of the difference between the lengths of two sides is less than the length of the third side,. The sum of the angles lies between π and 3π, π <. sin α = sin b sin β = sin c. of...
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The transformation of rectangular Cartesian coordinates by the rotation of axes is given by the formulas. x = ˆ x cos α – ˆ y sin α,. Suppose that two rectangular Cartesian coordinate systems OXY and O $ X $ Y $ are given and the first system is taken to the second by the translation of the origin O(0,...
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Parametric equations of a curve on the plane have the form. x = ϕ(t), y = ψ(t where x and y are treated as the coordinates of some point A for each value of the variable parameter t. In general, the variables x and y vary with t, and the point A moves on the plane.. Eliminating the parameter t...
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Angle between two straight lines.. We consider two straight lines given by the equations. y = k 1 x + b 1 and y = k 2 x + b where k 1 = tan ϕ 1 and k 2 = tan ϕ 2 are the slopes of the respective lines (see Fig. The angle α between these lines can...
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In polar coordinates (ρ, ϕ), the equation of an ellipse becomes ρ = p. Definition and canonical equation of hyperbola.. A curve on the plane is called a hyperbola if there exists a rectangular Cartesian coordinate system OXY in which the equation of this curve has the form. 0 (see Fig. The coordinates in which the equation of a hyperbola...