**Unit 3: Integration on Chains**
*The n-dimensional space **R^{n}** may contain, for any
dimension k smaller than n, subspaces that locally look just like
**R ^{k}**. Such subspaces are examples of k-dimensional
manifolds (think of the two-dimensional sphere inside

**R**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

^{3}**R**.

^{n}Unfortunately, the adaptation of integration on

**R**to integration over arbitrary manifolds is far from being a trivial task. Part of the reason for this is that manifolds in

^{n}**R**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

^{n}**R**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

^{2}**R**, 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.

^{n}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**

Completing this unit should take you approximately 26 hours.

☐ Subunit 3.1: 5 hours

☐ Subunit 3.2: 7 hours

☐ Subunit 3.3: 5 hours

☐ Subunit 3.4: 5 hours

☐ Subunit 3.5: 4 hours

**Unit3 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- 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.

**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**
- **Activity: Cornell University: Dr. Reyer Sjamaar’s Manifolds and
Differential Forms: “Chapter 2, Exercises: Differential Forms on
Euclidean Space”**
Link: Cornell University: Dr. Reyer Sjamaar’s

*Manifolds and Differential Forms*: “Chapter 2, Exercises: Differential Forms on Euclidean Space” (PDF)

Instructions: Complete all the exercises presented in the section, which is on pages 27 through 30.

As you work, keep in mind that the concept of differential forms that you met at the beginning of this chapter – and the basic concepts related to it – form the computational heart of the rest of this unit of the course. Mastering these concepts now, which you can achieve by seriously attempting all the exercises on these pages, will facilitate your understanding of upcoming results and their proofs.

Working through these exercises should take approximately 3 hours and 30 minutes.

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**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**
- **Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and
Differential Forms: “Chapter 4, Section 1: Definition and
Elementary Properties of the Integral” and “Chapter 4, Section 2:
Integration of Exact 1-forms”**
Link: Cornell University: Dr. Reyer Sjamaar’s

*Manifolds and Differential Forms*: “Chapter 4, Section 1: Definition and Elementary Properties of the Integral” (PDF) and“Chapter 4, Section 2: Integration of Exact 1-forms” (PDF)

Instructions: Read Section 1 and Section 2 on pages 47 through 51, and stop when you reach the section entitled “The Global Angle Function and the Winding Number.”

This section is concerned with integrating 1-forms along smooth curves. You will learn about the geometric interpretation of this integral and prove some of its basic properties. The text presents the results concerning the integral of 1-forms in relation to the Fundamental Theorem of Calculus. It is very important that you understand this analogy, since the main purpose of the remainder of this unit of the course is to generalize the Fundamental Theorem of Calculus.

Reading these sections should take approximately 30 minutes.

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**3.4.2 The Global Angle Function and the Winding Number**
- **Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and
Differential Forms: “Chapter 4, Section 3: The Global Angle
Function and the Winding Number”**
Link: Cornell University: Dr. Reyer Sjamaar’s

*Manifolds and Differential Forms*: “Chapter 4, Section 3: The Global Angle Function and the Winding Number” (PDF)

Instructions: Read Section 3, entitled “The Global Angle Function and the Winding Number,” on pages 51 through 56. Work through all of the exercises presented on pages 53 through 56.

This section will introduce you to the concept of the winding number, a geometric notion associated with curves that can be used to solve many problems. You will see it at work as you work through the exercises that accompany this section.

Reading this section and working through the exercises should take approximately 4 hours and 30 minutes.

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**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**
- **Reading: Cornell University: Dr. Reyer Sjamaar’s Manifolds and
Differential Forms: “Chapter 5, Section 2: The Boundary of a Chain”
and “Chapter 5, Section 3: Cycles and Boundaries”**
Link: Cornell University: Dr. Reyer Sjamaar’s

*Manifolds and Differential Forms*: “Chapter 5, Section 2: The Boundary of a Chain” (PDF) and “Chapter 5, Section 3: Cycles and Boundaries” (PDF)

Instructions: Read Section 2 and Section 3 on pages 59 through 62, and stop when you reach the section entitled “Stokes’ Theorem.”

This section deals with the intricacies of higher dimensions and introduces you to cycles and boundaries for chains, allowing you to examine how chains of various dimensions interact with each other. The text also includes helpful illustrations that explain the geometric meaning of the algebraic concepts of boundary and cycle – aides that will help you understand and follow the proofs presented on these pages.

Reading these sections should take approximately 1 hour.

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**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**
- **Final Exam: The Saylor Foundation's “MA242 Final Exam”**

```
Link: The Saylor Foundation’s [“MA242 Final
Exam”](http://school.saylor.org/mod/quiz/view.php?id=1395)
Instructions: You must be logged into your Saylor Foundation School
account in order to access this exam. If you do not yet have an
account, you will be able to create one, free of charge, after
clicking the link.
```