The document summarizes key concepts from a chapter on vectors and geometry in 3D space. [1] It introduces three-dimensional coordinate systems using ordered triples (x,y,z) and defines the distance formula between two points in 3D space. [2] It also defines concepts like the sphere equation and vectors, including their representation, magnitude, addition/subtraction, and dot and cross products. [3] It concludes by covering lines, planes, and their equations, as well as cylindrical coordinates.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
1. Complex numbers can be represented as ordered pairs of real numbers (a,b) and have definitions for addition, multiplication, and multiplication by scalars.
2. Common notation for complex numbers includes the zero (0,0), unity (1,0), and the complex conjugate (a,-b). Functions like the real part, imaginary part, absolute value, and argument are also introduced.
3. Complex numbers are connected to trigonometry through Euler's formula eiθ = cosθ + i sinθ and relations involving exponentiation, differentiation, and roots of unity.
Lesson 19: Double Integrals over General RegionsMatthew Leingang
The document is notes from a math class on double integrals over general regions. It includes announcements about office hours and problem sessions. It defines double integrals over general regions as limits of integrals over unions of rectangles approximating the region. It discusses properties of double integrals and iterated integrals over curved regions of Type I and Type II. It provides examples and worksheets for students to practice evaluating double integrals.
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.
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
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document provides instructions and information for a mathematics exam. It includes:
1) Details about the exam such as the date, time allotted, and materials allowed.
2) Instructions for candidates on how to identify their work and provide their information.
3) Information for candidates about the structure of the exam including the total number and types of questions, and the total marks available.
4) Advice to candidates about showing their working and obtaining full credit.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
1. Complex numbers can be represented as ordered pairs of real numbers (a,b) and have definitions for addition, multiplication, and multiplication by scalars.
2. Common notation for complex numbers includes the zero (0,0), unity (1,0), and the complex conjugate (a,-b). Functions like the real part, imaginary part, absolute value, and argument are also introduced.
3. Complex numbers are connected to trigonometry through Euler's formula eiθ = cosθ + i sinθ and relations involving exponentiation, differentiation, and roots of unity.
Lesson 19: Double Integrals over General RegionsMatthew Leingang
The document is notes from a math class on double integrals over general regions. It includes announcements about office hours and problem sessions. It defines double integrals over general regions as limits of integrals over unions of rectangles approximating the region. It discusses properties of double integrals and iterated integrals over curved regions of Type I and Type II. It provides examples and worksheets for students to practice evaluating double integrals.
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.
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
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document provides instructions and information for a mathematics exam. It includes:
1) Details about the exam such as the date, time allotted, and materials allowed.
2) Instructions for candidates on how to identify their work and provide their information.
3) Information for candidates about the structure of the exam including the total number and types of questions, and the total marks available.
4) Advice to candidates about showing their working and obtaining full credit.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
This document provides notes and formulae on additional mathematics for Form 5. It covers topics such as progressions, integration, vectors, trigonometric functions, and probability. For progressions, it defines arithmetic and geometric progressions and gives the formulas for calculating the nth term and sum of terms. For integration, it provides rules and formulas for integrating polynomials, trigonometric functions, and expressions with ax+b. It also defines vectors and their operations including vector addition and subtraction. Other sections cover trigonometric functions, their definitions, relationships and graphs, as well as probability topics such as calculating probabilities of events and distributions like the binomial.
(1) The student solved several integrals and derivatives.
(2) They sketched regions bounded by curves and found the areas.
(3) Properties of functions like extremes and concavity were examined.
This document discusses solving quadratic equations. It begins by defining quadratic equations as equations of the second degree in the form ax2 + bx + c = 0, where a ≠ 0. It then discusses:
- The roots or solutions of a quadratic equation are the values that make the equation equal to 0 when substituted for the variable.
- Quadratic equations can be solved by factorization, splitting the middle term into two factors and setting each factor equal to 0 to find the roots.
- Examples are provided to demonstrate solving quadratic equations by factorization, finding the two roots.
- A more general quadratic equation with parameters a, b is also factorized to find its two equal roots.
This document contains a mid-term examination paper for Class VIII students. It tests their knowledge in the subjects of Mathematics, Computer Science, and English.
The Mathematics section contains 20 multiple choice questions testing concepts like sets, square roots, radicals, number systems, and algebraic expressions. The Computer Science section has 12 multiple choice questions on topics such as hexadecimal conversion, binary addition, word processing functions, and programming basics.
The English section begins with 2 sample multiple choice comprehension questions. Sections B and C of each subject contain longer form questions to be answered in paragraphs, involving explanations, calculations, and problem solving. Students have 3 hours to complete the entire exam which is worth a total of 100 marks.
11.some fixed point theorems in generalised dislocated metric spacesAlexander Decker
This document introduces the notion of a generalized dislocated metric space and establishes some of its topological properties. Some fixed point theorems are obtained for self-maps on these spaces that satisfy contractive conditions. Specifically:
1) A generalized dislocated metric is defined on a set that satisfies non-negativity, identity of indiscernibles, symmetry, and a generalized triangle inequality conditions.
2) A topology called the d-topology is shown to be induced by a generalized dislocated metric, and properties of this topology are established.
3) Some fixed point theorems are proved for self-maps on generalized dislocated metric spaces that satisfy contractive conditions, including analogues of Banach's contraction
Some fixed point theorems in generalised dislocated metric spacesAlexander Decker
This document presents some fixed point theorems in generalized dislocated metric spaces. It introduces the notion of a generalized dislocated metric space and establishes its topological properties. It proves an analogue of Seghal's fixed point theorem for self-maps that satisfy contractive conditions in these spaces. This allows deriving generalized dislocated analogues of fixed point theorems of Banach, Kannan, Bianchini, and Reich. The document defines generalized dislocated metrics and establishes properties like uniqueness of limits. It shows the topology induced by such a metric satisfies Kuratowski's closure axioms.
This document contains a mid-term examination paper for Class VIII mathematics. It consists of 3 sections - Section A with 20 multiple choice questions to be completed in 30 minutes, Section B with 10 long-form questions worth 4 marks each, and Section C with 5 long-form questions worth 8 marks each. The paper tests students on various mathematics concepts including sets, radicals, exponents, averages, percentages, and algebraic expressions. Students are asked to solve problems, simplify expressions, find sums and products, and more. The paper is designed to evaluate students' understanding of core Class VIII math topics.
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.
The document discusses how queries work in sharded MongoDB environments. It explains that MongoDB collections are partitioned into chunks based on a shard key, and each chunk is assigned to a particular shard. When a query is executed, the mongos process routes it to the correct shard(s) based on the shard key range in the query. Queries involving only the shard key are efficient, targeting specific shards. Queries on non-shard keys require scattering and gathering across all shards, but secondary indexes can help efficiency on each shard.
This document provides an overview of Support Vector Data Description (SVDD), which finds the minimum enclosing ball that encapsulates a set of data points. It discusses how SVDD can be formulated as a quadratic programming problem and outlines its dual formulation. The document also notes that SVDD generalizes to non-linear settings using kernels, and discusses variations like adaptive SVDD and density-induced SVDD. Key points covered include the representer theorem, KKT conditions, and how the radius of the enclosing ball can be determined from the Lagrangian.
Lesson 30: Duality In Linear Programmingguest463822
Every linear programming problem has a dual problem, which in many cases has an interesting interpretation. The original ("primal") problem and the dual problem have the same extreme value.
Triangle ABC is given, with altitudes CD and BE from vertices C and B to opposite sides AB and AC respectively being equal. It is proved that triangle ABC must be isosceles by showing that triangles CBD and BCE are congruent by the right angle-hypotenuse (RHS) criterion, implying corresponding angles are equal, and then using corresponding parts of congruent triangles to show sides AB and AC are equal, making triangle ABC isosceles.
Working principle of dda and bresenham line drawing explaination with exampleAashish Adhikari
Bresenham's Line Algorithm
This algorithm is used for scan converting a line. It was developed by Bresenham. It is an efficient method because it involves only integer addition, subtractions, and multiplication operations. These operations can be performed very rapidly so lines can be generated quickly.
In this method, next pixel selected is that one who has the least distance from true line.
This document is an examination paper for Class VII students. It contains questions in three sections - Section A with 20 multiple choice questions worth 20 marks to be completed in 30 minutes, Section B with 10 long answer questions worth 40 marks, and Section C with 5 long answer questions worth 40 marks. The paper tests students on their knowledge of mathematics, general science, and computer science. It provides instructions on time limits, answering questions directly on the paper or in a separate book, and the total marks for each section and the exam overall.
MCQ's for class 7th, mcqs for class 7th, mcq for 7th, mcqs oxford book 7th class, mcqs for class 7th fazaia inter college lahore, mcq's for oxford book, mcq's countdown 7th class
The document provides a summary of mathematics formulae for Form 4 students. It includes:
1) Common functions and their derivatives such as absolute value, inverse, quadratic, and fractional functions.
2) Key concepts in algebra including the quadratic formula, nature of roots, and forming quadratic equations from roots.
3) Essential statistics measures like mean, median, variance, and standard deviation.
4) Formulas for coordinate geometry topics like distance, gradient, parallel and perpendicular lines, and locus equations.
5) Rules for differentiation including algebraic, fractional, and chain rule.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Mid term paper of Maths class VI 2011 Fazaia Inter collegeAsad Shafat
This document contains a mid-term examination for 6th class mathematics from Fazia Schools & Colleges. The exam has 3 sections: Section A with 20 multiple choice questions, Section B with 10 short answer questions worth 4 marks each, and Section C with 5 long answer questions worth 8 marks each. The exam covers topics in mathematics including sets, numbers, operations, ratios, and word problems. Students are asked to show their work, find sums, quotients, greatest common factors, least common multiples, and solve other mathematical problems.
Keynote Address presented by Winemaker and President-for-life Randall Grahm of Bonny Doon Vineyard, delivered to European Wine Bloggers Conference, Nov. 9, 2012, Izmir, Turkey
http://www.beendoonsolong.com/2012/10/digital-wine-communications-conference-speech-izmir-turkey/
The document contains multiple passages with different purposes:
- Some passages contain facts or true information and are intended to inform readers.
- Other passages contain fictional stories meant to entertain readers.
- Several passages attempt to persuade readers to purchase a product or do something through advertising.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
This document provides notes and formulae on additional mathematics for Form 5. It covers topics such as progressions, integration, vectors, trigonometric functions, and probability. For progressions, it defines arithmetic and geometric progressions and gives the formulas for calculating the nth term and sum of terms. For integration, it provides rules and formulas for integrating polynomials, trigonometric functions, and expressions with ax+b. It also defines vectors and their operations including vector addition and subtraction. Other sections cover trigonometric functions, their definitions, relationships and graphs, as well as probability topics such as calculating probabilities of events and distributions like the binomial.
(1) The student solved several integrals and derivatives.
(2) They sketched regions bounded by curves and found the areas.
(3) Properties of functions like extremes and concavity were examined.
This document discusses solving quadratic equations. It begins by defining quadratic equations as equations of the second degree in the form ax2 + bx + c = 0, where a ≠ 0. It then discusses:
- The roots or solutions of a quadratic equation are the values that make the equation equal to 0 when substituted for the variable.
- Quadratic equations can be solved by factorization, splitting the middle term into two factors and setting each factor equal to 0 to find the roots.
- Examples are provided to demonstrate solving quadratic equations by factorization, finding the two roots.
- A more general quadratic equation with parameters a, b is also factorized to find its two equal roots.
This document contains a mid-term examination paper for Class VIII students. It tests their knowledge in the subjects of Mathematics, Computer Science, and English.
The Mathematics section contains 20 multiple choice questions testing concepts like sets, square roots, radicals, number systems, and algebraic expressions. The Computer Science section has 12 multiple choice questions on topics such as hexadecimal conversion, binary addition, word processing functions, and programming basics.
The English section begins with 2 sample multiple choice comprehension questions. Sections B and C of each subject contain longer form questions to be answered in paragraphs, involving explanations, calculations, and problem solving. Students have 3 hours to complete the entire exam which is worth a total of 100 marks.
11.some fixed point theorems in generalised dislocated metric spacesAlexander Decker
This document introduces the notion of a generalized dislocated metric space and establishes some of its topological properties. Some fixed point theorems are obtained for self-maps on these spaces that satisfy contractive conditions. Specifically:
1) A generalized dislocated metric is defined on a set that satisfies non-negativity, identity of indiscernibles, symmetry, and a generalized triangle inequality conditions.
2) A topology called the d-topology is shown to be induced by a generalized dislocated metric, and properties of this topology are established.
3) Some fixed point theorems are proved for self-maps on generalized dislocated metric spaces that satisfy contractive conditions, including analogues of Banach's contraction
Some fixed point theorems in generalised dislocated metric spacesAlexander Decker
This document presents some fixed point theorems in generalized dislocated metric spaces. It introduces the notion of a generalized dislocated metric space and establishes its topological properties. It proves an analogue of Seghal's fixed point theorem for self-maps that satisfy contractive conditions in these spaces. This allows deriving generalized dislocated analogues of fixed point theorems of Banach, Kannan, Bianchini, and Reich. The document defines generalized dislocated metrics and establishes properties like uniqueness of limits. It shows the topology induced by such a metric satisfies Kuratowski's closure axioms.
This document contains a mid-term examination paper for Class VIII mathematics. It consists of 3 sections - Section A with 20 multiple choice questions to be completed in 30 minutes, Section B with 10 long-form questions worth 4 marks each, and Section C with 5 long-form questions worth 8 marks each. The paper tests students on various mathematics concepts including sets, radicals, exponents, averages, percentages, and algebraic expressions. Students are asked to solve problems, simplify expressions, find sums and products, and more. The paper is designed to evaluate students' understanding of core Class VIII math topics.
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.
The document discusses how queries work in sharded MongoDB environments. It explains that MongoDB collections are partitioned into chunks based on a shard key, and each chunk is assigned to a particular shard. When a query is executed, the mongos process routes it to the correct shard(s) based on the shard key range in the query. Queries involving only the shard key are efficient, targeting specific shards. Queries on non-shard keys require scattering and gathering across all shards, but secondary indexes can help efficiency on each shard.
This document provides an overview of Support Vector Data Description (SVDD), which finds the minimum enclosing ball that encapsulates a set of data points. It discusses how SVDD can be formulated as a quadratic programming problem and outlines its dual formulation. The document also notes that SVDD generalizes to non-linear settings using kernels, and discusses variations like adaptive SVDD and density-induced SVDD. Key points covered include the representer theorem, KKT conditions, and how the radius of the enclosing ball can be determined from the Lagrangian.
Lesson 30: Duality In Linear Programmingguest463822
Every linear programming problem has a dual problem, which in many cases has an interesting interpretation. The original ("primal") problem and the dual problem have the same extreme value.
Triangle ABC is given, with altitudes CD and BE from vertices C and B to opposite sides AB and AC respectively being equal. It is proved that triangle ABC must be isosceles by showing that triangles CBD and BCE are congruent by the right angle-hypotenuse (RHS) criterion, implying corresponding angles are equal, and then using corresponding parts of congruent triangles to show sides AB and AC are equal, making triangle ABC isosceles.
Working principle of dda and bresenham line drawing explaination with exampleAashish Adhikari
Bresenham's Line Algorithm
This algorithm is used for scan converting a line. It was developed by Bresenham. It is an efficient method because it involves only integer addition, subtractions, and multiplication operations. These operations can be performed very rapidly so lines can be generated quickly.
In this method, next pixel selected is that one who has the least distance from true line.
This document is an examination paper for Class VII students. It contains questions in three sections - Section A with 20 multiple choice questions worth 20 marks to be completed in 30 minutes, Section B with 10 long answer questions worth 40 marks, and Section C with 5 long answer questions worth 40 marks. The paper tests students on their knowledge of mathematics, general science, and computer science. It provides instructions on time limits, answering questions directly on the paper or in a separate book, and the total marks for each section and the exam overall.
MCQ's for class 7th, mcqs for class 7th, mcq for 7th, mcqs oxford book 7th class, mcqs for class 7th fazaia inter college lahore, mcq's for oxford book, mcq's countdown 7th class
The document provides a summary of mathematics formulae for Form 4 students. It includes:
1) Common functions and their derivatives such as absolute value, inverse, quadratic, and fractional functions.
2) Key concepts in algebra including the quadratic formula, nature of roots, and forming quadratic equations from roots.
3) Essential statistics measures like mean, median, variance, and standard deviation.
4) Formulas for coordinate geometry topics like distance, gradient, parallel and perpendicular lines, and locus equations.
5) Rules for differentiation including algebraic, fractional, and chain rule.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Mid term paper of Maths class VI 2011 Fazaia Inter collegeAsad Shafat
This document contains a mid-term examination for 6th class mathematics from Fazia Schools & Colleges. The exam has 3 sections: Section A with 20 multiple choice questions, Section B with 10 short answer questions worth 4 marks each, and Section C with 5 long answer questions worth 8 marks each. The exam covers topics in mathematics including sets, numbers, operations, ratios, and word problems. Students are asked to show their work, find sums, quotients, greatest common factors, least common multiples, and solve other mathematical problems.
Keynote Address presented by Winemaker and President-for-life Randall Grahm of Bonny Doon Vineyard, delivered to European Wine Bloggers Conference, Nov. 9, 2012, Izmir, Turkey
http://www.beendoonsolong.com/2012/10/digital-wine-communications-conference-speech-izmir-turkey/
The document contains multiple passages with different purposes:
- Some passages contain facts or true information and are intended to inform readers.
- Other passages contain fictional stories meant to entertain readers.
- Several passages attempt to persuade readers to purchase a product or do something through advertising.
This document provides a technical summary of hydraulic design considerations for bridges. It discusses regulatory topics, approaches for one-dimensional and two-dimensional hydraulic modeling of bridges, considerations for unsteady flow analysis, bridge scour concepts and countermeasure analysis, and sediment transport concepts. The goal is to provide guidance to design bridges that are as safe as possible while optimizing costs and limiting environmental impacts.
The document provides an agenda for a training session on using iPads in education, including getting familiar with the device, discovering classroom apps, and discussing questions and concerns about using iPads. Attendees will work in pairs at their own pace to master objectives, report conclusions, and get help from neighbors. The trainer will check in periodically and different slides provide directions, facts, and activities for pairs or groups.
Distracted driving refers to any activity that takes a driver's attention away from actually operating the vehicle. Some common distractions include using a cell phone, eating, talking to passengers, or adjusting the radio. The biggest risks are mental distractions that take the driver's eyes and attention off the road.
Tungku memiliki beberapa bagian utama yaitu bagian bercahaya, konveksi, dan pembakar. Bagian bercahaya menerima panas dari api, sedangkan bagian konveksi memulihkan panas tambahan. Pembakar mengandung api dan mengontrol aliran udara. Tungku digunakan dalam proses metalurgi untuk mengurangi bijih logam menjadi bentuk yang dapat dicairkan.
The document summarizes the vision and principles of the Arthur Academy. It discusses how the academy uses Direct Instruction curriculum and 10 learning principles to ensure all students learn to read, spell, write and do math at or above grade level. This includes principles like guided oral practice, engaging lessons students can succeed in, small reading groups, actively teaching students, and positive motivation. The goal is to prepare students for success in middle and high school by developing focused behavior, a strong work ethic, a sense of earned success, and belief in themselves.
The document is a series of short passages each with a multiple choice question about identifying the meaning of underlined words in context. Each passage provides a context clue about the intended definition, and the correct answer is always revealed and defined at the end in a concise manner.
This document provides materials for a week of lessons on Martin Luther King Jr. and civil rights leaders. It includes vocabulary words and activities, a reading about MLK's childhood, and reflection questions. The core topics are MLK, the civil rights movement, segregation, and letter writing. Students are asked to make inferences, identify an author's purpose, and summarize the main ideas.
The third document instructs students to read a news article, sequence at least 5 events, and draw pictures representing each event. It then provides a follow-up activity where students can arrange comic strips or news stories
The document provides guidance on copyreading and headline writing. It discusses preparing copy by typing on one side of the page and identifying stories. It offers tips for polishing copy such as asking what the story is about and whether the news is high enough. The duties of a copy editor are outlined, including following style rules, checking facts, spelling, grammar, organization, and watching for editorializing or libelous text. Copyreading symbols are used to suggest corrections without erasing text.
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.
The quadratic formula provides a method to solve quadratic equations of the form ax^2 + bx + c = 0. It expresses the solutions for x in terms of the coefficients a, b, and c as x = (-b ± √(b^2 - 4ac))/2a. The document demonstrates applying the quadratic formula to solve the equation 7x^2 + 14x - 3 = 0, obtaining the two solutions x1 = 0.2 and x2 = -2.2.
The quadratic formula provides a method to solve quadratic equations of the form ax^2 + bx + c = 0. It expresses the solutions for x in terms of the coefficients a, b, and c as x = (-b ± √(b^2 - 4ac))/2a. The document demonstrates applying the quadratic formula to solve the equation 7x^2 + 14x - 3 = 0, obtaining the two solutions x1 = 0.2 and x2 = -2.2.
Bowen prelim a maths p1 2011 with answer keyMeng XinYi
This document consists of 13 multiple choice mathematics questions testing concepts such as:
1) Solving quadratic, logarithmic, and trigonometric equations.
2) Finding gradients, derivatives, integrals, and curve equations.
3) Analyzing graphs of functions and solving simultaneous equations.
The questions cover a wide range of mathematics topics and require showing steps to find exact solutions or simplify expressions. Answers are provided in the form of a detailed answer key.
1) The document discusses representing complex numbers geometrically using the Argand diagram. Complex numbers a + ib can be represented as a point (a,b) on the Argand plane, with the real part a on the x-axis and imaginary part b on the y-axis.
2) Examples are given of representing different complex numbers as points on the Argand plane, such as 2 + 3i as point (2,3). It is shown that a + bi is not the same as -a - bi, a - bi, or -z.
3) The modulus (absolute value) of a complex number a + ib is defined as the distance from the point (a,b) representing
1. Basic algebra involves variables, algebraic expressions, and equations. Variables represent unknown values.
2. Algebraic expressions contain variables, numbers, and operators. They can be simplified by combining like terms or using properties of exponents.
3. Equations set two algebraic expressions equal to each other and can be solved algebraically to find the value of variables. There are methods for solving different types of equations like linear, fractional, and simultaneous equations.
This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.
The document contains an answer key for a mathematics assignment on quadratic functions. It includes:
1) Graphing quadratic equations and identifying vertex, zeros, and y-intercept.
2) Solving quadratic equations by factoring.
3) Identifying true statements about quadratic functions.
4) Solving quadratic equations using the quadratic formula.
5) Setting up and solving an optimization word problem involving quadratic sales based on number of items sold.
6) Answering true/false questions based on a graph of a quadratic function.
7) Writing the quadratic equation for an age relationship problem.
8) Setting up the Pythagorean theorem to solve for side lengths of
1. The document contains the answer key to a math test on quadratic functions.
2. It includes graphing quadratic equations, solving by factoring, identifying properties, using the quadratic formula, modeling real world situations, and applying the Pythagorean theorem and properties of parabolas.
3. Several questions involve finding the vertex, x-intercepts, maximum/minimum values, and distance or drop measures for quadratic equations describing real world motions or sales situations.
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.
A quadratic equation is an equation equivalent to the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. To solve a quadratic equation, we first get it into standard form and then either factor if possible or use the quadratic formula. If factoring results in a negative number under a square root, there are no real solutions. Completing the square is another method that allows us to factor a quadratic expression and solve for the roots.
- The document discusses calculating integrals using substitution and breaking them into simpler integrals.
- It provides an example of using constants A and B to rewrite an integral in terms of simpler integrals I1 and I2.
- The integrals I1 and I2 are then evaluated using substitution and integral formulas to arrive at the final solution for the original integral.
This document contains 3 problems:
1) Showing that a sequence is increasing and less than 3.
2) Calculating the average value of a function over an interval.
3) Finding the solution to a partial differential equation of the form 2xyy' – y^2 + x^2 = 0. The solution is found to be y = √cx - x^2.
1. The cross product of two vectors gives a vector perpendicular to both vectors, with magnitude equal to the area of the parallelogram formed by the two vectors.
2. If two adjacent sides of a parallelogram are given by vectors a and b, the area of the parallelogram is |a x b|.
3. If the position vectors of three vertices of a triangle are given, the area of the triangle can be found as 1/2 times the magnitude of the cross product of any two sides of the triangle.
This document discusses quadratic equations and functions. It explains how to solve quadratic equations by factoring, completing the square, and using the quadratic formula. It also discusses using the discriminant to determine the number and type of roots. Properties of quadratic functions such as the sum and product of roots are covered. Methods for constructing quadratic equations and functions given certain properties are provided. Finally, it briefly discusses sketching the graph of a quadratic function.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be repeated or complex.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the equation in terms of the roots.
3) Symmetrical functions of the roots remain the same if the roots are swapped, and can be written in terms of the sum and product of the roots.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression.
3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.
This document discusses number theory and modulo operations. It covers topics like modular arithmetic, congruence, the modulo operator, and modular multiplicative inverses. Modular arithmetic involves doing arithmetic operations and reducing results modulo a number n. Finding a modular multiplicative inverse involves finding an integer x such that ax is congruent to 1 modulo n.
This document contains examples of using the Pythagorean theorem to solve for the lengths of sides of right triangles. It provides two examples:
1) For a right triangle with sides of lengths 9, 12, and c, it uses the Pythagorean theorem (c^2 = 9^2 + 12^2) to calculate that c = 15.
2) For a right triangle with sides of lengths 11, 60, and c, it uses the Pythagorean theorem (c^2 = 11^2 + 60^2) to calculate that c = 61.
1. Chapter 12 Vectors and the Geometry of Space
12.1 Three-dimensional Coordinate systems
A. Three dimensional Rectangular Coordinate Sydstem:
The Cartesian product
R3 = R × R × R = {(x, y, z) : x, y, z ∈ R},
where (x, y, z) iscalled ordered triple.
B. Distance:
The distance |P1 P2 | between two points P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ) is
|P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
C. Sphere:
An equation of a sphere with center C(h, k, ) and radius r is
(x − h)2 + (y − k)2 + (z − )2 = r2 a.
12.2 Vectors
A. Vector: a quantity that has both of magnitude and direction.
−→
B. (Re)presentation and Notation: − , < a1 , a2 , a3 >, AB.
→
a
Definition 1. A two dimensional vector is an ordered pair − =< a1 , a2 > with a1 and
→
a
a2 real numbers. A three dimensional vector is an ordered triple − =< a1 , a2 , a3 >
→
a
with a1 , a2 and a3 real numbers.
C. Magnitude: Let − =< a1 , a2 , a3 >. Then, the magnitude of − is
→
a →
a
|− | =
→
a a2 + a2 + a2 .
1 2 3
D. Multiplication of a vector by a scalar:
Let − =< a1 , a2 , a3 > and c be a scalar (real number). Then,
→
a
c− =< ca1 , ca2 , ca3 > and |c− | = |c||− |.
→a →
a →
a
→
−
E. Vector addition/difference : Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. Then,
→a
→ →
− + − =< a + b , a + b , a + b >,
a b 1 1 2 2 3 3
2. → →
− − − =< a − b , a − b , a − b >
a b 1 1 2 2 3 3
F. Standard Basis vectors :
In R2 ,
→
− →
−
i =< 1, 0 >, j =< 0, 1 > .
In R3 ,
→
− →
− →
−
i =< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 > .
→
− →
− →
−
If − =< a1 , a2 , a3 >, then − = a1 i + a2 j + a3 k .
→
a →
a
G. Unit Vector :
A vector whose length is 1. If − = 0, then the unit vector that has the same direction as
→
a
→
− is
a
→
−
− = a .
→u
|− |
→
a
12.3 The Dot Product
→
− →
−
Definition 2. If − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, the dot product of − and b
→a →
a
→ −→
is the number − · b given by
a
→ →
− ·− =a b +a b +a b .
a b 1 1 2 2 3 3
Note 1: A result of the dot product of two vectors is a scalar (not a vector)
Note 2: The dot product is sometimes called inner product or scalar product.
A. Properties of the dot product
(1) → →
− · − = |− |2
a a →a
→ −
→ − − → →
− · b = b · a
(2) a
(3) → → →
− · (− + − ) = − · − + − · −
a b c → → → →
a b a c
(4) → →
− ) · − = c(− · − ) = − · (c− )
(c a b → →
a b →
a
→
b
→ →
− −
(5) 0 · a =0
→
−
Theorem 1. Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. If θ is the angle between the
→a
→
−
vectors − and b , then
→
a
→ →
− · − = |− ||− | cos θ.
a b → →
a b
2
3. →
−
Corollary 1. Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. If θ is the angle between the
→
a
→
−
vectors − and b , then
→
a
→ →
− ·−
a b
cos θ = .
→ →
− ||− |
|a b
B. Direction Angles and Direction Cosine:
Let the angles betwees − and x-axis, y-axis, and z-axis are α, β, and γ respectively. Then,
→
a
→
− =< a , a , a >,
if a 1 2 3
a1 a2 a3
cos α = − , cos β = −
→| →| cos γ = |− | .
→
|a |a a
Then,
a
− | =< cos α, cos β, cos γ > .
→
|a
C. Projection :
→
−
C.1. Scalar Projection : component of b along − →
a
→
− →
− → →
− ·−
a b
comp− b = | b | cos θ = − .
→
a
|→|
a
C.2. Vector Projection:
→
− →→
− −a → →
− ·−
a b →
−
a
proj− b = comp− b − =
→ → →| − |.
a a
|a |− |
→
a →
|a
12.4 The Cross Product
→
−
Definition 3. If − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, the cross product of − and
→a →
a
→
− → →
− × − given by
b is the vector a b
→ →
− × − =< a b − a b , a b − a b , a b − a b > .
a b 2 3 3 2 3 1 1 3 1 2 2 1
→ −
a
→
Theorem 2. The cross product − × b is orthogonal to both of − =< a1 , a2 , a3 > and
→
a
→
−
b =< b1 , b2 , b3 >.
Note 1: A result of the cross product of two vectors is a vector (Not a scalar). So, the
cross product is sometimes called vector product.
3
4. →
−
Theorem 3. If θ, 0 ≤ θ ≤ π, is the angle between − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >,
→
a
then
→ −
a
→ → →
a
−
|− × b | = |− || b | sin θ.
→ −→
Note 2: the length of the cross product − × b is equal to the area of the parallelogram
a
→
−
determined by − and b .
→
a
A. Scalar Triple Product :
→ −
a
→
The volume of the parallelopiped determined by the vectors − , b , and − is the magnitude
→
c
of the scalar triple product
→ − →→
|− · ( b × − )|.
a c
12.5 Equations of Lines and Planes
A. Equation of a line L :
A line in a space is determined by a point P0 (x0 , y0 , z0 ) a vector − that is parallel to the
→
n
line. Let − =< a, b, c >, − =< x, y, z >, − 0 =< x0 , y0 , z0 >, and t is a scalar.
→
v →
r →
r
A.1. Vector Equation : − = − 0 + t− .
→ →
r r →
v
A.2. Parametric Equation : x = x0 + at y = y0 + bt z = z0 + ct.
A.3. Symmetric Equation :
x − x0 y − y0 z − z0
= = .
a b c
B. Equation of a Plane :
A plane in a space is determined by a point P0 (x0 , y0 , z0 ) a vector − (normal vector) that
→
n
is orthogonal to the plane. Let − =< x, y, z >, − 0 =< x0 , y0 , z0 >, and − =< a, b, c >.
→r →r →n
B.1. Vector Equation of a Plane:
→ → →
− · (− − − ) = 0.
n r r0
B.2.. Scalar Equation of a plane :
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
(or ax + by + cz + d = 0 with d = −ax0 − by0 − cz0 ).
4
5. C. Distance D from a point P1 (x1 , y1 , z1 ) to the plance ax + by + cz + d = 0:
|ax1 + by1 + cz1 + d|
D= √ .
a2 + b 2 + c 2
12.6/7 Cylinder and Cylindrical Coordinates
Identify and sketch the surfaces
(1) x2 + y 2 = 1 (2) y 2 + z 2 = 1.
To convert from cylindrical to rectangular Coordinate,
x = r cos θ y = r sin θ z = z.
To convert from rectangular to cylindrical Coordinate,
y
r2 = x2 + y 2 tan θ = z = z.
x
Chapter 13 Vector Functions
13.1 Vector Functions and Space Curves
A. Vector Function: A vector (valued) function (e.g, in R3 ) is of the form
→
− (t) =< f (t), g(t), h(t) >,
r
where all the component functions f , g, and h are real valued function.
B. Limit of a vector function − : If − (t) =< f (t), g(t), h(t) >, then
→
r →
r
→
− (t) =< lim f (t), lim g(t), lim h(t) >
lim r
t→a t→a t→a t→a
provided the limits of f (t) ,g(t), and h(t) exist.
In particular, a vector function − is continuous at t = a
→
r
lim − (t) = − (a).
→
r →
r
t→a
C. Space Curve: Suppose that f , g, and h are continuous functions on an interval I. Let
C = {(x, y, z) : x = f (t) y = g(t) z = h(t)},
where t varies through the interval I, is called a space curve.
The equations x = f (t), y = g(t), z = h(t) are called parametric equations of C and t
is called a parameter.
5
6. 13.2 Derivatives and Integrals of Vector Functions
A. Derivative : The derivative of a vector (valued) function − is defined by
→
r
d−
→r →
− (t + h) − − (t)
r →
r
= − (t) = lim
→
r
dt h→0 h
if the limit exists.
The vector − (t) is called the tangent vector of − , and it unit tangent vector is given
→
r →r
by
→
− (t)
r
T(t) = − → (t)| .
|r
Theorem 4. If r→
− (t) =< f (t), g(t), h(t) >, where f , g, and h are differentiable functions,
then
→
− (t) =< f (t), g (t), h (t) > .
r
Theorem 5. Suppose − and − are differentiable vector functions, c is a scalar, and f is
→
u →
v
a real valued function. Then,
d −
(1) [→(t) + − (t)] = − (t) + − (t)
u →v →u →
v
dt
d −
(2) [c→(t)] = c− (t)
u →u
dt
d
(3) [f (t)− (t)] = f (t)− (t) + f (t)− (t)
→
u →u →
u
dt
d −
(4) [→(t) · − (t)] = − (t) · − (t) + − (t) · − (t)
u →v →
u →v →u →
v
dt
d −
(5) [→(t) × − (t)] = − (t) × − (t) + − (t) × − (t)
u →v →u →
v →
u →
v
dt
d −
(6) [→(f (t))] = f (t)− (f (t)) (Chain Rule)
u →u
dt
• See Example 1-5
B. Integrals : If − (t) =< f (t), g(t), h(t) >, where f , g, and h are integrable in [a, b], then
→r
the definite integral if the vector function − (t) can be defined by
→
r
b b b b
→
− (t)dt = →
− →
− →
−
r f (t)dt i + g(t)dt j + h(t)dt k.
a a a a
• See Example 6
6
7. 13.3 Arc Length and Curvature
A. Arc Length : Let a ≤ t ≤ b, and let − (t) =< f (t), g(t), h(t) > where f , g , and h
→
r
are continuous on I. Then, the length of the space curve (arc length) from t = a to t = b
is defined by
b
L = →
− (t) 2 dt
r
a
b
= [f (t)]2 + [g (t)]2 + [h (t)]2 dt
a
b 2 2 2
dx dy dz
= + + dt
a dt dt dt
• See Example 1.
13.4 Motion in Space :Velocity and Acceleration
A. Velocity vector : Suppose a particle moves through space so that its position vector
at time t is − (t). The the velocity vector at time t is defined by
→
r
→
− →
−
− (t) = lim r (t + h) − r (t) = − (t).
→v →r
h→0 h
The velocity vector − (t) is also the tangent vector and points in the direction of the tangent
→v
line. Further, the speed of the particle at time t is
→
− (t) = − (t) .
v →r
B. Acceleration vector : The acceleration of the particle at time t is
→
− (t) = − (t) = − (t).
a →
v →r
• See Examples 1-3.
C. Newton’s Second Law of Motion : If, at any time t, a force F(t) acts on an object
of mass m producing an acceleration − (t), then
→
a
F(t) = m− (t).
→
a
• See Examples 4 and 5.
7
8. Chapter 14 Partical derivatives
14.1 Functions of Several Variables
A. Functions of two varialbes:
Definition 4. A function f of two variables is a rule of the form
f : (x, y) ∈ D → z = f (x, y).
Here the set D is the domain of f and its range is the set {f (x, y) : (x, y) ∈ D}.
• See Examples 1, 4
B. Graph : Let f is a function of two variables with domain D. Then, the graph of f is
{(x, y, z) ∈ R3 : z = f (x, y), (x, y) ∈ D}.
• See Examples 6, 8
14.2 Limits and Continuity
• Let’s think about the two limits
x2 − y 2 x2 − y 2
lim lim and lim lim .
x→0 y→0 x2 + y 2 y→0 x→0 x2 + y 2
A. Limit :
Definition 5. Let f be a function of two variables with domain D and (a, b) ∈ closure(D).
We say that the limit of f (x, y) as (x, y) → (a, b) is L, i.e.,
lim f : (x, y) = L
(x,y)→(a,b)
if for any number > 0 there is number δ > 0 such that f (x, y) − L < whenever
(x, y) − (a, b) < δ and (x, y) ∈ D.
Remark: Let f (x, y) → L1 as (x, y) → (a, b) along a path C1 and f (x, y) → L2 as
(x, y) → (a, b) along a path C2 . Thn, if L1 = L2 , the limit lim(x,y)→(a,b) f : (x, y) does not
exist.
• See Examples 1-4.
8
9. B. Continuity :
Definition 6. Let f be a function of two variables with domain D and (a, b) ∈ closure(D).
The function f (x, y) is called continuous at (a, b) if
lim f : (x, y) = f (a, b).
(x,y)→(a,b)
We say f (x, y) is continuous on D if f is continuous at every points (a, b) in D.
14.3 Partial Derivatives
A. Definition : Let f be a function of two variables. Its partial derivatives are functions
fx and fy defined by
f (x + h, y) − f (x, y)
fx (x, y) = lim
h→0 h
f (x, y + h) − f (x, y)
fy (x, y) = lim
h→0 h
B. Notations: If z = f (x, y),
∂f ∂ ∂z
fx (x, y) = fx = = f (x, y) = = Dx f
∂x ∂x ∂x
∂f ∂ ∂z
fy (x, y) = fy = = f (x, y) = = Dz f
∂y ∂y ∂x
• See Examples 1-5.
Theorem 6. Let f be a function of two variables and defined on D. Let (a, b) ∈ D. Then,
if fxy and fyx are both continuous on D,
fxy (a, b) = fyx (a, b).
14.4 Tangent Planes and Linear Approximation
A. Definition : Let f be a function of two variables, and assume that fx and fy are
continuous. An equation of tangent plane to the surface z = f (x, y) at P (x0 , y0 , z0 ) is
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).
Define L(x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ). Then L(x, y) is called a
linearization of f at (x0 , y0 ). Also, the approximation
f (x, y) ≈ L(x, y)
is called the linear approximation and the thangent plane approximation.
9
10. Definition 7. Let z = f (x, y) and ∆z = f (x0 + ∆x, y0 + ∆y). Then the function f is
differentiable at (x0 , y0 ) if ∆z can be written
∆z = fx (x0 , y0 )∆x + fy (x0 , y0 )∆y + 1 δx + 2 δxy,
where 1 , 2 → 0 as δx, δy → 0.
Theorem 7. Asume that fx and fy exist near (x0 , y0 ) and are continuous at (x0 , y0 ). Then
f is differentiable at (x0 , y0 ).
Total differential: The differential or total differential is is defined by
∂z ∂z
dz = fx (x, y)dx + fy (x, y) = dx + dy.
∂x ∂y
• See Examples 1,2, 4
14.5 The Chain Rule
The Chain Rule (Case 1): Assume that z = f (x, y) is a differentiable function of x and
y, where x = g(t) and y = h(t) are both differentiable of t. Then
dz ∂f dx ∂f dy
= + .
dt ∂x dt ∂y dt
The Chain Rule (Case 2): Assume that z = f (x, y) is a differentiable function of x and
y, where x = g(s, t) and y = h(s, t) are both differentiable functions of s and t. Then
∂z ∂z dx ∂f dy
= +
∂s ∂x ds dy ds
∂z ∂z dx ∂f dy
= + .
∂t ∂x dt dy dt
• See Examples 1,3, 5( For General Version)
Implicit Differentiation : Assume that z = f (x, y) is given implicitly as a function of
∂F
the form F (x, y, z) = 0. If F and f are differentiable and = 0, then
∂z
∂F ∂F
∂z ∂z ∂y
= − ∂x =−
∂x ∂F ∂y ∂F
∂z ∂z
• See Examples 9
10
11. 14.6 Directional Derivatives and the Gradient Vector
A. Definition : Let f be a function of two variables. The directional derivatives of
f (x0 , y0 ) in the direction of unit vector − =< a, b > is
→
u
f (x0 + ha, y + hb) − f (x0 , y0 )
D− f (x0 , y0 ) = lim
→
u
h→0 h
if the limit exists.
In the case of three variable function, we can define the directional derivatives in a similar
manner.
Theorem 8. Let f be a differentialbe function of x and y. Then f has directional deriva-
tives in the direction of unit vector − =< a, b > and
→
u
D− f (x, y) = fx (x, y)a + fy (x, y)b.
→
u
The Gradient Vector : Let f be a function of several (say three) variables. The Gradient
of f is the vector function f fdefined by
→ ∂f − ∂f −
∂f − → →
f (x, y, z) =< fx (x, y, z), fy (x, y, z), fz (x, y, z) >= i + j k
∂x ∂x ∂z
Note that for any − =< a, b, c >, D− f (x, y, z) =
→
u →
u f (x, y, z) · − .
→
u
• See Examples 2, 3, 4
14.7 Maximum and Minimum Values
Definition 8. Let f be a function of two variables. Then, f (a, b) is called local maximum
value if f (a, b) ≥ f (x, y) when (x, y) is near (a, b). Also, f (a, b) is called local minimum
value if f (a, b) ≤ f (x, y) when (x, y) is near (a, b).
Theorem 9. If f has local maximum or minimum value at (a, b) and fx and fy exist, then
fx (a, b) = 0 and fy (a, b) = 0.
A point (a, b) is called a critical point of f if fx )a, b) = 0 and fy (a, b) = 0.
Second Derivatives Test: Assume that the second partial derivatives of f are continuous
on a disk with center (a, b), and assume that fx (a, b) = 0 and fy (a, b) = 0. Let
D = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 .
(a) If D > 0 and fxx (a, b) > 0, then f (a, b) is a local minimum.
(b) If D > 0 and fxx (a, b) < 0, then f (a, b) is a local maximum.
11
12. (c) If D < 0, then f (a, b) is not a local maximum or minimum. (the point (a,b) is called a
saddle point of f ).
• See Examples 1, 2, 3, 6.
Theorem 10. If f is continuous on a closed, bounded set D in R2 , then f attains an
absolute maximum value f (x1 , y1 ) and an absolute minimum value f (x2 , y2 ) at some points
(x1 , y1 ) and (x2 , y2 ) in D.
To find an absolute maximum and minimum values of f on a closed, bounded set D :
(1) Find the values of f at the critical points of f in D.
(2) Find the extreme values of f on the boundary of D.
(3) The largest of the values from steps 1 and 2 is the absolute maximum value; The
smallest of the values is the absolute minimum value.
• See Example 7.
14.8 Lagrange Multipliers
Method of Lagrange Multifiers : In order to find the maximum and minimum values
of f (x, y, z) subject to the constraint g(a, y, z) = k
(a) Find all values of x, y, z, and λ such that
f (x, y, z) = λ g(x, y, z)
g(a, y, z) = k
(b) Evaluate f at all the points (x, y, z) that results from step (a). The largest of the values
is the absolute maximum value; The smallest is the absolute minimum value of f .
• See Examples 2, 3
12
13. Chapter 15 Multiple Integrals
15.1 Double Integrals over Rectangles
Definition 9. The double integral of f over R = [a, b] × [c, d] is
∞ ∞
f (x, y)dA = lim f (xi , yj )∆A (1)
R m,n→∞
i=1 j=1
if the limit exists, where (xi , yj ) is in
Rij = [xi−1 , xi ] × [yj−1 , yj ].
Here the right-hand side of (1) is called s double Riemann sum.
If f (x, y) ≥ 0, the volume V of the solid that lies above the rectangle R and below the
surface z = f (x, y) is
V = f (x, y)dA.
R
15.2 Iterated Integrals
Theorem 11. (Fubini) Let f be a continuous function on R = [a, b] × [c, d].
b d d b
f (x, y)dA = f (x, y)dydx = f (x, y)dxdy.
R a c c a
The two integrals in the right-hand side of the above identity are called iterated integrals.
More generally, this theorem is true if f is bounded on R, f is discontinuous only on a
finite number if snmooth curves, and the iterated integrals exist.
Special Cases : If f (x, y) = g(x)h(y) on R = [a, b] × [c, d],
b d d b
f (x, y)dA = f (x, y)dydx = h(y)dy · g(x)dx .
R a c c a
• See Examples 1-5.
15.3 Double Integrals over General Regions
Type I : Let f be a continuous on a type I region D such that
D = {(x, y)|a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}
then
b g2 (x)
f (x, y)dA = f (x, y)dydx.
R a g1 (x)
13
14. Type II : Let f be a continuous on a type II region D such that
D = {(x, y)|c ≤ y ≤ d, h1 (x) ≤ x ≤ h2 (x)}
then
d h2 (x)
f (x, y)dA = f (x, y)dxdy.
R c h1 (x)
• See Examples 1-5.
Properties of Double Integrals : Assume that all of the following integrals exist. Then,
(1) [f (x, y) + g(x, y)]dA = f (x, y)dA + [g(x, y)]dA
D D D
(2) cf (x, y)dA = c [f (x, y)]dA
D D
(3) f (x, y)dA ≥ [g(x, y)]dA, if f (x, y) ≥ g(x, y)
D D
(4) 1dA = A(D)
D
(4) f (x, y)dA ≥ [g(x, y)]dA, + [g(x, y)]dA, if D = D1 ∪ D2 .
D D1 D2
Here D1 and D2 don’t overlap except (perhaps) on the boundary. Also, if m ≤ f (x, y) ≤ M
for all (x, y) ∈ D, then
mA(D) f (x, y)dA ≤ M A(D).
D
15.4 Double Integrals over Polar Coordinates
Change to Polar Coordinates in a Double Integral : Let f be a continuous on a po-
lar rectangle
R = {(r, θ) | 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β}
then
β b
f (x, y)dA = f (r cos θ, r sin θ)rdrdθ).
R α a
• See Examples 1,2.
15.7 Triple Integrals
14
15. Theorem 12. (Fubini’s Theorem for Triple Integrals) Let f be a continuous function on
B = [a, b] × [c, d] × [r, s].
s d b d b
f (x, y, z)dV = f (x, y, z)dxdydz = f (x, y)dxdy.
B r c a c a
Triple Integrals over General Regions:
Type I : Let f be a continuous on region D (type I or type II in double integral) such
that
E = {(x, y, z) | (x, y) ∈ D u1 (x, y) ≤ z ≤ u2 (x, y)},
then
u2 (x,y)
f (x, y, z)dV = f (x, y, z)dz dA.
E D u1 (x,y)
Type II : Let f be a continuous on region D (type I or type II in double integral) such
that
E = {(x, y, z) | (y, z) ∈ D u1 (y, z) ≤ z ≤ u2 (y, z)},
then
u2 (y,z)
f (x, y, z)dV = f (x, y, z)dx dA.
E D u1 (y,z)
• See Examples 1-3.
15.8 Triple Integrals in Cylindrical and Spherical Coordinates
Formula for triple integration in cylindrical coordinates:
u2 (r cos θ,r sin θ)
f (x, y, z)dV = f (r cos θ, r sin θ, z)rdzdrdθ.
E D u1 (r cos θ,r sin θ)
• See Examples 1,2.
15
16. Chapter 16 Vector Calculus
16.1 Vector Fields
Definition 10. Let E be a subset of R3 . A vector field on R3 is a function F that assigns
to each (x, y, z) ∈ E a three-dimensional vector F(x, y, z). We can write F as follows:
→
− →
− →
−
F(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z) k .
Gradient Fields: If f is a scalar function of three (or two) variables, its gradient is a
vector field on R3 given by
→
− →
− →
−
f (x, y, z) = fx (x, y, z) i + fy (x, y, z) j + fz (x, y, z) k .
• See Examples 1, 2, 6.
16.2 Line Integrals
Definition 11. Let C be a smooth curve given by the parametric equation
x = x(t) y = y(t), a ≤ t ≤ b.
If f is defined on the curve C, then the line integral of f along C is defined by
b 2 2
dx dy
f (x, y)ds = f (x(t), y(t)) + dt.
C a dt dt
Remark: If C is a piecewise-smooth curve, that is, C is a finite union of smooth
curves C1 , · · · Cn , then
f (x, y)ds = f (x, y)ds + · · · + f (x, y)ds.
C C1 Cn
Line integral of f along C with respect to x and y:
b
f (x, y)dx = f (x(t), y(t))x (t)dt
C a
b
f (x, y)dy = f (x(t), y(t))y (t)dt
C a
• See Examples 1, 2, 4.
Line integrals in Space: Suppose that C is a smooth curve given by the parametric
equation
x = x(t) y = y(t) z = z(t), a ≤ t ≤ b.
16
17. If f is defined on the curve C, then the line integral of f along C is defined by
b 2 2 2
dx dy dz
f (x, y)ds = f (x(t), y(t), z(t)) + + dt.
C a dt dt dt
Compact Notation :
b
f (x, y, z)ds = f (− (t))|− (t)|dt.
→
r →
r
C a
• See Examples 5, 6.
Line integrals of Vector Fields: Let F be a continuous vector field defined on a smooth
curve C given by a vector function − (t), a ≤ t ≤ b. Then, the line integral of F along
→
r
C is
b
F · d− =
→
r F(− (t)) · − (t)dt =
→r →
r F · T ds,
C a C
where T (x, y, z) is the unit tangent vector at the point (x, y, z).
• See Examples 7, 8.
16.3 The Fundamental Theorem for Line Integrals
Theorem 13. Let C be a smooth curve given by the vector function − (t), a ≤ t ≤ b. Let
→
r
f be a continuous function and its f is continuous on C. Then,
f · d− = f (− (b)) − f (− (a)).
→
r →
r →
r
C
Note: We can evaluate f · d− by knowing the value of f at the end of points of C.
→
r
C
Definition 12. A vector field F is called a conservative vector field if there is a scalar
function f such that F = f . Here f is called a potential function of F.
Note: Line integrals of conservative vector fields are independent of path.
Theorem 14. f · d− is independent of path in a domain D iff
→
r f · d− = 0 for
→
r
C C
every closed path in D.
17
18. →
− →
−
Theorem 15. Let F = P i + Q j be a conservative vector field, where all the partial
derivatives are continuous. Then,
∂P ∂Q
= , in D.
∂y ∂x
→
− →
−
Theorem 16. Let F = P i + Q j be a vector fields on an open simply-connected region
D. Supppose that all the partial derivatives are continuous and
∂P ∂Q
= in D.
∂y ∂x
Then F is conservative.
16.4 Green’s Theorem
Theorem 17. Let C be a positively oriented, piecewise-smooth, simple closed curve in the
plane and let D be the region bounded by C. Supppose that all the partial derivatives of P
and Q are continuous on an open region contains D, then
∂Q ∂P
P dx + Qdy = − dA
C D ∂x ∂y
Application: Line The formulas to find the area of D :
1
A= xdy = − ydxa = xdy − ydx.
C C 2 C
• See Examples 1,2,3
16.5 Curl and Divergence
→
− →
− →
−
Curl: If F = P i + Q j + R k is a vector field on R3 and partial derivatives of P , Q, and
R all exist, then the curl of F is the vector field defined by
curl F = ×F
∂R ∂Q →
− ∂R ∂P →
− ∂Q ∂P →
−
= − i − − j + − k
∂y ∂z ∂x ∂z ∂x ∂y
Theorem 18. If f is a function of three variables that has continuous second-order partial
derivatives, then
curl ( f ) = 0.
Theorem 19. If F is a vector field defined on all of R3 whose component functions have
continuous partial derivatives and curl F = 0, then F is a conservative vector field.
18
19. • See Examples 1,2,3
→
− →
− →
−
Divergence: If F = P i + Q j + R k is a vector field on R3 and partial derivatives of P ,
Q, and R all exist, then the divergence of F is the function of three variables defined by
divF = ·F
∂P ∂Q ∂R
= + +
∂x ∂y ∂z
→
− →
− →
−
Theorem 20. If F = P i + Q j + R k is a vector field defined on all of R3 and P , Q,
and R have continuous second-order partial derivatives, then
div curl F = 0.
• See Examples 4,5
19