This document summarizes the contents of a presentation on geometry and integrability. It discusses topics like Sophie Kowalevski's work, billiards within ellipses, the Poncelet theorem, generalizations of the Darboux theorem, the Kowalevski top, and discriminantly separable polynomials. It also presents theorems regarding properties of billiard trajectories within ellipsoids and the geometric interpretation of the Kowalevski fundamental equation.
This document discusses solving a differential equation using the Frobenius method. It presents the equation xy'' + (1 - 2x)y' + (x - 1)y = 0 and provides steps to find the indicial equation and power series solutions. These include determining coefficients, setting coefficients of like powers of x equal to 0, and solving the resulting equations to obtain the solutions as a power series expansion in terms of x.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
Howard, anton cálculo ii- um novo horizonte - exercicio resolvidos v2Breno Costa
This document provides 40 examples of solving initial value problems for ordinary differential equations using separation of variables. The examples cover both first and second order linear differential equations with various forcing functions. Solutions are obtained by separating variables, integrating, and applying initial conditions to determine constants of integration. Special cases where the standard procedure fails due to singularities are also discussed.
1. A differential equation is an equation that relates an unknown function with some of its derivatives. The document provides a step-by-step example of solving a differential equation to find the xy-equation of a curve with a given gradient condition.
2. The key steps are: (1) write the derivative term as a fraction, (2) integrate both sides, (3) apply the initial condition to determine the constant term, (4) write the final function relationship.
3. Common types of differential equations discussed are separable first order equations, where the derivative terms can be isolated by dividing both sides.
Matematika Ekonomi Diferensiasi fungsi sederhanalia170494
This document discusses differentiation rules for simple functions including:
- Constant functions have a derivative of 0
- Polynomial functions have derivatives that are the polynomial with the exponent decreased by 1 and multiplied by the exponent
- The product rule, quotient rule, and chain rule for differentiation
- Examples of applying these rules to differentiate a variety of functions
This document discusses affine functions. It defines affine functions as functions of the form f(x) = ax + b, where a and b are real numbers. It provides examples of linear functions where b = 0, constant functions where a = 0, and the identity function where a = 1 and b = 0. It discusses the angular coefficient a and the linear coefficient b. It explains that the graph of an affine function is a straight line that can be increasing or decreasing. It also discusses finding the zero or root of an affine function and studying the sign of an affine function.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
This document discusses solving a differential equation using the Frobenius method. It presents the equation xy'' + (1 - 2x)y' + (x - 1)y = 0 and provides steps to find the indicial equation and power series solutions. These include determining coefficients, setting coefficients of like powers of x equal to 0, and solving the resulting equations to obtain the solutions as a power series expansion in terms of x.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
Howard, anton cálculo ii- um novo horizonte - exercicio resolvidos v2Breno Costa
This document provides 40 examples of solving initial value problems for ordinary differential equations using separation of variables. The examples cover both first and second order linear differential equations with various forcing functions. Solutions are obtained by separating variables, integrating, and applying initial conditions to determine constants of integration. Special cases where the standard procedure fails due to singularities are also discussed.
1. A differential equation is an equation that relates an unknown function with some of its derivatives. The document provides a step-by-step example of solving a differential equation to find the xy-equation of a curve with a given gradient condition.
2. The key steps are: (1) write the derivative term as a fraction, (2) integrate both sides, (3) apply the initial condition to determine the constant term, (4) write the final function relationship.
3. Common types of differential equations discussed are separable first order equations, where the derivative terms can be isolated by dividing both sides.
Matematika Ekonomi Diferensiasi fungsi sederhanalia170494
This document discusses differentiation rules for simple functions including:
- Constant functions have a derivative of 0
- Polynomial functions have derivatives that are the polynomial with the exponent decreased by 1 and multiplied by the exponent
- The product rule, quotient rule, and chain rule for differentiation
- Examples of applying these rules to differentiate a variety of functions
This document discusses affine functions. It defines affine functions as functions of the form f(x) = ax + b, where a and b are real numbers. It provides examples of linear functions where b = 0, constant functions where a = 0, and the identity function where a = 1 and b = 0. It discusses the angular coefficient a and the linear coefficient b. It explains that the graph of an affine function is a straight line that can be increasing or decreasing. It also discusses finding the zero or root of an affine function and studying the sign of an affine function.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
1. The document contains an assignment on mathematics involving ordinary and partial differential equations, matrices, vector calculus, and their applications. It includes solving various types of differential equations, finding eigenvectors and eigenvalues of matrices, evaluating line, surface and volume integrals using Green's theorem and Stokes' theorem. The assignment contains 20 problems spanning these topics.
2. The assignment covers key concepts in differential equations, linear algebra, and vector calculus including solving ordinary differential equations, partial differential equations, systems of linear equations, eigenproblems, line integrals, surface integrals, divergence, curl, gradient, Laplacian, and theorems like Green's theorem and Stokes' theorem.
3. Students are required to solve 20 problems involving these
1. The document provides 14 problems involving partial differential equations (PDEs). The problems involve forming PDEs by eliminating arbitrary constants from functions, finding complete integrals, and solving PDEs.
2. Methods used include taking partial derivatives, finding auxiliary equations, and making substitutions to isolate the PDE or solve it.
3. The document covers a range of techniques for working with PDEs, including eliminating constants, finding trial solutions, integrating subsidiary equations, and solving auxiliary equations to find complete integrals.
Pt 3&4 turunan fungsi implisit dan cyclometrilecturer
This document discusses implicit differentiation and provides examples of taking the derivative of implicit functions. It begins by presenting the general form of an implicit function as f(x,y)=0 and provides some examples. It then shows how to take the derivative dy/dx of several implicit functions by applying the chain rule and implicit differentiation. Formulas for the derivatives of inverse trigonometric functions are also provided, along with examples of finding dy/dx for functions involving inverse trigonometric functions.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
The document defines:
1) Powers and exponentiation, including rules for multiplying, dividing, and raising powers of numbers.
2) Examples are provided to illustrate the rules.
3) It notes that care must be taken with negative bases and even/odd exponents.
The summary provides the high level definition of powers/exponentiation and notes some key rules and examples are given to illustrate, highlighting the need to consider signs with negative bases. It does not include details of the specific examples or problems shown in the document.
1. The radius of curvature at a point on a curve is defined as the reciprocal of the curvature at that point. It represents the radius of the circle that best approximates the curve near that point.
2. For the circle x^2 + y^2 = 25, the radius of curvature at any point is equal to the radius of the circle, which is 25.
3. For the curve xy = c^2, the radius of curvature at the point (c, c) is c.
1. The document contains 10 problems involving exact differential equations, 10 problems involving homogeneous differential equations, and 10 other groups of problems involving various types of differential equations.
2. The problems are in Spanish and involve identifying the appropriate differential equation based on given information, and providing the solution steps to solve for an integral or constant.
3. Solutions are provided in the form of integral or constant expressions in response to each problem.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
This document summarizes key topics from a lesson on quadratic forms, including:
1) It defines a quadratic form in two variables as a function of the form f(x,y) = ax^2 + 2bxy + cy^2.
2) It classifies quadratic forms as positive definite, negative definite, or indefinite based on the sign of f(x,y) for all non-zero (x,y) points.
3) It gives examples of quadratic forms and classifies them, such as f(x,y) = x^2 + y^2 being positive definite.
This document contains the answer key for an exam on differential equations. It provides the solutions to 4 problems:
1) Solving a separable ODE to find an implicit solution.
2) Solving a Bernoulli equation by substitution to find the general solution.
3) Finding an integrating factor to solve an exact ODE and determine the general implicit solution.
4) Solving a linear ODE initial value problem to model salt in a tank over time.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
This section introduces differential equations and their use in mathematical modeling. It provides examples of verifying solutions to differential equations by direct substitution. Typical problems show finding an integrating constant to satisfy an initial condition. Differential equations are derived from descriptions of real-world phenomena involving rates of change. The section establishes foundational knowledge of differential equations and their solution methods.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document discusses the persecution of Slavic peoples during World War 2. It notes that over 5 million Russians, 3 million Ukrainians, and 1.5 million Belarusians died. The Nazis viewed Slavs as subhuman and intended to transform them into a race of laborers and servants under German control. Hitler invaded the Soviet Union in part due to his hatred of Slavs, whom he wanted to eliminate to make room for the superior German race in an effort to gain Lebensraum, or living space.
Isolation of genes differentially expressed during the defense response of Ca...CIAT
Genes differentially expressed in cassava in response to whitefly attack were identified through microarray analysis. Several genes involved in defense responses were found to be upregulated, including genes related to pathogenesis, proteasomes, chitinases, peroxidases, lipases, and heat shock proteins. Pathway analysis indicated that jasmonic acid/ethylene and salicylic acid signaling pathways regulating various defense mechanisms were activated. The microarray analysis provides insight into cassava's gene expression responses when defending against whitefly attack.
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
1. The document contains an assignment on mathematics involving ordinary and partial differential equations, matrices, vector calculus, and their applications. It includes solving various types of differential equations, finding eigenvectors and eigenvalues of matrices, evaluating line, surface and volume integrals using Green's theorem and Stokes' theorem. The assignment contains 20 problems spanning these topics.
2. The assignment covers key concepts in differential equations, linear algebra, and vector calculus including solving ordinary differential equations, partial differential equations, systems of linear equations, eigenproblems, line integrals, surface integrals, divergence, curl, gradient, Laplacian, and theorems like Green's theorem and Stokes' theorem.
3. Students are required to solve 20 problems involving these
1. The document provides 14 problems involving partial differential equations (PDEs). The problems involve forming PDEs by eliminating arbitrary constants from functions, finding complete integrals, and solving PDEs.
2. Methods used include taking partial derivatives, finding auxiliary equations, and making substitutions to isolate the PDE or solve it.
3. The document covers a range of techniques for working with PDEs, including eliminating constants, finding trial solutions, integrating subsidiary equations, and solving auxiliary equations to find complete integrals.
Pt 3&4 turunan fungsi implisit dan cyclometrilecturer
This document discusses implicit differentiation and provides examples of taking the derivative of implicit functions. It begins by presenting the general form of an implicit function as f(x,y)=0 and provides some examples. It then shows how to take the derivative dy/dx of several implicit functions by applying the chain rule and implicit differentiation. Formulas for the derivatives of inverse trigonometric functions are also provided, along with examples of finding dy/dx for functions involving inverse trigonometric functions.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
The document defines:
1) Powers and exponentiation, including rules for multiplying, dividing, and raising powers of numbers.
2) Examples are provided to illustrate the rules.
3) It notes that care must be taken with negative bases and even/odd exponents.
The summary provides the high level definition of powers/exponentiation and notes some key rules and examples are given to illustrate, highlighting the need to consider signs with negative bases. It does not include details of the specific examples or problems shown in the document.
1. The radius of curvature at a point on a curve is defined as the reciprocal of the curvature at that point. It represents the radius of the circle that best approximates the curve near that point.
2. For the circle x^2 + y^2 = 25, the radius of curvature at any point is equal to the radius of the circle, which is 25.
3. For the curve xy = c^2, the radius of curvature at the point (c, c) is c.
1. The document contains 10 problems involving exact differential equations, 10 problems involving homogeneous differential equations, and 10 other groups of problems involving various types of differential equations.
2. The problems are in Spanish and involve identifying the appropriate differential equation based on given information, and providing the solution steps to solve for an integral or constant.
3. Solutions are provided in the form of integral or constant expressions in response to each problem.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
This document summarizes key topics from a lesson on quadratic forms, including:
1) It defines a quadratic form in two variables as a function of the form f(x,y) = ax^2 + 2bxy + cy^2.
2) It classifies quadratic forms as positive definite, negative definite, or indefinite based on the sign of f(x,y) for all non-zero (x,y) points.
3) It gives examples of quadratic forms and classifies them, such as f(x,y) = x^2 + y^2 being positive definite.
This document contains the answer key for an exam on differential equations. It provides the solutions to 4 problems:
1) Solving a separable ODE to find an implicit solution.
2) Solving a Bernoulli equation by substitution to find the general solution.
3) Finding an integrating factor to solve an exact ODE and determine the general implicit solution.
4) Solving a linear ODE initial value problem to model salt in a tank over time.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
This section introduces differential equations and their use in mathematical modeling. It provides examples of verifying solutions to differential equations by direct substitution. Typical problems show finding an integrating constant to satisfy an initial condition. Differential equations are derived from descriptions of real-world phenomena involving rates of change. The section establishes foundational knowledge of differential equations and their solution methods.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document discusses the persecution of Slavic peoples during World War 2. It notes that over 5 million Russians, 3 million Ukrainians, and 1.5 million Belarusians died. The Nazis viewed Slavs as subhuman and intended to transform them into a race of laborers and servants under German control. Hitler invaded the Soviet Union in part due to his hatred of Slavs, whom he wanted to eliminate to make room for the superior German race in an effort to gain Lebensraum, or living space.
Isolation of genes differentially expressed during the defense response of Ca...CIAT
Genes differentially expressed in cassava in response to whitefly attack were identified through microarray analysis. Several genes involved in defense responses were found to be upregulated, including genes related to pathogenesis, proteasomes, chitinases, peroxidases, lipases, and heat shock proteins. Pathway analysis indicated that jasmonic acid/ethylene and salicylic acid signaling pathways regulating various defense mechanisms were activated. The microarray analysis provides insight into cassava's gene expression responses when defending against whitefly attack.
Protect your identity by shredding credit card offers, using a locked mailbox, and mailing envelopes inside the post office. Install a firewall and virus protection software, and never respond to phishing emails. Be wary of 'shoulder surfers' by shielding your hand when entering your PIN number, as thieves may watch you with binoculars.
B. Nikolic/ B. Sazdovic: Hamiltonian Approach to Dp-brane NoncommutativitySEENET-MTP
This document summarizes a talk on Hamiltonian approaches to Dp-brane noncommutativity in string theory. It discusses:
1) Basic facts about strings and superstrings, including open and closed strings, Dp-branes, and the 5 consistent superstring theories.
2) Boundary conditions for strings ending on Dp-branes and their treatment as canonical constraints. This leads to noncommutativity on the Dp-brane.
3) Details of the Type IIB superstring theory in 10 dimensions and the model considered, which involves graviton, gravitinos, and R-R fields but no dilatinos or dilaton.
M. Dimitrijevic/ V. Radovanovic: D-deformed Wess-Zumino Model and its Renorma...SEENET-MTP
The document summarizes a presentation on D-deformed Wess-Zumino model and its renormalizability properties. It discusses deforming spaces and symmetries using twist formalism. Specifically, it describes how an Abelian twist can deform Poincaré symmetry to obtain noncommutative spacetime while preserving associativity. This allows for construction of D-deformed supersymmetric theories like the Wess-Zumino model, along with investigating their renormalizability.
The document discusses the EN TASC program and Partnership Plus model for serving Ticket to Work beneficiaries. It provides 3 key benefits:
1) Partnership Plus allows beneficiaries to receive upfront vocational rehabilitation services from state VR agencies and ongoing support from employment networks after case closure to help maintain employment.
2) It benefits beneficiaries by providing individualized sequential services and a wider scope of supports.
3) It benefits both VR agencies and employment networks by promoting increased coordination and collaboration between service providers to better serve beneficiaries.
Este documento proporciona análisis técnicos y recomendaciones de posiciones largas en varias acciones del mercado español y europeo, incluyendo precios de entrada, stops y objetivos. Incluye gráficos de líneas de soporte y resistencia para cada valor analizado. El autor advierte que el análisis se proporciona solo con fines informativos y no debe considerarse como una recomendación de inversión.
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
The document discusses Taylor series and their applications. It introduces Taylor series as a way to approximate functions using their derivatives. Examples are provided for linear, quadratic, and higher order Taylor approximations. Applications discussed include using Taylor series in physics for concepts like special relativity equations.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
The document is notes for a lesson on partial derivatives. It introduces partial derivatives and their motivation as slopes of curves through a point on a multi-variable function. It defines partial derivatives mathematically and gives an example. It also discusses second partial derivatives and notes that mixed partials are always equal due to Clairaut's Theorem when the function is continuous. Finally, it provides an example of calculating second partial derivatives.
1. Find the line tangent to the curve of intersection between the surfaces F(x,y,z)=x^2+y^2+z^2=1 and g(x,y,z)=x+y+z+5 at the point P=(1,2,2).
2. Find the values of constants a and b that will make the expression zxy-zx-zy identically zero, given z=U(x,y)e(ax+by) where Uxy=0.
3. Find the points on the sphere x^2+y^2+z^2=9 whose distance from the point (4,-8,8) are
Assignment For Matlab Report Subject Calculus 2Laurie Smith
This document provides the requirements and assignments for a Calculus 2 Matlab report. It includes topics such as: finding partial derivatives of various functions, studying extrema of functions, evaluating double and triple integrals, and calculating mass and centers of mass of solids. Students are divided into groups and will be randomly assigned a topic involving solving concrete problems numerically using Matlab.
This document contains a chapter on functions with 30 math exercises. The exercises involve evaluating functions, determining domains and ranges, analyzing graphs of functions, and solving word problems involving functions.
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
The document defines different types of polynomials and their key properties. It discusses linear, quadratic, and cubic polynomials, and defines them based on their highest degree term. It also covers the degree of a polynomial, zeros of polynomials, and the relationship between the zeros and coefficients of quadratic and cubic polynomials. Finally, it discusses the division algorithm for polynomials.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
This document provides summaries of common special functions and polynomials, including Legendre polynomials, associated Legendre functions, Bessel functions, spherical Bessel functions, Hermite polynomials, and Laguerre polynomials. It defines the differential equations that govern each function, provides generating functions, discusses orthogonality properties, and lists some important recurrence relations and examples. The document is intended as a short survey of essential properties of these important mathematical functions.
02Application of Derivative # 1 (Tangent & Normal)~1 Module-4.pdfRajuSingh806014
If y = f(x) be a given function, then the differential coefficient f' (x) or dy at the point P (x , y ) is
(i) If the tangent at P (x1,y1) of the curve y = f(x) is parallel to the x- axis (or perpendicular to y- axis) then = 0 i.e. its slope will be
zero.
FGdy J
dx 1 1
m = H K = 0
the trigonometrical tangent of the angle (say) which the positive direction of the tangent to the curve at P makes with the positive direction of x- axis Gdy J, therefore represents the slope of the
tangent. Thus
dx (x1,y1)
The converse is also true. Hence the tangent at (x1,y1) is parallel to x- axis.
GJ = 0
(x1,y1)
(ii) If the tangent at P (x , y ) of the curve y =
1 1
f (x) is parallel to y - axis (or perpendicular
to x-axis) then = / 2 , and its slope will be infinity i.e.
dy
m = =
dx (x ,y )
The converse is also true. Hence the tangent at (x1, y1) is parallel to y- axis
Fdy
(x1,y1)
Thus
(i) The inclination of tangent with x- axis.
dy
(iii) If at any point P (x1, y1) of the curve y = f(x), the tangent makes equal angles with the
axes, then at the point P, = / 4 or 3 / 4 ,
= tan–1
GHdxJK
Hence at P, tan = dy/dx = 1. The
(ii) Slope of tangent = dy
dx
(iii) Slope of the normal = – dx/dy
Ex.1 Find the following for the curve y2 = 4x at point (2,–2)
(i) Inclination of the tangent
(ii) Slope of the tangent
(iii) Slope of the normal
Sol. Differentiating the given equation of curve, we get dy/dx = 2/y = –1 at (2,–2)
so at the given point.
(i) Inclination of the tangent = tan–1(–1) = 135º
(ii) Slope of the tangent = –1
(iii) Slope of the normal = 1
converse of the result is also true. thus at
(x1,y1) the tangent line makes equal angles with the axes.
GJ = 1
(x1,y1)
Ex.2 The equation of tangent to the curve y2 = 6x at (2, – 3).
(A) x + y – 1 = 0 (B) x + y + 1 = 0 (C) x – y + 1 = 0 (D) x + y + 2 = 0
Sol. Differentiating equation of the curve with respect to x
(a) Equation of tangent to the curve y = f(x) at A (x1,y1) is
2y dy = 6
dx
FGdy J
dx (2,3)
= 3 = –1
3
y – y1 =
FGdy J
(x1,y1)
(x–x1)
Therefore equation of tangent is y + 3 = – (x – 2)
x + y + 1 = 0 Ans. [B]
Ex.3 The equation of tangent at any of the curve x = at2, y = 2at is -
(A) x = ty + at2 (B) ty + x + at2 = 0
(C) ty = x + at2 (D) ty = x + at3
2 a 1
Sol. dy/dx = (dy/dt)/(dx/dt) = 2 at = t
equation of the tangent at (x,y) point is
(y – 2 at) = 1 (x – at2)
t
ty = x + at2 Ans.[C]
Ex.4 The equation of the tangent to the curve x2 (x – y) + a2 (x + y) = 0 at origin is-
(A) x + y + 1 = 0 (B) x + y + 2 = 0 (C) x + y = 0 (D) 2x – y = 0
Sol. Differentiating equation of the curve w.r.t. x
dy/dx = – y x
(i) If tangent line is parallel to x - axis, then dy/dx = 0 y = 0 and x = a
Thus the point is (a,0)
(ii) If tangent is parallel to y – axis , then dy/dx = x = 0 and y = a
Thus the point is (0,a)
(iii) If tangent line makes equal angles with both axis , then dy/dx = 1
y =
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Chapter 2
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Chapter 3
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V. Dragovic: Geometrization and Generalization of the Kowalevski top
1. Sophie’s world: geometry and integrability
Vladimir Dragovi´
c
Mathematical Institute SANU, Belgrade
Math. Phys. Group, University of Lisbon
SSSCP2009
Niˇ, 4 April 2009
s
2. Contents
1 Sophie’s world
2 Billiard within ellipse
3 Poncelet theorem
4 Generalization of the Darboux theorem
5 Higher-dimensional generalization of the Darboux theorem
6 A new view on the Kowalevski top
4. Sophie’s world: the story of the history of integrability
Acta Mathematica, 1889
5. Addition Theorems
sin(x + y ) = sin x cos y + cos x sin y
cos(x + y ) = cos x cos y − sin x sin y
sn x cn y dn y + sn y cn x dn x
sn (x + y ) =
1 − k 2 sn 2 x sn 2 y
cn x cn y − sn x sn y dn x dn y
cn (x + y ) =
1 − k 2 sn 2 x sn 2 y
2
1 ℘′ (x) − ℘′ (y )
℘(x + y ) = −℘(x) − ℘(y ) +
4 ℘(x) − ℘(y )
6. The Quantum Yang-Baxter Equation
′ 23
R 12 (t1 −t2 , h)R 13 (t1 , h)R (t2 , h) = R 23 (t2 , h)(R 13 (t1 , h)R 12 (t1 −t2 , h)
R ij (t, h) : V ⊗ V ⊗ V → V ⊗ V ⊗ V
t – spectral parameter
h – Planck constant
10. Billiard within ellipse
A trajectory of a billirad within an ellipse is a polygonal line with
vertices on the ellipse, such that successive edges satisfy the billiard
reflection law: the edges form equal angles with the to the ellipse at
the common vertex.
16. Poncelet theorem (Jean Victor Poncelet, 1813.)
Let C and D be two given conics in the plane. Suppose there exists
a closed polygonal line inscribed in C and circumscribed about D.
Then, there are infinitely many such polygonal lines and all of them
have the same number of edges. Moreover, every point of the conic
C is a vertex of one of these lines.
17. Mechanical interpretation of the Poncelet theorem
Let us consider closed trajectory of billiard system within ellipse E.
Then every billiard trajectory within E, which has the same caustic
as the given closed one, is also closed. Moreover, all these
trajectories are closed with the same number of reflections at E.
19. Generalization of the Darboux theorem
Тheorem
Let E be an ellipse in E2 and (am )m∈Z , (bm )m∈Z be two sequences
of the segments of billiard trajectories E, sharing the same caustic.
Then all the points am ∩ bm (m ∈ Z) belong to one conic K,
confocal with E.
20. Moreover, under the additional assumption that the caustic is an
ellipse, we have:
if both trajectories are winding in the same direction about the
caustic, then K is also an ellipse; if the trajectories are winding in
opposite directions, then K is a hyperbola.
21. For a hyperbola as a caustic, it holds:
if segments am , bm intersect the long axis of E in the same
direction, then K is a hyperbola, otherwise it is an ellipse.
22. Grids in arbitrary dimension
Тheorem
Let (am )m∈Z , (bm )m∈Z be two sequences of the segments of billiard
trajectories within the ellipsoid E in Ed , sharing the same d − 1
caustics. Suppose the pair (a0 , b0 ) is s-skew, and that by the
sequence of reflections on quadrics Q1 , . . . , Qs+1 the minimal
billiard trajectory connecting a0 to b0 is realized.
Then, each pair (am , bm ) is s-skew, and the minimal billiard
trajectory connecting these two lines is determined by the sequence
of reflections on the same quadrics Q1 , . . . , Qs+1 .
26. Тheorem
(i) There exists a polynomial P = P(x) such that the
discriminant of the polynomial F in s as a polynomial in
variables x1 and x2 separates the variables:
Ds (F )(x1 , x2 ) = P(x1 )P(x2 ). (1)
(ii) There exists a polynomial J = J(s) such that the discriminant
of the polynomial F in x2 as a polynomial in variables x1 and s
separates the variables:
Dx2 (F )(s, x1 ) = J(s)P(x1 ). (2)
Due to the symmetry between x1 and x2 the last statement
remains valid after exchanging the places of x1 and x2 .
27. Lemma
Given a polynomial S = S(x, y , z) of the second degree in each of
its variables in the form:
S(x, y , z) = A(y , z)x 2 + 2B(y , z)x + C (y , z).
If there are polynomials P1 and P2 of the fourth degree such that
B(y , z)2 − A(y , z)C (y , z) = P1 (y )P2 (z), (3)
then there exists a polynomial f such that
Dy S(x, z) = f (x)P2 (z), Dz S(x, y ) = f (x)P1 (y ).
28. Discriminantly separable polynoiamls – definition
For a polynomial F (x1 , . . . , xn ) we say that it is discriminantly
separable if there exist polynomials fi (xi ) such that for every
i = 1, . . . , n
Dxi F (x1 , . . . , xi , . . . , xn ) =
ˆ fj (xj ).
j=i
It is symmetrically discriminantly separable if
f2 = f3 = · · · = fn ,
while it is strongly discriminatly separable if
f1 = f2 = f3 = · · · = fn .
It is weakly discriminantly separable if there exist polynomials fi j (xi )
such that for every i = 1, . . . , n
Dxi F (x1 , . . . , xi , . . . , xn ) =
ˆ fji (xj ).
j=i
29. Тheorem
Given a polynomial F (s, x1 , x2 ) of the second degree in each of the
variables s, x1 , x2 of the form
F = s 2 A(x1 , x2 ) + 2B(x1 , x2 )s + C (x1 , x2 ).
Denote by TB 2 −AC a 5 × 5 matrix such that
5 5
ij i j−1
(B 2 − AC )(x1 , x2 ) = TB 2 −AC x1−1 x2 .
j=1 i =1
Then, polynomial F is discriminantly separable if and only if
rank TB 2 −AC = 1.
31. Geometric interpretation of the Kowalevski fundamental
equation
Тhеоrem
The Kowalevski fundamental equation represents a point pencil of
conics given by their tangential equations
ˆ
C1 : − 2w1 + 3l1 w2 + 2(c 2 − k 2 )w3 − 4clw2 w3 = 0;
2 2 2
2
C2 : w2 − 4w1 w3 = 0.
The Kowalevski variables w , x1 , x2 in this geometric settings are the
pencil parameter, and the Darboux coordinates with respect to the
conic C2 respectively.
32. Multi-valued Buchstaber-Novikov groups
n-valued group on X
m : X × X → (X )n , m(x, y ) = x ∗ y = [z1 , . . . , zn ]
(X )n symmetric n-th power of X
Associativity
Equality of two n2 -sets:
[x ∗ (y ∗ z)1 , . . . , x ∗ (y ∗ z)n ] и [(x ∗ y )1 ∗ z, . . . , (x ∗ y )n ∗ z]
for every triplet (x, y , z) ∈ X 3 .
Unity e
e ∗ x = x ∗ e = [x, . . . , x] for each x ∈ X .
Inverse inv : X → X
e ∈ inv(x) ∗ x, e ∈ x ∗ inv(x) for each x ∈ X .
33. Multi-valued Buchstaber-Novikov groups
Action of n-valued group X on the set Y
φ : X × Y → (Y )n
φ(x, y ) = x ◦ y
Two n2 -multi-subsets in Y :
x1 ◦ (x2 ◦ y ) и (x1 ∗ x2 ) ◦ y
are equal for every triplet x1 , x2 ∈ X , y ∈ Y .
Additionally , we assume:
e ◦ y = [y , . . . , y ]
for each y ∈ Y .
34. Two-valued group on CP1
The equation of a pencil
F (s, x1 , x2 ) = 0
Isomorphic elliptic curves
Γ1 : y 2 = P(x) deg P = 4
Γ2 : t 2 = J(s) deg J = 3
Canonical equation of the curve Γ2
Γ2 : t 2 = J ′ (s) = 4s 3 − g2 s − g3
Birational morphism of curves ψ : Γ2 → Γ1
ˆ
Induced by fractional-linear mapping ψ which maps zeros of the
polynomial J ′ and ∞ to the four zeros of the polynomial P.
35. Double-valued group on CP1
There is a group structure on the cubic Γ2 . Together with its
soubgroup Z2 , it defines the standrad double-valued group
structure on CP1 :
2 2
t1 − t2 t1 + t2
s1 ∗c s2 = −s1 − s2 + , −s1 − s2 + ,
2(s1 − s2 ) 2(s1 − s2 )
where ti = J ′ (si ), i = 1, 2.
Тheorem
The general pencil equation after fractional-linear transformations
ˆ ˆ
F (s, ψ −1 (x1 ), ψ −1 (x2 )) = 0
defines the double valued coset group structure (Γ2 , Z2 ).
36. iΓ1 ×iΓ1 ×m ψ−1 ×ψ−1 ×id
C4 - Γ1 × Γ1 × C - Γ2 × Γ2 × C
Q
Q ia ×ia ×m
iΓ1 ×iΓ1 ×id×id Q p1 ×p1 ×id
Q
? Q s
Q ?
Γ1 × Γ1 × C × C CP1 × C
p1 ×p1 ×id
ϕ1 ×ϕ2 ˆ ˆ
ψ−1 ×ψ−1 ×id
? ?
+
C×C CP1 × C
m2 mc ×τc
? ?
2 f 2
CP CP × C/ ∼
37. Double-valued group CP1
Тhеоrem
Associativity conditions for the group structure of the
double-valued coset group (Γ2 , Z2 ) and for its action on Γ1 are
equivalent to the great Poncelet theorem for a triangle.
38. References
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c
walevski top
2. V. Dragovi´, Marden theorem and Poncelet-Darboux curves
c
arXiv:0812.48290 (2008)
3. V. Dragovi´, M. Radnovi´, Hyperelliptic Jacobians as Billiard
c c
Algebra of Pencils of Quadrics: Beyond Poncelet Porisms, Adv.
Math., 219 (2008) 1577-1607.
4. V. Dragovi´, M. Radnovi´, Geometry of integrable billiards and
c c
pencils of quadrics, Journal Math. Pures Appl. 85 (2006),
758-790.
5. V. Dragovi´, B. Gaji´, Systems of Hess-Appel’rot type Comm.
c c
Math. Phys. 265 (2006) 397-435
6. V. Dragovi´, B. Gaji´, Lagrange bitop on so(4) × so(4) and
c c
geometry of Prym varieties Amer. J. Math. 126 (2004),
981-1004
7. V. I. Dragovich, Solutions of the Yang equation with rational
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