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Real Analysis II

Purpose of Course  showclose

Real Analysis II is the sequel to Saylor’s Real Analysis I, and together these two courses constitute the foundations of real analysis in mathematics. In this course, you will build on key concepts presented in Real Analysis I, particularly the study of the real number system and real-valued functions defined on all or part (usually intervals) of the real number line. The main objective of MA241 was to introduce you to the concept and theory of differential and integral calculus as well as the mathematical analysis techniques that allow us to understand and solve various problems at the heart of science—namely, questions in the fields of physics, economics, chemistry, biology, and engineering. In this course, you will build on these techniques with the goal of applying them to the solution of more complex mathematical problems. As long as a problem can be modeled as a functional relation between two quantities, each of which can be expressed as a set of real numbers, the techniques used for real-valued functions of one variable should suffice. However, most practical problems cannot be modeled via functions of a single real variable. For instance, modeling a moving particle in space requires three real variables in the three-dimensional coordinate system of real numbers. In another example from physics, the altitude a projectile will reach—a quantity described by one real variable—depends on two factors: the weight of the projectile as well as the initial velocity it has acquired from some external force. Sometimes, depending on the answer desired, a problem may be modeled as a single-variable or a multivariable function. For example, a particle moving in three-dimensional space through a force field (think of a dust particle floating in the air as it is blown by gusts of wind) may be modeled either as a function of time (a single-variable function) to describe the coordinates of the particle at each instance of time; or, if one is interested in the final resting place of the particle as a function of its initial position, the same problem may be modeled as a multivariable function that requires three inputs (the coordinates of the initial position) in order to produce three outputs (the coordinates of the resting place). In this course, you will learn about some of the intricacies of the geometry of higher-dimensional spaces, how the theory of multivariable functions is developed, and how to apply the advanced techniques of differentiation and integration to such functions. Finally, you will explore applications of these advanced techniques to the solution of complex mathematical problems.

Course Information  showclose

Welcome to MA242. General information on this course and its requirements can be found below.
 
Course Designer: Ittay Weiss, Ph.D.
 
Primary Resources: This course draws on a range of different free, online educational materials, with primary use of the following online textbooks:
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned readings and materials. Be sure to follow carefully the instructions for each unit and assignment, as these guidelines are designed to lead you in an efficient study of the material. As instructed, you also will need to complete all the assigned problem sets within specific units and subunits of this course. Finally, you must successfully complete and pass the course’s Final Exam.
 
Please note that you will receive an official grade only on your Final Exam. In order to pass this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
 
Time Commitment: This course should take you a total of 137 hours to complete. Each unit includes time advisories that list the amount of time you are expected to spend on each subunit and assignment. These time advisories should help you plan your coursework accordingly. It may be useful for you to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take you approximately 66 hours to complete. Perhaps you can sit down with your calendar and decide to complete Subunit1.1 (a total of 4 hours) on Monday and Tuesday nights; Subunit 1.2 (a total of 6 hours) on Tuesday and Wednesday nights; etc.
 
Tips/Suggestions: The course is a continuation of Saylor’s MA241, Real Analysis I, and thus assumes a level of mathematical maturity that would be expected upon mastery of MA241. Please keep in mind that the indicated time advisories for each unit, subunit, and assignment of this course are not absolute, and heavily depend on your own personal mastery of the material in MA241. For example, you may find yourself spending more time on certain topics than is suggested in the time advisory for each unit; or you may need to spend more time on reviewing the background material presented in MA241. If this happens, just keep in mind that it is not necessarily a problem, but rather another opportunity to improve, enhance, and hone your previous knowledge. Please note that you do not need a calculator for this course, but be sure to have plenty of scrap paper and a writing utensil available to you at all times, since you will be performing calculations, stating proofs, and working through problem sets throughout each unit of this course and during the Final Exam.
 
Please note that understanding the exercises presented in the below course materials is an absolutely essential part of internalizing the new subject matter. Although it is not always necessary that you complete each and every exercise for a given assignment, it is imperative that you take the exercises supplementing each reading seriously. To aid you in the learning process, the instructions for each unit and assignment may indicate which exercises are more or less crucial than others. In particular, the time advisory for each assignment, unless stated otherwise, is not meant to indicate how long it should take you to solve all the exercises for a particular reading, but rather how much time you should spend on both the reading and your attempts to solve the exercises. When certain exercises are imperative, or when solving many exercises is required to understand fully the material, this requirement (and additional allotted time) will be indicated clearly in the time advisory.
 
A general resource that may be helpful to you in this course is a mathematics question-and-answer website hosted by StackExchange and found here. This site allows you to pose subject-matter questions to be answered by other site users. In general, these site users provide high-quality answers, often within minutes.

Learning Outcomes  showclose

Upon successful completion of this course, you will be able to:
  • define and compute directional and partial derivatives;
  • prove properties of the directional and partial derivatives;
  • summarize the role of matrices in the theory of differentiation;
  • define differentiable functions;
  • prove properties of the differential;
  • state, use, and prove the chain rule and the Cauchy invariant rule;
  • compute and prove properties related to repeated differentiation;
  • use and prove Taylor’s Theorem in higher dimensions;
  • define, compute, and prove properties related to the Jacobian;
  • state, prove, and use the Inverse Function Theorem;
  • define and use the technique of a Baire category;
  • locate local and global maxima and minima;
  • apply the method of Lagrange multipliers to locate conditional maxima and minima;
  • define the Riemann integral of a function ;
  • define and recognize sets of zero content;
  • state and prove the basic properties of the Riemann integral;
  • identify Riemann integrable functions;
  • relate multiple and iterated integrals;
  • compute Riemann integrals using iterated integrals;
  • define and recognize Jordan measurable sets;
  • summarize the role of Jordan measurable sets in the theory of the Riemann integral;
  • formulate and prove the Change of Variables formula;
  • compute integrals with the Change of Variables formula;
  • use polar and spherical coordinates;
  • define differential forms, their product, the exterior derivative, and the Hodge star operation;
  • compute with differential forms;
  • define n-chains and identify boundaries and cycles;
  • define, compute, and use the winding number of a curve;
  • define the integral of k-forms on n-chains;
  • state and prove Stokes’ Theorem; and
  • obtain the formulas of Green, Gauss, and Stokes as special cases.

Course Requirements  showclose

In order to take this course, you must:
 
√    have access to a computer;
 
√    have continuous broadband Internet access;
 
√    have the ability and permission to install plug-ins and/or software (e.g., Adobe Reader or Flash).
 
√    have the ability to download and save files and documents to a computer;
 
√    have the ability to open Microsoft files and documents (.doc, .docx, .ppt, .xls, etc.);
 
√    have competency in the English language;
  
√    have read the Saylor Student Handbook; and
 
√    have successfully completed the following Saylor courses, or their equivalents: MA101, MA102, MA103, MA211, and MA221 from The Core Program in Saylor’s Mathematics discipline, as well as Saylor’s MA241 (Real Analysis I).

Unit Outline show close


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  • Unit 1: Differentiation on Rn  

    A note about the mathematical notation used in this course: Throughout this course, we will be concerned with the space Rn. Some authors refer to this space as En. As such, some of the resources provided in this course use one notation, while others use the other notation. For all intents and purposes, you may treat these two notations as interchangeable names used for the same space.
     
    The derivative of a function at a point p, as developed in MA241, can be thought of as the slope of the line (if it exists) tangent to the graph of the function f at the point (p,f(p)). But generalizing this idea to a function is not straightforward. To see why, consider a function . Its graph is an object in four dimensions and it is not at all clear what a tangent line would be for such an object.
     
    Recall that in MA241 you learned how the derivative of a function at a point, which is just a single real number, can be used to discern quite a lot of information about the function     – at least in a small interval containing the given point. For instance, you saw that the derivative can be used to locate local maxima and minima of the function, to determine whether the function increases or decreases, to determine whether a function is locally invertible, and, through repeated derivations and Taylor’s Theorem, to obtain powerful approximations of the function.
     
    In the first unit of this course, you will explore how to extend these techniques to functions of several real variables. Indeed, a fundamentally different idea is required in order to describe what the correct analog of the derivative of a function is for functions of several variables. You will learn that the correct notion to use is that of the differential of a function, which is nothing but a linear map. By approaching multivariable functions in this way, you will build a strong bridge between multivariable analysis and linear algebra. You will see how the techniques of differential calculus from MA241 can be developed further to obtain powerful techniques in higher dimensions. In particular, you will learn how to use the differential map to determine local maxima and minima and local invertiblity, how to generalize Taylor’s Theorem, and how to apply the technique of Lagrange multipliers to solve more general problems, such as locating extremal points.
     
    From the very beginning of this unit, you will see that higher dimensions introduce many new difficulties and subtleties     – complex aspects that will accompany you throughout this course. As you approach these subtleties, it may be helpful to keep the following analogy in mind: Riding a bicycle is relatively easy, since generally there are only two directions one can fall in – to either side of the wheels. But riding a unicycle is far more difficult, since there are now infinitely many directions one can fall in. The transition of going from single-variable functions to multivariable functions is comparable to learning to ride a unicycle when, up to this point, you have learned only how to ride a bicycle.

    Unit 1 Time Advisory   show close
    Unit 1 Learning Outcomes   show close
  • 1.1 Directional and Partial Derivatives  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 1: Directional and Partial Derivatives”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 1: Directional and Partial Derivatives” (PDF)
       
      Instructions: Read Section 1, “Directional and Partial Derivatives,” and stop on page 7. You will return to this textbook throughout this unit, so you may prefer to save this PDF to your computer for quick reference.
       
      The idea behind the directional derivative is to eliminate dimensions so as to obtain a function of a single variable. Following this step, you immediately can apply the techniques of differential calculus that you encountered in MA241. In this reading, you will explore how to reduce to one dimension and establish the first properties of the directional derivative. Be sure to follow carefully the examples presented in this reading and to work through all the problem exercises presented on pages 6-7.
       
      Reading this section and completing the exercises should take approximately 4 hours.
       
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  • 1.2 Linear Maps, Functionals, and Matrices  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 2: Linear Maps and Functionals. Matrices”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 2: Linear Maps and Functionals. Matrices” (PDF)
       
      Instructions: Read Section 2, “Linear Maps and Functionals. Matrices,” and stop on page 16.
       
      The directional derivative that you encountered in subunit 1.1 is not, on its own, a sufficient tool for approaching higher-dimensional differential calculus. In preparation for a more refined notion of differentiable functions, this reading concentrates on those properties of linear maps that are relevant to the coming definitions you will encounter in this unit of the course. Please note that some aspects of this reading will be a review of concepts you already know from MA211 Linear Algebra. Be sure to work through all the problem exercises presented on pages 14-16 of this reading.
       
      Reading this section and completing the exercises should take approximately 6 hours.
       
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  • 1.3 Differentiable Functions  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 3: Differentiable Functions”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 3: Differentiable Functions” (PDF)
       
      Instructions: Read Section 3, titled “Differentiable Functions,” stopping on page 28.
       
      This section introduces you to the heart of differential calculus. In sharp contrast with the case in MA241 – in which the derivative of a function at a point was simply a real number – the consequences of higher dimensions prohibit this kind of simplicity for the differentiability of functions of higher dimensions. Instead, you will see that the correct notion of the differential at a point for higher dimensional functions is a linear map that satisfies certain properties. Some aspects of this reading may not be easy to understand at first, but the concepts presented here are absolutely crucial for you to follow and comprehend. Be sure to review along the way, as necessary, and work through all the problem exercises presented on pages 25-28 of this reading.
       
      Reading this section and completing the exercises should take approximately 8 hours.
       
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  • 1.4 The Chain Rule and the Cauchy Invariant Rule  
    • Reading: Professor Elias Zakon's Mathematical Analysis: Volume II: “Chapter 6, Section 4: The Chain Rule. The Cauchy Invariant Rule”

      Link: Professor Elias Zakon's Mathematical Analysis: Volume II: “Chapter 6, Section 4: The Chain Rule. The Cauchy Invariant Rule” (PDF)
       
      Instructions: Read Section 4, titled “The Chain Rule. The Cauchy Invariant Rule,” and stop on page 35.
       
      Recall that in MA241 you learned about the chain rule for computing the derivative of a composite function. This section focuses on the generalization of this result to the composition of functions in higher dimensions. You will see that while the chain rule from MA241 requires only the multiplication of two numbers to obtain the correct derivative, acquiring the result for higher dimensions is somewhat more intricate. However, you may remember the essence of this process with a helpful slogan: The differential of a composite function is the composite of the differential functions. Be sure to work through all the problem exercises presented on pages 33-35.
       
      Reading this section and completing the exercises should take approximately 5 hours.
       
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  • 1.5 Repeated Differentiation and Taylor’s Theorem  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 5: Repeated Differentiation. Taylor’s Theorem”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 5: Repeated Differentiation. Taylor’s Theorem” (PDF)
       
      Instructions: Read Section 5, titled “Repeated Differentiation. Taylor’s Theorem,” and stop on page 47.
       
      In this reading, you will encounter your first application of differentiation in higher dimensions. Recall that in MA241 you worked with Taylor’s Theorem, which enables the approximation of functions by repeatedly computing their derivatives at a point. This section focuses on extending this technique to higher dimensions. The generalization presented here is a direct one. The only difficulties lie in keeping track of the indices, of which there are many due to the high number of dimensions. Be sure to work through all the problem exercises presented on pages 44-47.
       
      Reading this section and completing the exercises should take approximately 7 hours.
       
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  • 1.6 Determinants, Jacobians, and Bijective Linear Operators  
  • 1.7 Inverse and Implicit Functions; Open and Closed Maps  
  • 1.8 Baire Categories; More on Linear Maps  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 8: Baire Categories. More on Linear Maps”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 8: Baire Categories. More on Linear Maps” (PDF)

      Instructions: Read Section 8, titled “Baire Categories. More on Linear Maps,” and stop on page 79.
       
      This section introduces you to a powerful concept in analysis, the notion of a Baire category. It was introduced and first used by the French mathematician René-Louis Baire, and it constitutes a fundamental tool in modern analysis. A Baire category is a topological and set theoretic concept and thus constitutes a technique with a somewhat different flavor from what you have seen so far in this course. This section will guide you through the concepts that you need to understand and illustrate the applicability of a Baire category. Be sure to work through all the problem exercises presented on pages 76-79 of this reading.
       
      Reading this section and completing the exercises should take approximately 8 hours.
       
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  • 1.9 Local Extrema, Maxima, and Minima  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 9: Local Extrema. Maxima and Minima”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 9: Local Extrema. Maxima and Minima” (PDF)
       
      Instructions: Read Section 9, titled “Local Extrema. Maxima and Minima,” and stop on page 87.
       
      As you saw in MA241, the derivative of a function is a very powerful tool that can be used to locate maxima and minima of a function. Locating extremal values of a function is crucial for solving many problems in science, since often a problem can be stated as finding a maximum or a minimum of a function. Moreover, historically such problems have constituted one of the main driving forces in developing the calculus. In this reading, you will learn how to apply the techniques developed so far in this course to help you locate the extremal values of a function in higher dimensions. Be sure to work through all the problem exercises presented on pages 84-87.
       
      Reading this section and completing the exercises should take approximately 6 hours.
       
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  • 1.10 More on Implicit Differentiation and Conditional Extrema  
    • Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 10: More on Implicit Differentiation. Conditional Extrema”

      Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 10: More on Implicit Differentiation. Conditional Extrema” (PDF)

      Instructions: Read Section 10, titled “More on Implicit Differentiation. Conditional Extrema,” stopping on page 96.
       
      This section establishes further the technique of implicit differentiation and introduces you to the technique of using Lagrange multipliers. This technique is a very powerful method for obtaining extremal points of a function whose variables are conditioned on some smaller subset in the higher-dimensional space. This is a new phenomenon that you did not see in MA241 – simply because when considering only one dimension, such restrictions are useless. In contrast, when considering higher dimensions, these are very relevant problems. The technique of using Lagrange multipliers – named after the eighteenth-century French-Italian mathematician Joseph-Louis Lagrange – is a general method that may be used for solving such problems. This section represents the culmination point of the first unit of this course, which has focused on differentiation.
       
      For this reading, it is recommended that you read carefully through the text to understand each and every detail and concentrate on the proofs and examples provided. Following your reading, work through the exercises at the end of the section, on pages 94-96, reviewing the text as needed. Complete as many of the exercises as you feel are needed for you have a good understanding of how the method of Lagrange multipliers works.
       
      Reading this section and completing the exercises should take approximately 6 hours.
       
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  • Unit 2: Riemann Integration in Higher Dimensions  

    As you learned in MA241, the integral is a form of a continuous sum that is developed and used to measure the area between the graph of a function and the X-axis. The graph of a “nice-enough” function encloses an area in which, at least intuitively, has volume. The Riemann integral for such functions turns this idea into a concrete definition that is a direct generalization of the concept of the Riemann integral for functions . Unlike the case of differentiability, which we explored in the previous unit of this course, the passage to higher dimensions presents relatively little difficulty in the case of the Riemann integral. In this unit you will define the n-dimensional Riemann integral; study the basic properties of this integral; see the subtleties that do arise for this case; and learn two important computation techniques: repeated integrals and the Change of Variables formula. The latter is a generalization of the Change of Variable formula that you learned about in MA241.

    Unit 2 Time Advisory   show close
    Unit 2 Learning Outcomes   show close
  • 2.1 A Review of Riemann Integration in  
    • Reading: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 3, Section 1: Definition of the Integral” and “Chapter 3, Section 2: Existence of the Integral”

      Link: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 3, Section 1: Definition of the Integral” (PDF) and “Chapter 3, Section 2: Existence of the Integral” (PDF)
       
      Instructions: Read Chapter 3, titled “Integral Calculus of Functions of One Variable,” on pages 113 through 134. Skip the subsection on page 125 regarding the Riemann-Stieltjes Integral. You will return to this textbook throughout this unit, so you may prefer to save this PDF to your computer for quick reference.
       
      The aim of this reading is to refresh your memory regarding the definition and intuitive geometric meaning of the integral of functions of one variable. As you read, please pay particular attention to the notions of intervals, lower and upper sums, and the rigorous definition of the Riemann integral and its geometric interpretation. Remember that this reading comprises a review of material you already have learned, and thus you do not need to complete the reading’s supplementary exercises unless you feel they will benefit your studies. Please keep in mind that you may find yourself returning to this reading to review material as you approach the remaining topics of this unit.
       
      Taking this review reading seriously will provide you with a sound foundation for understanding the material in the rest of this unit. Unless you feel completely fluent with the details of the Riemann integral from MA241, you should spend up to 3 hours on this reading.
       
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  • 2.2 Riemann Integration in Rn  
  • 2.2.1 Integrals over Rectangles  
  • 2.2.2 The Volume of Rectangles; Upper and Lower Sums  
    • Reading: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral”

      Link: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)
       
      Instructions: Read Section 1, titled “Definition and Existence of the Multiple Integral.” Start on page 441 at the section titled “Upper and Lower Integrals.” Read through page 448 and stop at the section titled “Sets with Zero Content.”
       
      In this reading you will encounter many elementary properties of the Riemann integral in higher dimensions. To help you interpret these results in the right context, please note that the text refers to results about the Riemann integral of functions of one variable, which you reviewed at the beginning of this unit. In this sense, the new results about the Riemann integral in higher dimensions are analogous to familiar properties of the Riemann integral from MA241. Note that many of the proofs presented in these pages are left to you, the reader, to complete. Please be sure to work through these exercises as you read.
       
      Reading this section and completing the exercises should take approximately 3 hours.
       
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  • 2.2.3 Sets of Zero Content  
    • Reading: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral”

      Link: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)
       
      Instructions: Read Section 1, entitled “Definition and Existence of the Multiple Integral.” Start on page 448 at the section titled “Sets of Zero Content.” Read through page 453 and stop when you reach the section titled “Differentiable Surfaces.”
       
      In this reading, you will encounter the geometric subtleties of higher dimensions that relate to integration. Intuitively, sets of zero content are sets that do not affect the Riemann integral, and thus these sets are important in stating properties of the Riemann integral and understanding its behavior. Sets of zero content also are relevant for the Riemann integral of functions of one variable. However, in higher dimensions, sets of zero content can be very complicated and include many interesting cases. You will encounter some of these sets in the next sub-subunit of this course.
       
      Reading this material should take approximately 1 hour and 30 minutes.
       
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  • 2.2.4 Differentiable Surfaces  
  • 2.2.5 Properties of the Riemann Integrals  
    • Reading: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral”

      Link: Trinity University: Dr. William F. Trench’s Introduction to Real Analysis: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)
       
      Instructions: Read Section 1, entitled “Definition and Existence of the Multiple Integral.” Start on page 455 at the section titled “Properties of Multiple Integrals.” Read through page 459 and stop when you reach the section entitled “7.1 Exercises.”
       
      In this reading, you will explore the many basic properties that are true of the Riemann integral, which was developed at the beginning of this unit. Notice that the results presented here in the text are direct analogues of the properties of the Riemann integral for functions of one variable. Also note that most of the proofs presented on these pages are left as exercises for you, the reader, to complete. It is vital that you complete the proofs presented here in order to cement your understanding of these concepts. If you’ve carefully studied the material developed so far in this course, especially in the context of the familiar results of MA241, then you should be able to complete all the proofs presented on these pages.
       
      Reading this material should take approximately 1 hour and 30 minutes.
       
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  • 2.3 Techniques of Integration  
  • 2.3.1 Preparatory Exercises  
  • 2.3.2 Iterated Integrals and Repeated Integrals  
    • Reading: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 2: Iterated Integrals and Multiple Integrals”

      Link: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 2: Iterated Integrals and Multiple Integrals” (PDF)
       
      Instructions: Read Section 2: “Iterated Integrals and Multiple Integrals” on pages 462 through 484.
       
      As you read this section, you will be introduced to a very important technique for reducing the computation of a Riemann integral in dimension n to the computation of n ordinary Riemann integrals. This new technique allows all the techniques for computations of integrals that you learned in MA241 to be used in higher dimensions as well. This section is quite lengthy, not because the technique is very complicated, but rather because many examples are worked out in great detail. You will see that in practice, this is a rather simple method to implement. Be sure to work through all the problem exercises presented on pages 480-484 of this reading.
       
      Reading this section and completing the exercises should take approximately 10 hours.
       
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  • 2.3.3 The Change of Variables Formula  

    This sub-subunit is concerned with the Change of Variables formula for the Riemann integral in higher dimensions. Recall that the Change of Variable formula for the ordinary Riemann integral, which you met in MA241, involved changing the bounds of integration – in effect, changing the set over which the function is integrated. A similar phenomenon occurs in higher dimensions as well, but in higher dimensions the change required can be far more elaborate and subtle than it is in the one-dimensional case. In the following sub-subunits of this course, prepare yourself to be confronted with rather complicated geometrical difficulties before arriving at the sought formula.

  • 2.3.3.1 Change of Variables in Multiple Integrals  
    • Reading: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals”

      Link: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)
       
      Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on pages 484 through 494, and stop when you reach the section entitled “Formulation of the Rule for Change of Variables.”
       
      This section will guide you through the concepts of Jordan measurable sets, transformations of such sets, and the change of content that occurs under a linear transformation. It turns out that Riemann integration works well when integrating over what are known as Jordan measurable sets. Anticipating that a change of variables will distort the set upon which the function is integrated, you first must understand how Jordan measurable sets are transformed – in particular, the effect of linear transformations on the content of such sets.
       
      Reading this section should take approximately 1 hour.
       
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  • 2.3.3.2 Formulation of the Rule for Change of Variables  
    • Reading: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals”

      Link: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)
       
      Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on page 494 through 496, and stop when you reach the section entitled “The Main Theorem.”
       
      This section will guide you through the geometric intuition that leads to the Change of Variables formula. Since the formula itself takes some time to get used to, it would be a good idea for you to read through this section at least twice. On your first reading, try to absorb the geometric intuition, but do not stop to contemplate any details. Following your first reading, spend some time considering and interpreting the Change of Variables formula. Then, read these pages again, this time spending as much time as is required for you to understand each step.
       
      Reading this section should take approximately 1 hour.
       
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  • 2.3.3.3 The Main Theorem  
    • Reading: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals”

      Link: Trinity University: Dr. William F. Trench's Introduction to Real Analysis: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)
       
      Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on page 496 through 505, and stop when you reach the section entitled “Polar Coordinates.”
       
      This section is concerned with the proof of the Change of Variables formula. This proof is quite involved and certainly not easy to comprehend – it is therefore recommended that you tackle this proof in two stages. For this assignment, try not to spend more than one hour reading over the material, keeping in mind that it is acceptable not to understand each step of the proof at this point. You can refer back to this reading after you have progressed through the next subunit of this course and completed a few exercises that apply the Change of Variables formula. After you have worked through some exercises using this formula, you likely will find this proof more palatable and easier to understand.
       
      Reading this section should take approximately 2 hours.
       
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  • 2.3.3.4 Polar Coordinates, Spherical Coordinates, and Exercises  
  • Unit 3: Integration on Chains  

    The n-dimensional space Rn may contain, for any dimension k smaller than n, subspaces that locally look just like Rk. Such subspaces are examples of k-dimensional manifolds (think of the two-dimensional sphere inside R3 as the set of points of distance 1 from the origin). It is natural and desirable to be able to integrate suitable quantities defined over k-dimensional manifolds inside Rn.
     
    Unfortunately, the adaptation of integration on Rn to integration over arbitrary manifolds is far from being a trivial task. Part of the reason for this is that manifolds in Rn can be extremely complicated as soon as n>2. The case n=1 presents little difficulty since the only possible manifold in  that is of smaller dimension than that of is a point. The case n=2 already presents some difficulties, since now manifolds inside R2 include curves. This situation still can be managed rather straightforwardly because the boundary of a curve is a rather simple thing: either a point or a pair of points, or else an empty set. In contrast, things get significantly more complicated in higher dimensions since boundaries of arbitrary manifolds in Rn, for n>2, can be very wild indeed. Thus, a more systematic approach is required to allow us to circumvent these geometric subtleties in order to arrive at a unifying theory.
     
    Historically, it took mathematicians quite some time to develop the unifying approach that is presented in this final unit of the course. This approach requires quite a large dose of linear algebra involving some very abstract notions. However, it also allows us to obtain a unification of very complicated geometric notions as well as a very easy proof of the Fundamental Theorem of Calculus in its most general form – encompassing all notions of integrals on manifolds in the context of this course. 

    Unit 3 Time Advisory   show close
    Unit 3 Learning Outcomes   show close
  • 3.1 Manifolds  
    • Reading: Cornell University: Dr. Reyer Sjamaar's Manifolds and Differential Forms: “Chapter 1, Sections 1-4: Introduction”

      Link: Cornell University: Dr. Reyer Sjamaar's Manifolds and Differential Forms: “Chapter 1, Sections 1-4: Introduction” (PDF)
       
      Instructions: Read the first four sections of Chapter 1, entitled “Introduction,” on pages 1 through 15. You will return to this textbook throughout this unit, so you may prefer to save this PDF to your computer for quick reference.
       
      These sections will introduce you to the notion of manifolds in Rn. The numerous illustrations on these pages are very helpful for understanding the relevant geometric concepts, and the examples here are carefully chosen to indicate the importance of this notion. When you have completed this reading, you should attempt to solve all the exercises presented on pages 13-15. While it is advisable that you work through all the given exercises, you may also choose to concentrate only on the more crucial ones at this point in time, returning to the rest of the exercises later. If this is the case, make sure that you are able to solve exercises 1.1-1.6 at this time, and return to the remainder of the exercises as you progress through this unit, particularly when you feel that you need some experiential groundwork for the abstract notions being discussed.
       
      Reading these sections and solving the exercises should take approximately 5 hours.
       
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  • 3.2 Differential Forms on Euclidean Space  
  • 3.2.1 Elementary Properties  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 1: Elementary Properties”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 1: Elementary Properties” (PDF)
       
      Instructions: Read Section 1, entitled “Elementary Properties,” on pages 17 through 20, and stop when you reach the section entitled “The Exterior Derivative.”
       
      This section introduces you to the concept of the differential form. In stark contrast with Chapter 1, which was very geometric in approach, this chapter is very algebraic in approach. It is likely that you will feel as if you do not understand how the abstract notions presented on these pages are at all related to integration. Don’t worry – things will become clearer as you progress through the upcoming subunits of this course. For now, focus on fully absorbing the algebraic concepts presented here so that you are prepared to see them in action in future readings. Remember that the way forward is by understanding all the arguments presented in the proofs on these pages. If a concept is not clear in this reading, try going back to earlier subunits of this course to review relevant definitions or previous results.
       
      Reading this section should take approximately 1 hour.
       
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  • 3.2.2 The Exterior Derivative  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 2: The Exterior Derivative”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 2: The Exterior Derivative” (PDF)
       
      Instructions: Read Section 2, entitled “The Exterior Derivative,” on pages 20 through 22, and stop when you reach the section entitled “Closed and Exact Forms.”
       
      This section introduces you to the exterior derivative operation, which as you will see, establishes a connection between differential forms and the gradient of a function (which you’ve met before). With this connection in place, you will begin to understand the relevance of differential forms to the theory of integration. As you read, remember that you will need to become proficient in performing computations with differential forms and the exterior derivative – both for concrete instances and for generic ones.
       
      Reading this section should take approximately 30 minutes.
       
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  • 3.2.3 Closed and Exact Forms  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 3: Closed and Exact Forms”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 3: Closed and Exact Forms” (PDF)
       
      Instructions: Read Section 3, entitled “Closed and Exact Forms,” on pages 22 through 23, and stop when you reach the section entitled “The Hodge Star Operator.”
       
      In this reading, you will discover that, upon using the exterior derivative introduced in the previous subunit, a differential form may exhibit certain properties, called closed and exact. The significance of these terms will become clearer to you later in this course. For now, you may consider these notions abstractly and focus on the definitions as they are presented in the reading. Please pay special attention to Example 2.10 on page 23, as it is central to your understanding of these concepts and will appear again later in this course.
       
      Reading this section should take approximately 30 minutes.
       
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  • 3.2.4 The Hodge Star Operator  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 4: The Hodge Star Operator”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 4: The Hodge Star Operator” (PDF)
       
      Instructions: Read Section 4, entitled “The Hodge Star Operator,” on pages 23 through 24, and stop when you reach the section entitled “Div, Grad and Curl.”
       
      This section describes the Hodge star operator on differential forms. The Hodge star operator is a combinatorial construction and thus computable. It has an elegant interpretation relating to vector fields and 1-forms. As such, it serves to relate the abstract notions of differential forms with more familiar geometric concepts of integration. As you read this section, focus on understanding the definition of the Hodge star operator. You will relate this abstract construction to more familiar concepts in the next subunit of this course.
       
      Reading this section should take approximately 30 minutes.
       
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  • 3.2.5 Divergence, Gradient and Curl  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 5: Div, Grad and Curl”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 2, Section 5: Div, Grad and Curl” (PDF)
       
      Instructions: Read Section 5, entitled “Div, Grad and Curl,” on pages 24 through 27, and stop when you reach the section entitled “Exercises.”
       
      This section relates differential forms to divergence, gradient, and curl operators. This relationship is established by means of the Hodge star operator and is worked out in detail on these pages. In particular, the three equations surrounded by boxes on pages 26 and 27 illustrate concepts that are fundamental to your work in this course. Please aim to understand these equations fully before proceeding to the next subunit.
       
      Reading this section should take approximately 1 hour.
       
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  • 3.2.6 Exercises  
  • 3.3 Pulling Back Forms  
  • 3.3.1 Determinants  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 3, Section 1: Determinants”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 3, Section 1: Determinants” (PDF)
       
      Instructions: Read Section 1, entitled “Determinants,” on pages 31 through 36, and stop when you reach the section entitled “Pulling Back Forms.”
       
      This section is a review of the concept of the determinant in linear algebra. In particular, this text approaches the determinant as a volume function and deduces its basic properties. The appearance of the determinant in the context of integration should not come as a surprise to you here, since you already have seen the determinant of the Jacobian play a prominent role in the Change of Variables formula for the Riemann integral. As you read, note how this section emphasizes the geometric interpretation of the determinant – particularly its effect on the concept of volume.
       
      Reading this material, depending on your current understanding of the determinant, should take, at most, 1 hour.
       
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  • 3.3.2 Pulling Back Forms  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 3, Section 2: Pulling Back Forms”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 3, Section 2: Pulling Back Forms” (PDF)
       
      Instructions: Read Section 2, entitled “Pulling Back Forms,” on pages 36 through 45.
       
      This section will introduce you to the important operation of pulling back differential forms along certain functions. You will learn the precise meaning of this operation as well as recount and prove its basic properties. Intuitively, pulling back a differential form is the result of changing the variables appearing in the form to new ones, with the old variables being expressed as smooth functions of the new ones. In this situation, there are some technical difficulties – or, to describe them more accurately, some unfamiliar direction reversals – that make grasping the results a bit more difficult here. But, if you have carefully followed the material presented in Chapter 2 and Chapter 3 of this textbook and have completed all the exercises assigned from those chapters, then you should be able to follow this section proficiently. The section ends by relating the pull-back operation to the determinant, thereby illuminating the geometric nature of the pull-back operation. Be sure to work through all the problem exercises presented on pages 42 through 45 of this section.
       
      Reading this section and working through the exercises should take approximately 4 hours.
       
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  • 3.4 Integration of 1-forms  
  • 3.4.1 Definition and Elementary Properties of the Integral  
  • 3.4.2 The Global Angle Function and the Winding Number  
  • 3.5 Integration and Stokes' Theorem  
  • 3.5.1 Integration of Forms Over Chains  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 5, Section 1: Integration of Forms Over Chains”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 5, Section 1: Integration of Forms Over Chains” (PDF)
       
      Instructions: Read Section 1, entitled “Integration of Forms Over Chains,” on pages 57 through 59, and stop when you reach the section entitled “The Boundary of a Chain.”
       
      This section will generalize the integral of a 1-form to the integral of a k-form for arbitrary, non-negative k. When integrating k-forms, it is very convenient to introduce the notion of chains, which are also discussed in this reading. The text examines in great detail some concrete examples of these concepts for low values of k. It is important for you to thoroughly understand these cases in order to develop your intuition regarding these concepts before you continue with the rest of this unit.
       
      Reading this section should take approximately 30 minutes.
       
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  • 3.5.2 Cycles and Boundaries of Chains  
  • 3.5.3 Stokes' Theorem  
    • Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 5, Section 4: Stokes’ Theorem”

      Link: Cornell University: Dr. Reyer Sjamaar’s Manifolds and Differential Forms: “Chapter 5, Section 4: Stokes’ Theorem” (PDF)
       
      Instructions: Read Section 4, entitled “Stokes’ Theorem,” on pages 63 through 65. Work through all the exercises presented on these pages.
       
      This final reading of the course is concerned with the statement and proof of Stokes’ Theorem, also known as the Fundamental Theorem of Calculus. With all the mathematical machinery in place, after all the hard work you have done in the previous units of this course, the final statement of Stokes’ Theorem that is presented here comes across as being elegant, with the proof itself very short – almost trivial.
       
      Reading this section and working through the exercises should take approximately 2 hours and 30 minutes.
       
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  • Final Exam  

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