# MA242: Real Analysis II

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.

Completing this unit should take you approximately 66 hours.

☐    Subunit 1.1: 4 hours

☐    Subunit 1.2: 6 hours

☐    Subunit 1.3: 8 hours

☐    Subunit 1.4: 5 hours

☐    Subunit 1.5: 7 hours

☐    Subunit 1.6: 6 hours

☐    Subunit 1.7: 10 hours

☐    Subunit 1.8: 8 hours

☐    Subunit 1.9: 6 hours

☐    Subunit 1.10: 6 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, you should 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; and
• apply the method of Lagrange multipliers to locate conditional maxima and minima.

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: Scroll down to the “Download the Book” section and download your preferred version by clicking the appropriate “I accept” link. 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.

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.

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.

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.

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.

1.6 Determinants, Jacobians, and Bijective Linear Operators   - Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 6: “Determinants. Jacobians. Bijective Linear Operators” Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 6: “Determinants. Jacobians. Bijective Linear Operators” (PDF)

Instructions: Read Section 6, titled “Determinants. Jacobians. Bijective Linear Operators,” and stop on page 57.

Because the differential of a function is a linear map and because linear maps are closely related to matrices, the differential calculus in higher dimensions is also closely related to matrices. This section introduces you to the concept of the Jacobian of a function and then studies some of the Jacobian’s properties. In the next subunit, you will use these properties to establish the very important Inverse Function Theorem. Be sure to work through all the problem exercises presented on pages 55-57.

Reading this section and completing the exercises should take approximately 6 hours.

1.7 Inverse and Implicit Functions; Open and Closed Maps   - Reading: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 7: Inverse and Implicit Functions. Open and Closed Maps” Link: Professor Elias Zakon’s Mathematical Analysis: Volume II: “Chapter 6, Section 7: Inverse and Implicit Functions. Open and Closed Maps” (PDF)

Instructions: Read Section 7, titled “Inverse and Implicit Functions. Open and Closed Maps,” and stop on page 70.

In MA241, you learned that if a function has a non-zero derivative at a point, then it is invertible, at least in a small open interval containing that point. This case is an example of how you can infer local information about a function from knowledge of its derivative at just one single point. This section takes this technique further, focusing on the generalization of this result (and some related results) and its application to higher dimensions. Be sure to work through all the problem exercises presented on 67-70.

Reading this section and completing the exercises should take approximately 10 hours.

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.

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](http://www.saylor.org/courses/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.