Math 110.211: Honors Multivariable Calculus
Fall 2016 Course Page
Instructor: Dr. Richard Brown
MW 12:00pm - 1:15pm
Room: Maryland 201
|Jordan Paschke||Friday||12:00pm - 12:50pm||Maryland 201|
Vector Calculus, 4th Ed. by Susan Jane Colley
The first midterm has been postponed until Monday, October 17!
Course Syllabus and Homework Assignment Schedule
Welcome to the Fall 2016 version of 110.211 Honors Multivariable Calculus. Most of the informational aspects of the course, as well as its logistics will be documented here and in the linked syllabus above. As you know, this is an honors course. The prerequisite for this course is a full year of simgle variable calculus. But there is also a co-requisite of linear algebra. A co-requisite means that either you have already taken a course in linear algebra, or you are currently enrolled also in a linear algebra course. The link above takes you to a page which details the basic structure of the course along with an idea of the week-by-week schedule. As need be, this syllabus and schedule will be amended to reflect the current focus of the course. Also, it will be updated regularly with the homework assignments and such. For now, let's leave it at that.
Some extra stuff:
So do you want to start thinking like a mathematician? Here is a good place to start:
Prerequisite material for this course:
Calculus: This is to be considered the third semester of our course series 110.108-9 Calculus for the Physical Sciences and Engineering. The techniques you will see and the theory we will develop all stem from appropriate generalizations of what we consider to be Single Variable Calculus. Should you need, the official department syllabi for this series can be found here:
If you did not take 110.108 and/or 110.109 here at Hopkins, you should acquaint yourself with the material that I will assume you already know. Unfortunately, this will have to be done on your own. Take some time to review these syllabi and make sure you have covered ALL of this material. Feel free to consult with me and/or your Section TA about this prerequisite material.
Linear Algbra: Linear Algebra is considered a co-requisite for this course. Hence I will make the assumption that any material up to and including the 7th week of our standard 110.201 Linear Algebra course is known. We can certainly make adjustments in class. But this will help a lot. Here is the syllabus for what we call linear algebra:
How to write up Homework Solutions: Constructing homework solutions is a vital way to explore and strengthen your understanding of the theoretical underpinnings and practical applications of the material in this course. There is no better way to fully comprehend the mathematical content of this course than to attempt to explain in full detail just how a mathematical problem is posed, presented and solved via the conceptual and practical application of technique and theory. Besides developing a great tool for continued study, both in this course and in future courses, constructing comprehensive and detailed solutions to mathematical problems develops your ability to communicate mathematical ideas effectively, rather than simply to calculate. The construction of your solutions, in effect the story you tell that convinces the reader that your solution is indeed correct, will be an important part of all grading criteria regarding homework assignments.
Some relevant deadlines, calendars and schedules to keep in mind:
Friday, September 2: Some Linear Algebra backstory: As an introduction to the course, I thought to play with the structure of Euclidean space and linear algebra just to establish notation and begin the conversation. In the notes attached, I got to page 7, although the rest is straightforward. Take a look and send any questions along. I also used a bit of Mathematica for visualization. It is here.
Wednesday, September 7: Section 2.1. Again, covering this section is mostly for notation and viewpoint. Pay close attention to why and how we visualize functions, though parameterizations, graphs, slices and sections. These will expose the visual clues to how we analyze functions. Some Mathematica to help in visualization: Parameterized curves, surfaces, and sections and slices of graphs.
Monday, September 12: Section 2.2. We define and investigate the notion of a limit in more than one dimension. This is much more subtle than in the Calculus I case, and much harder to fully investigate using the definition alone. Fortunately, all of the nice functions from Calculus I, as well as all of the properties of limits we already learned are still valid. We will not go deep in this section, but just survey some ideas which we will explore in more detail in the context of more advanced material. Here is some Mathematica detailing some of the more basic pathological functions, where limits do not exist even as intuition indicates they should.
Wednesday, September 14: Section 2.3. Today, we finish our discussion on limits and pass through the concept of continuity. Really, there is little to add to the mix since the only new idea is that the limit of a function not only exists but equals the function value at a point of continuity. But there are a few rules and extensions that we talk about here. Then on to differentiability, where things start to diverge from single variable calculus. Here we define what differentiability is for a vector-valued function on more than one variable, both from an analytical as well as geometric perspective, and start the discussion on its properties. Here is a Mathematica notebook giving some geometric meaning to the derivative of a real-valued function on two variables and how the tangent plane to its graph in three space is defined and constructed.
Monday, September 19: Section 2.4. Here, we finish Section 2.3 by defining the derivative of a general function from n-space to m-space, and discuss its properties and interpretations. Then we move on into Section 2.4, where we bring back the rules for differentiation (used to derive new functions constructed using various combinations of other functions) from Calculus I and use them in our new context. The basic frame for this discussion is, "the rules are the same, but only precisely when they actually make sense." What this means is the focus of this class. Also, we will look at higher derivatives and the notion of a function being differentiable more than once.
Wednesday, September 21: Section 2.5. Really, we just finished Section 2.4 today and gave the basic structure of the Chain Rule offered in Theorem 2.5.3. The bulk of today was to work out some examples of the Product Rule (Product of a real-valued and a vector valued function, and the dot product of two vector-valued functions of the same size. We then moved on the notion of what a kth partial was of a real-valued function was and when mixed partials of differentiable functions are equal. The differentiable class of a function was discussed, along with just what kind of object the kth derivative of a real-valued function on n variables was and how it encompasses its nk partials.
Monday, September 26: Section 2.6. Today, we finish up with the Chain Rule and move into directional derivatives, a generalization of a partial derivative where we look for how a function is changing at a point in any single direction in the domain. This gives a powerful tool, both conceptually as well as technically, to discuss the role the derivative of a function plays in exposing the properties of both functions on and sets within Euclidean space. We define the gradient of a real-valued function (finally) and its interpretations and usefulness, and move toward one of the most powerful theorems of multivariable calculus, the Implicit Function Theorem.
Wednesday, September 28: Section 2.6 (extra): Here, we finish up with the properties of the gradient (for now), and give a treatment of both the Implicit Function Theorem (for real-valued functions), and the Inverse Function Theorem. These are very powerful theorems that expose some of the hidden structure of real-valued and vector-valued functions of more than one variable. We will study the ideas in class, and here is a proof of the Implicit Function Theorem for a function on (a subset of) three space. And here is a Mathematica Notebook for this class.
Monday, October 3: Section 3.1 and 3.2: Today we begin the study of Chapter 3 on vector-valued functions. For the most part, there are only two topics of discussion here: paths or curves and vector fields, respectively defined as functions from the real line into n-space, or functions from n-space into itself. The reason for an entire chapter on these two items is that they play a huge role in a solid general understanding of all of the calculus of vector-valued functions of more than one variable. They also introduce the idea of a geometric object begin completely defined by a function, allowing us to fold geometry into the analysis of functions in a fundamental way. This is one of the core principles of higher mathematics. Today, curves in n-space and some of their properties. One defining characteristic of a curve in n-space is that its length should be independent of its paramaterization, even though we calculate the length using the parameterization. Here is a write up of why this is so.
Wednesday, October 5: Section 3.3: Vector fields, as geometric objects and/or functions, provide a backbone in which all of physics and engineering, really mathematical modeling is structured on. From force fields in physics to slope fields in differential equations and modeling, the notion of a vector field allows us to recover measureable quantities from models defined only by equations of motion. Here, we begin the study of their basic structure and properties.
Monday, October 10: Section 4.1. This was Jordan's Lecture on differentials and Taylor Series. I have no notes on this.
Wednesday, October 12: Section 4.2: Local and global extrema are much like their counterparts in single variable calculus. They are just points in the domain of a real-valued function where the function value is locally the lowest or highest. And they occur, if at all, at critical points of the function. If the function is differentiable everywhere, then extrema only occur at places where the derivative (matrix) has zeros as all of its elements. Thus all of the directional derivatives are 0 here also. But since directional derivatives are just derivative along slices through the function, we can also check the concavity of these slice functions along vector directions in the domain. This leads to a notion of a second directional derivative, and also to one major application of the Hessian matrix of second partials. Relating this to a quadratic form, we construct the Second Derivative Test for a C^2 real-valued function of more than one variable. We develop this today. We then end with the multidimensional counterpart of the Extreme Value Theorem, once we understand what closed and bounded mean for a domain in real n-space. Here is some Mathematica from today.
Monday, October 17: Midterm 1. No lecture.
Wednesday, October 19: Section 4.3:
Thursday, October 20: Sections 5.1 and 5.2 and 5.3:
Monday, October 24: Section 5.4: Today we continue the general idea of integration of a real-valued function on more than one variable by generalizing the 2-dimensional version to three dimensions. There is little that is new here except for the pattern of the generalization that leads to the n-dimensional version. Fubini's Theorem still holds, and switching the order of integration outside of a cuboid region still involves checking that the region is elementary in different permutations of the variables of integration and that, if so, one can rewrite the limits as functions of some of the variable properly.
Wednesday, October 26: Section 5.5: Today, we focus on the idea of changing the coordinates in an integral. In Calculus I, the Substitution Method was an actual change of coordinates used usually to make the integrand easier to play with. Here, and in more generality, changing the coordinate system on a region is used more to make the region easier to integrate over. Of course, it must be true that the value of the definite integral should be the same no matter the coordinates used. Hence one must be careful to properly account for the change, precisely as in the Substitution Method, where a change of variable creates a new variable corresponding to the "inside function" of the composition of functions in the integrand (this is a function of the old variable). The extra piece was the derivative of the inside function. This generalizes as the 1-dimensional version of a similar phenomena in higher dimensions. We detail this today. Here is a write-up of the volume of a 2-sphere of radius rho in 3-space, in both rectilinear coordinates and spherical coordinates.
Monday, October 31, 2016: Section 6.1: Here, we dive deeper into integration under the idea that in multiple directions, there are more ways to study the properties of functions. The first type is called the line integral, where one integrates over a curve. The two varieties, the scalar and vector line integrals, have interesting geometric interpretations as well as simple meaning on their own. Today, we define and study these two types. Here are two fun GIFs from the internet, meant to give some contextual meaning to line integrals; Scalar and vector line integrals.
Wednesday, November 2, 2016: Section 6.2 and 6.3: Today, we go directly into on of the three big theorem's of vector calculus, Green's Theorem. This theorem exposes a deep relationship between the aggregate behavior of a vector field along the boundary of a relatively nice region in the plane (the vector line integral), to the integral of a related function on the interior of the region. Since Green's Theorem is restricted to regions in the plane, there are a number of ways to craft the related integrals, giving different geometric meaning to the quantities. One interesting geometric interpretation is that the theorem relates the total twisting effect of a vector field in the region (measured by integrating the curl of the vector field as it sits in three space with no vertical component), to the total tangent component of the vector field along the closed boundary. Proving this theorem is neither deep nor long, and we go over the idea here in lecture. We finish with a general definition and discussion of the properties of a special kind of vector field that shows up in an lot of physical applications.
Monday, November 7, 2016: Section 7.1: The two dimensional counterpart to a curve in n-space is a surface in n-space, and today we define and discuss the properties of parameterized surfaces in (mostly) three-space (and sometimes n-space.) The parallels to curves will be obvious, and discussing these parallels will bring up very interesting contrasts, which we will highlight. Then we will begin the discussion of how a parameterization of a surface in space allows us to discuss the properties of the surface, including how functions behave when defined on the surface. Here is some Mathematica concerning surface parameterizations.
Wednesday, November 9, 2016: Section 7.2: We continue with the idea of understanding how the calculus of functions behaves along parameterized surfaces (instead of along parameterized curves.) Today, we define and study both scalar and vector surface integrals, of real-valued functions and vector fields along surfaces embedded in three space, respectively. We sill stick to surfaces in three space for the expediency of understanding these concepts without too much intricate machinery. But we will allude regularly to the idea that we can embed and parameterize a surface in n-space, (n>2), and play the same game. We also discuss the idea of reparameterizations, orientation of a surface, and geometric interpretations, all as a lead up to another of the three big theorems, Stoke's Theorem.
Monday & Wednesday, November 14 & 16, 2016: Section 7.3: This week, we finish the foundational material of what makes a vector calculus course with a full discussion of the two other Big Theorems, those of Stokes and Gauss. We run through the theorem, develop the intuition of what is going on, establish some main situations where each theorem is relevant and talk about proofs. Then we use these theorems to give a much more precise idea of just what the divergence and the curl of a vector field actually is and how to understand these concepts geometrically.
Monday, November 28, 2016: Recitation and review for Midterm 2. No Lecture.
Wednesday, November 30, 2016: Midterm II. No Lecture. Solutions here.
Friday, Monday and Wednesday, December 2, 5, 7, 2016: Sections 8.1, 8,2 and 8.3: During this last three lectures, I have decided to lecture on the structure of differential forms form the perspective of multi-linear algebra and n-forms on vector spaces. This is basically not done in the book. This allows me to give a much more foundational treatment of just what forms are and not just how they work. We learn their structure, how to integrate them and how to differentiate them, all with an eye toward what works regardless of the dimension. We show how many of the things we learned in the past, from the product rule and the Substitution Method in Calculus I to the Change of Variables Theorem and Fubini's Theorem in Calculus III, are all just examples of more general structure. We then finish with the Generalized Stoke's Theorem, and show how the various big theorems of Gauss, Stokes and Green are also simply particular examples. We end with the same result of the Fundamental Theorem of Calculus. In fact, one can easily say that the Generalized Stokes Theorem is just the Fundamental Theorem of Multivariable Calculus.
Problem Set 1 (Due Wednesday, September 14):
Section 2.1: 2, 4, 6, 11, 12, 16, 18, 20, 36, 38, 40
Problem Set 2 (Due Friday, September 23):
Section 2.2: 5, 10, 15, 28, 32, 43, 46, 50, 52
Section 2.3: 4, 8, 16
Problem Set 3 (Due Wednesday, September 28):
Section 2.3: 28, 32, 36, 40, 41, 58
Section 2.4: 4, 5, 22, 25, 27, 29a, 30
Section 2.5: 2, 4
Problem Set 4 (Due Wednesday, October 5):
Section 2.5: 12, 16, 20, 31, 34, 36, 37
Section 2.6: 1, 8, 20, 26, 34, 41, 42, 44, 46
Problem Set 5 (Due Wednesday, October 12):
Section 3.1: 5, 9, 12, 19, 28
Section 3.2: 4, 7, 14
Section 3.3: 4, 10, 24, 28
Section 3.4: 4, 7, 14, 24
Problem Set 6 (Due Thursday, October 20):
Section 4.1: 4, 12, 17, 18, 25
Section 4.2: 6, 16, 22, 30, 32, 43, 50
Problem Set 7 (Due Wednesday, October 26):
Section 4.3: 4, 6, 12, 21, 22, 32
Section 5.1: 2, 6, 8, 12
Section 5.2: 2, 3
Problem Set 8 (Due Wednesday, November 2):
Section 5.2: 8, 18, 35
Section 5.3: 4, 10, 12, 14
Section 5.4: 6, 14
Section 5.5: 4, 6, 10, 12, 15, 27, 36
Problem Set 9 (Due Wednesday, November 9):
Section 6.1: 4, 6, 13, 23, 24, 34
Section 6.2: 4, 8, 17, 25, 30
Prove Lemma 2 in the lecture on Section 6.2 (page 3)
Problem Set 10 (Due Wednesday, November 16):
Section 6.3: 2, 6, 10, 20, 30
Section 7.1: 4, 7, 10, 12, 24, 27
Section 7.2: 3, 6, 12, 14, 27
Problem Set 11 (Due Monday, November 28):
Section 7.3: 1, 8, 10, 12, 16, 17, 20, 21
Problem Set 12 (Not to be handed in...)
Section 8.1: 4, 8, 10, 18
Section 8.2: 7, 9, 12, 13
Section 8.3: 4, 5, 8, 9, 11
This page will be updated regularly when new information about the course arises. General information about course structure, requirements, as well as specific information related to your lecture or section, will be posted here and updated as needed.
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Last updated: 03/29/2012