Introductory Math Courses
Courses in Multivariable Mathematics
The department offers 3 sequences in multivariable mathematics. If you took multivariable calculus elsewhere, please click the button below:
Already took some multivariable calculus?
The Series:
See detailed list of topics.
Math Linear Algebra, Multivariable Calculus, and Modern Applications (5 units) covers linear algebra and multivariable differential calculus in a unified manner alongside applications related to many quantitative fields. This material includes the basic geometry and algebra of vectors, matrices, and linear transformations, as well as optimization techniques in any number of variables (involving partial derivatives and Lagrange multipliers).
The unified treatment of both linear algebra (beyond dimension 3 and including eigenvalues) and multivariable optimization is not covered in a single course accessible to nonmajors anywhere else. Many students who learn some multivariable calculus before arriving at Stanford find Math 51 to be instructive to take due to its broad scope and synthesis of concepts. If you want transfer credit to substitute for Math 51 then you will likely need two courses (one on multivariable calculus, one on linear algebra).
Math Integral Calculus of Several Variables (5 units)covers multivariable integration, and in particular Green’s Theorem and Stokes’ Theorem. This uses both linear algebra and matrix derivative material from Math
Math Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications (5 units)develops core concepts, examples, and results for ordinary differential equations, and covers important partial differential equations and Fourier techniques for solving them. This uses both linear algebra and matrix derivative material from Math
The unified treatment of both ordinary and partial differential equations (PDE’s) along with Fourier methods is not covered in a single course accessible to nonmajors anywhere else. For those who studied differential equations before arriving at Stanford, Math 53 will strengthen your skills and broaden your knowledge via its use of linear algebra and its coverage of PDE's and Fourier methods. If you want to transfer credit to substitute for Math 53 then you will likely need two courses (one on ordinary differential equations, and one on PDE/Fourier material).
**Math 52 and Math 53 can be taken in either order.
This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.
Math 51 Textbook Math 53 Textbook
The table of contents and page of applications near the start of the course texts provide more information; these course texts are freely available to anyone with an SUNetId.
For those with a strong interest in math and a preference for more conceptual and theoretical understanding we recommend the following two sequences:
The 60CMSeries
Math 61CMCMCM Modern Mathematics: Continuous Methods(5 units each) This prooforiented threequarter sequence covers the material of 51, 52, 53, and additional advanced calculus, higherdimensional geometry, and ordinary and partial differential equations. This provides a unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know singlevariable calculus very well and have an interest in a theoretical approach to the subject.
This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.
The 60DMSeries
This prooforiented threequarter sequence covers the same linear algebra and multivariable optimization material as the 60CMseries but draws its motivation from topics in discrete math rather than from the more analytic topics as in the 60CMseries. Its discrete math coverage includes combinatorics, probability, some basic group theory, number theory, and graph theory. Students should have an interest in a theoretical approach to the subject.
This series provides the necessary mathematical background for majors in Computer Science, Economics, Mathematics, Mathematical and Computational Science, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering majors requiring Math 50’s beyond Math 51 are recommended to take Math 60CM.
Math 51 
Course DescriptionLinear Algebra and Multivariable Calculus are two of the most widely used mathematical tools in quantitative work across all fields of study. This course develops conceptual understanding and problemsolving skills in both, highlighting how multivariable calculus is most naturally understood in terms of linear algebra, and the course text addresses a variety of realworld applications. Our focus is on teaching you skills that underlie a wide array of applications and preparing you for all courses involving advanced quantitative work (across all sciences, engineering, economics, computer science, statistics, and so on). By the end of this course, you should be able to:
For a detailed syllabus see the Syllabus page. 
First Day ChecklistWelcome to Math 51! This syllabus site details the course's policies, schedules, and expectations, including for assignments and grading calculation.Per University policy, your decision to take the course implies that you agree to these requirements and to the grading policies spelled out here; so be sure to read everything on these pages.

Class Structure and AssessmentMath 51 has an "active learning" structure; research has shown that preclass reading, combined with daily participation in class activities targeted to specific learning goals, improves student learning outcomes in math and science courses. Furthermore, active learning increases student performances and narrows achievement gaps for historically underserved students. Here's what this means for us: Both MWF class sessions and TuTh discussion sections are more interactive than traditional math classes:
Canvas questionnaire assignments on the twiceweekly preclass reading: a typical questionnaire consists of 1 checkin question (always the first question) and 3 to 5 "lowstress" questions. Except for the checkin question, you needn't answer more than one or two sentences per question, and you get full marks for ANY goodfaith answer. These assignments are intended to give the instructor feedback on how the reading went and how the course is going; think of them as surveys in which students are voting for which topics need more motivation in class (and which need less or none). Because we will have to review your feedback in a limited time period, the firm deadlines are:

Honor code policyBy Math Department policy, any student found to be in violation of the Honor Code on any assignment or exam in this course will receive a final course letter grade of NP. You are fully responsible to adhering to the requirements of the Honor Code document. In particular, it is forbidden to

Office hours and other resources for helpYou are encouraged to attend the office hours provided by the instructors and teaching assistants. You may attend the office hours of any teaching staff member inperson or online. No appointment is ever necessary for virtual office hours, just drop in at the scheduled virtual office hours on the Math 51 Nooks site with your questions! For inperson office hours, instructors and TAs may at their discretion impose office capacity limit and meet with students on a firstcome firstserved basis; a signup sheet will be posted outside the office to facilitate these inperson meetings. The scheduled inperson and online office hours for any given week can be found on the office hours page. Note that they might change slightly from week to week so it's always a good idea to check both the calendar on the office hours page and the Math 51 Nooks site for virtual office hours.The office hours page also lists some other help resources. 
COVID adaptationsIn accordance with University guidance, everyone must wear mask covering both nose and mouth during inperson classes and office hours. In particular, no eating is permitted during class.Students should go to the section (both lecture and discussion section) they're officially registered for ease of contact tracing, so they can be notified promptly if a particular section has to move online at short notice. Students with schedule conflicts should officially enroll in the section they can attend in the next 2 weeks and then go to the section they are officially enrolled in for the rest of the quarter. We all need to be prepared to pivot to remote instruction this quarter, possibly at very short notice. Please make sure that your Math 51 Canvas announcement notifications are on so you can receive remote instruction announcement promptly. Furthermore, the university has set clear guidelines on classroom and course policies. See in particular policies on lecture recording and student absence due to COVID or other illnesses. 
MATH 51 grades improve after new textbook, syllabus introduced in fall
Students’ median grades on MATH “Linear Algebra, Multivariable Calculus, and Modern Applications” exams rose at least 15 percent between spring and fall after a new textbook and syllabus were introduced with the aim of making the course slowerpaced, more engaging and more applicable.
In spring , the mean on the first midterm was 77 percent; in fall, the mean was 92 percent. On the second midterm, the mean was 68 percent in spring and 87 percent in fall.
MATH 51 is a core requirement for many majors at Stanford. Around students take the course each quarter, the majority of whom are freshmen. Though the course has gradually changed since its founding in , the fall redesign was one of the most drastic.
Students have claimed that there is a disconnect between the material taught and the material tested, that there is too much material being crammed into too little time and that students must teach themselves the material. The math department responded to such concerns in a June letter to the editor, in which they discussed the anticipated course overall.
The changes
During the past four years, professors from the mathematics department have been working on a new textbook to accompany MATH Previously, the course used two different textbooks, but there were issues with adapting the textbooks to the material.
According to professor Brian Conrad, the Director of Undergraduate Studies in Mathematics, MATH 51 required two textbooks in the past because its curriculum, which integrates multivariable calculus and linear algebra, is unique to Stanford.
“There’s pretty much no other pure institution … which teaches a kind of general audience multivariable calculus that incorporates linear algebra right from the start,” he said. “And that’s one of the reasons there’s no offtheshelf book that we could really use.”
According to mathematics professor Rafe Mazzeo, one of the faculty involved in the creation of the new textbook and syllabus, updating the course material involved communication with people across 15 different departments to determine what was most important for students to learn.
The math department decided not to reveal the textbook’s writers so that the book can be seen as a departmental effort with room for potential future revisions.
One faculty member took a quarter off from teaching MATH 51 to write a skeletal version of the text. Then the department worked together to fill in the chapters, finally finishing the full text and exercises in summer
“We have no intention of publishing this [textbook],” Mazzeo said. “We want it to be freely accessible to students. And we’re going to keep it easily updatable, because things change.”
One major change in the new textbook is the inclusion of more references to the material’s realworld applications.
“One goal of the wide array of realworld contexts discussed in the new book is to convey to students how ndimensional considerations with n much bigger than three are extremely relevant to many contemporary applications,” Conrad said. “This is something that was difficult to convey in the earlier incarnation of the course because the texts that were used never addressed realworld applications of higherdimensional linear algebra.”
Conrad noted that the new textbook includes information about how topics connect to various fields in order to encourage students to understand the importance of what they learn in the course. By writing their own book, the faculty could emphasize how different concepts interact with each other.
“I think the book is easier for students to access, it’s an easier read, it’s easier for them to find information and people were very positively responsive to that,” Mazzeo said.
A student who took MATH 51 in spring and retook it in autumn said the new textbook was “much more clear” because it was “written much more specifically for the course.”
“It was much easier to follow than the Colley and Levandoski books that they used last time,” said the student, who requested to not be named in this article.
In addition to replacing the course textbooks, the math department changed MATH 51’s pacing. Topics such as Eigenvalues and Eigenvectors, row reduction and determinants were either moved to Math 52 or 53 or removed entirely.
So far, the overall student response to MATH 51’s restructuring has been positive. According to course evaluations, 70 percent of students rated the new textbook as either “extremely useful” or “very useful.”
“I think the course was actually very well organized, and it felt like there was a lot of correspondence between what we learned in lecture, and what we saw in homework and what was in the textbook,” said Ophir Horowitz ’22, who took the course in fall “In general, the textbook was very coherent. It was easy to follow and I think just the fact that because they wrote it, it went in order.”
A more gradual change in MATH 51 over the past several years was the faculty’s application of active learning techniques. This involved the incorporation of iClickers in lecture and prereadings that have students read the textbook before lecture and ask questions about potentially confusing topics.
Additionally, MATH 51 discussion sections have shifted from a lecturebased structure to one in which students receive worksheets and go over problems together.
“We tried very hard to find a model that incorporates features of both [traditional lecture and flippedclassroom class structure] that would allow us to proceed and keep students engaged,” Mazzeo said.
More changes to come
The math department looks to continue improving MATH 51 as it receives feedback from students and continues to adjust the new textbook. In addition, Conrad and Mazzeo said they aim to eventually write textbooks for MATH 52 and
Inspired by mathematics education professor Jo Boaler’s course EDUC “How To Learn Mathematics,” ASSU Senator Melissa Loupeda ’21 put forth a proposal to slow down the course by splitting its material between two quarters.
Loupeda said taking a challenging course like MATH 51 can decrease students’ confidence, especially if they come from a high school without a strong math background. She emphasized the importance of selfconfidence in effective learning.
“If your first experience with math in particular is really negative, then that carries on,” said Loupeda. “Because that class is the basis for your next stats class, next ECON 50 … And if you feel not confident about the material because it was too fast, you’re going to carry that with you.”
While the project may take time, Loupeda’s overall goal is to improve accessibility and confidence, and her project has already been endorsed by the First Generation and LowIncome Partnership (FLIP) as well as the Stanford Students of Color Coalition (SOCC).
Loupeda’s other ideas include creating an experience similar to the Stanford Summer Engineering Academy (SSEA) but for mathematics, creating a preparatory course for MATH 51 or creating an online course to further supplement students. However, this idea doesn’t just apply to MATH 51; Loupeda is working with the Office of the Vice Provost for Teaching and Learning and the Office of Institutional Research to improve accessibility for STEM courses in general, which may possibly lead to sloweddown versions of other courses in the future.
In defense of MATH 51
Buzzing with a sense of renewed enthusiasm and excitement after (high school) senioritis has (hopefully) left us, each freshman is faced with an intimidating choice even before even setting foot on campus. “Which math class, if any at all, should I take?” The placement test is hopelessly inadequate at imbuing us with a sense of confidence in making that choice. Many take the online test with an attitude of resigned compromise (Who wants to sit down for a placement test which spits out evaluations about their academic abilities immediately after high school has ended?) and is hilariously incomprehensive at making those evaluations in the first place.
Nevertheless, invigorated with innocent eagerness to start college, we venture into the depths of Carta and Reddit in hopes of making an informed decision.
Carta is flooded with cautionary tales of the old MATH “Linear Algebra and Differential Calculus of Several Variables.” As one student recounts, “It is very difficult to understand the textbook and moves at a very fast pace if you have no background in linear algebra. Unless you need the class I wouldn’t recommend taking it, but it is doable and office hours help a lot.” Another user gives an equally comprehensive and impactful review, simply reading, “Don’t.” The Daily itself has published an editorial that is titled “Is MATH 51 all you wanted out of your Stanford education?” As one might reasonably predict, it does not sing songs of praise in favor of MATH
These various reviews gesture at the reality that students generally do not take MATH 51 intending to dive head first into intellectual challenge, but rather to fulfill requirements. I am willing to admit that I, too, initially decided to take this class because I felt that my skills in mathematics would only deteriorate if I put it off for another quarter.
As of last year, MATH 51 has undergone a significant revision, employing a new textbook written by Stanford math professors and new methods of increasing engagement. But MATH 51 has continued to retain its horrid reputation from past iterations, and on campus, during International Student Orientation and New Student Orientation, anecdotes of utter horror were swapped when I asked upperclassmen what they thought about MATH
Despite the negative experiences I was recounted, I calmed myself with the thought that I could always “shop and drop,” and thought I might as well attend class for the first few weeks. The arduous process of enrolling in the class only seemed like a premonition for the class itself: before the first day of class, I had to hurriedly arrange an iclicker, download the textbook and give myself a pep talk. I grabbed a seat in lecture, and little did I know at the time, sat down in the class that would soon become one of my favorite classes of the quarter.
The first couple weeks are deceptively straightforward and unrepresentative of the course in general. But even when the course picked up, it was more of a gradual speedup than a sudden jump. The class is structured to help the student at every turn: a preclass questionnaire, administered to survey topics that students find most confusing, tangibly results in changes the course staff makes to how the class is taught.
I have noticed the textbook changes in places where ambiguous explanations have given way to more accessible ones based on popular feedback. Professors are engaging and at times, even crack jokes. They operate with the consideration that this is a hard class and that there is a lot to cover, and do their best to explain things as simply as possible. They are also aware that most students don’t take this class because they love math, but rather because they need to, and don’t spend time unnecessarily expounding on complex proofs.
For me, they do a great job at piquing interest in how the concepts actually function, while also keeping pace with the class. My professor even put up a survey to gather information about how confident students felt with their skills and offered to bring candy in exchange for the completion of the survey. Office hours and tutors are accessible, and during discussion section we are given worksheets that concretize our understanding of the material.
I am by no means saying that the class is easy or requires a lowerthanaverage commitment to work, but the class does a great job at constant evaluation and modifying itself to suit itself to the student’s interest best. MATH 51 brings with it the realization that understanding of these concepts is directly linked to utility in the real world. Homework is sometimes difficult, but always empowers understanding of the concept, and the wide variety of skills that the class develops come together and complement each other at the end.
MATH 51 is notoriously known for not being consistent across quarters, but taking this course in fall quarter, one of the quarters that is considered to have a harder curve, I don’t feel the sense of dread that people have often associated with this class. It is the class that I look most forward to, and I am glad that I took this class with an open mind and a resolve to work hard. To future incoming freshmen, if you feel confident in your foundations in math and feel prepared, consider taking this class. Yes, it’s a lot of work, but it’s also extremely enjoyable.
Textbook stanford math 51
Math Linear Algebra and Differential Calculus of Several Variables, Summer
Section 01 (Class #): MondayFriday pmpm in
Section 02 (Class #): MondayFriday pmpm in
Teaching Staff
Instructor  Sheel Ganatra  Course Assistant  Valentin Buciumas  Course Assistant  Zev Rosengarten 
Office  E  Office  A  Office  N 
ganatra (at) math (dot) stanford (dot) edu  buciumas (at) stanford (dot) edu  zevr (at) math (dot) stanford (dot) edu 
Course Description and Prerequisites
Linear Algebra and Multivariable Calculus are two of the most widely used mathematical tools across all scientific disciplines. This course seeks to develop background in both and highlight the ways in which multivariable calculus can be naturally understood in terms of linear algebra.
By the end of this course, you should be able to:
 Understand notions in linear algera, such as vectors and their spans, matrices and their relationship to linear transformations, subspaces and their bases.
 Solve and analyze systems of linear equations, and relate the solution set to properties of algebraic objects (matrix null space, column space) associated to the system.
 Analyze the rate of change of a multivariable function in coordinate directions via partial derivatives. Aggregate partial derivatives (derivative matrix, gradients) to get local information such as: tangent planes, linear approximation, and directional derivatives.
 Use differential calculus techniques such as critical points and Lagrange multipliers to solve for constrained local and global extrema.
Prerequesites (from the Stanford Course catalogue): Math 21, or 42, or a score of 4 on the BC Advanced Placement exam or 5 on the AB Advanced Placement exam, or consent of instructor. Note that our prerequesite for the above AP exam scores is not strict, so if you believe you have comparable requesite background, you are welcome to enroll.
Textbook and Learning Resources (Office Hours, Tutoring)
The textbook is a special combined edition of Levandosky's Linear Algebra book and parts of Colley's Vector Calculus. (We will use Chapter 2 and 4 from the 4th edition of Colley.) Hardcopy versions of the text should be available at the campus bookstore. An electronic version is also available. If you are interested in one, read these instructions and then go to the publisher's site.
Calculators are neither required nor recommended for Math 51 (and will not be allowed on exams anyway).
Each member of the course staff will hold office hours every week in which you can discuss concepts covered in classes, homework problems, or other classrelated questions. You may attend the office hours of any member of the course staff, and no appointment is ever necessary.
Stanford Summer Tutor Program (STP) offers FREE tutoring and academic skills coaching to students enrolled in Stanford's Summer Quarter; just drop in. Math tutoring is held in the Organic Chemistry Building, and the weekly schedule is posted here.
Here is some advice for succeeding in Math 51, given by Professor Brumfiel (slightly modified to account for the different structure of the class in summer).
Statement from the Registrar concerning students with documented disabilities:
"Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Office of Accessible Education (OAE). Professional staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty dated in the current quarter in which the request is being made. Students should contact the OAE as soon as possible since timely notice is needed to coordinate accommodations. The OAE is located at Salvatierra Walk (phone: )."
Course Grade
The course grade will be based on the following:
 % Homework assignments (assigned and due weekly),
 % Midterm Exam 1 (on Wednesday, July 6 in class),
 % Midterm Exam 2 (on Wednesday, July 27 in class),
 % Final Exam (on Saturday, August 13 from pmpm, same time for both sections).
Homework Assignments
Homeworks will be posted here on an ongoing basis. Please read carefully the Weekly Homework Policy before submitting your homework assignment; it gives important instructions in how to approach the homework assignments and how to write up (for instance, you are required to show all your work for full credit; just answers given will often receive no credit).
HW Submission Instructions:
 Section 01 (pmpm) students: Submit your assignments to Zev Rosengarten, either (a) in person at his office, (b) under his office door, or (c) via email (no photos please! only typed or scanned solution sets).
 Section 02 (pmpm) students: Submit your assignments to Valentin Buciumas, either (a) in person at his office, (b) under his office door, or (c) via email (no photos please! only typed or scanned solution sets).
Late homeworks will not be accepted. In order to accomodate exceptional situations such as serious illness, your lowest homework score will be dropped at the end of the quarter. You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions. The Honor Code applies to this and all other aspects of the course.
As homeworks are completed, solutions (in PDF) will be uploaded here.
Once you pick up your graded HWs, it is your responsibility to look over your assignment (and CourseWork grade), along with the posted solutions, to make sure it is correctly graded. Here are instructions for how to submit a regrade request in case there is any issue with your grade.
Midterm and Final Examinations
All exams for Math 51 are closedbook, closednotes exams, with no calculators or other electronic aids permitted. Furthermore, they are all cumulative (so Midterm 2 will test on everything covered in class to that point, not just everything covered since Midterm 1, and so on). Your overall exam average is the weighted average of all scores on all exams; this average counts % towards the final grade. No scaling or curving is applied to individual exam scores.
A good way to prepare for these exams is to make sure you can solve all of the homework problems, and to solve problems from old exams from previous incarnations of Math 51; see also below.
Once you pick up your graded midterms, it is your responsibility to look over your assignment (and CourseWork grade), along with the posted solutions, to make sure it is correctly graded. Here are instructions for how to submit a regrade request in case there is any issue with your grade.
Midterm exam 1
The first midterm was held in class (the usual room, ) on Wednesday, July 6. The topics ranged through topics covered in class by the end of Friday, July 1.
Here is the exam. Here are solutions to the exam.
Midterm exam 2
The second midterm exam was held in class (the usual room, ) on Wednesday, July 27. The topics ranged through topics covered in class by the end of Friday, July 22.
Here is the exam. Here are solutions to the exam.
Final Exam
The Final Exam was held on Saturday, August 13 from pm  pm.
Here is the exam (with a small typo fixed in 13g  because of this typo, everyone received credit for that part of problem 13). Here are solutions to the exam.
Previous Exams (for practice)
Here is a page with links to old exams given in previous incarnations of Math
Lecture Plan
Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary. L refers to The Levandosky, Linear Algebra part of your text (Part 1) and C refers to the Colley, Vector Calculus part of your text (Part 2).
Note: no recording of lectures is permitted. You are welcome to take pictures of the whiteboard though.
Day  Lecture topics  Book chapters  Remarks 

June 20  Welcome and overview of class. Vectors in R^n and linear combinations.  L1, L2.  
June 21  Linear combinations and spans.  L2.  
June 22  Parametric representations of lines and planes. Linear independence and dependence. A brief mention of the dot product.  L3, L4.  
June 23  Dot products and cross products.  L4.  
June 24  Systems of linear equations. Matrices. Augmented matrices associated to systems of linear equations, row operations and reduced row echelon form (rref).  L5, L6.  
June 27  Matrixvector products (and matrix multiplication). The relationship to systems of linear equations: the augmented system corresponding to [Ab] describes solutions to the equation Ax = b. The null space N(A) of a matrix A.  L7, L8.  See also L15 p. for general matrix multiplication. 
June 28  Homogeneous and inhomogeneous systems of linear equations. The column space C(A) of a matrix A. The relationship between N(A), C(A), and solutions to the system of linear equations Ax = b.  L9.  
June 29  Subspaces of R^n. If A is an m x n matrix, the null space N(A) is a subspace of R^n and the column space C(A) is a subspace of R^m.  L  
June 30  Basis for a subspace. How find the basis of the null space and column space of a matrix A. The dimension of a subspace. Properties of dimension. The ranknullity theorem relating the dimension of the column space (or rank) and the dimension of the null space (or nullity) of a matrix A.  L11, L  
July 1  Linear transformations. Every linear transformation T has an associated matrix A; how to go back and forth. Examples and nonexamples of linear transformations. The kernel and image of a linear transformation.  L13, L  
July 4  No class (holiday)  
July 5  (Guest lecture by Professor Ralph Cohen) More examples of linear transformations. Composition of linear transformations and matrix multiplication.  L14, L  
July 6  Midterm exam 1 (in class).  
July 7  Inverses.  L  
July 8  Determinants  L  
July 11  Systems of coordinates. The change of basis matrix (and its inverse!) as a way of going between different systems of coordinates. Systems of coordinates with respect to orthonormal bases.  L21 (p. ) for systems of coordinates, and L22 (p. ) for orthonormal bases.  
July 12  Systems of coordinates II: the matrix of a linear transformation in different coordinates. Similar matrices.  L  
July 13  Eigenvectors and Eigenvalues I. The notion of an eigenbasis. If a matrix has an eigenbasis then it is similar to a diagonal matrix, meaning it is diagonalizable (and vice versa).  L  
July 14  Eigenvectors and Eigenvalues II. Cases in which we can find an eigenbasis. Symmetric matrices and the Spectral Theorem.  L23, L18 (for transpose), L25  
July 15  Multivariable functions, graphs, and level sets. Contour maps (which are drawings of collections of level sets). Parametric curves.  C + additional notes on parametric curves (handout given in class)  Note that Colley doesn't use the term contour map but you will be expected to know what this term means. 
July 18  Limits, continuity; partial derivatives; differentiability.  C (up to page , not including addendum), C (up to page , not including addendum)  Limits will only appear on the final exam in a minimal way; you should know the formal definition and intuitive meaning, and how to show certain limits do *not* exist, but you will not need to check a limit exists directly from the definition. You should however know the definition of continuity and differentiability in terms of limits. 
July 19  Partial derivatives and higher order derivatives. Clairaut's theorem. Differentiability and applications: tangent planes to graphs, linear approximations of multivariable functions. The gradient vector of a realvalued function.  C (up to page , not including addendum), C (up to page , not including addendum)  
July 20  Tangent lines to images of parametric curves. The matrix of partial derivatives of a general multivariable function. Differentiability and the total dervative linear transformation (whose associated matrix is the matrix of partial derivatives).  C (up to page , not including addendum), C (up to page , not including addendum), paper handout given earlier in class (for finding tangent lines to images of parametric curves)  
July 21  The chain rule. A brief mention of directional derivatives and the gradient.  C (up to page , not including addendum), C (up to page , for the gradient), C (up to page , not including implicit/inverse function theorems)  
July 22  (Guest lecture by Oleg Lazarev) Directional derivatives and the gradient. Application: tangent planes to implicit surfaces, and more generally tangent planes to level sets.  C (up to page , not including implicit/inverse function theorems), C (for definition of gradient)  
July 25  No class.  
July 26  Review session during regular class hours in regular classroom. Students are welcome to come to both classes' review sessions; problems covered will be different!  
July 27  Midterm exam 2 (in class).  
July 28  Quadratic forms.  L26, not including Prop.  Any verification of definiteness of a quadratic form should be either directly from definitions or using Prop. of Levandosky, checking the signs of the eigenvalues of the corresponding matrix. Solutions on exams which use Prop for 2x2 matrices will not receive credit. 
July 29  First and secondorder Taylor approximations for multivariable functions. The Hessian of a realvalued function, which is (in most nice cases) a symmetric matrix.  C (up to page , not including addendum)  
August 1  The first and second derivative tests for finding local extrema (minimum/maximum) of multivariable functions.  C and L26 (to check definiteness of the quadratic form associated to the Hessian, we will use Levandosky Prop. , not the method in Colley p. and not the method in Levadosky Prop.  Any verification of definiteness of the quadratic form associated to the Hessian should be either directly from definitions or using Prop. of Levandosky, checking the signs of the eigenvalues of the corresponding matrix. Solutions on exams which use Prop for 2x2 matrices or Colley p's "principal minors" will not receive credit. 
August 2  Finding global extrema (minima/maxima) of functions. The Extreme Value theorem which ensures that global extrema of continuous functions always exist when the domain is closed and bounded (also called compact ). Using the first and second derivative tests for interior local extrema, and finding boundary local extrema via parametrizing the boundary.  C p. for definitions of a ball and a closed set. C Definition for the definition of a compact set, and Theorem  
August 3  More closed and bounded domains. Finding boundary extrema via parametrizing the boundary , and finding global extrema in examples.  C as above, handout on parametrizing the boundary given two days earlier  
August 4  Lagrange Multipliers with one constraint. Worked Examples.  C, up to page (not including "A Hessian Criterion for Constrained Extrema")  
August 5  Lagrange Multipliers continued. More examples and applications: calculating the distance from a point in R^n to various sets.  C (as above)  
August 8  Lagrange Multipliers with multiple constraints.  C (as above)  
August 9  A conceptual exercise in finding extrema. An application of finding extrema to Least Squares Solutions, in particular for linear regressions.  C (note: C doesn't have any particularly new mathematics, it just applies existing concepts to various problems; so you could think of this section as "applied problem solving exercises involving Colley Chapter 4 concepts").  This class day might only appear on the final exam in an inessential way: specific formulae appearing in C will not need to be memorized for the final exam. If anything from this section appears it will be carefully explained so that all you need to know is the calculus done in C and C 
August 10  An applications of Linear Algebra: Google's PageRank algorithm.  None (though you can search to learn more about this topic online!)  This is an optional lecture which will not appear on the final exam. 
August 11  Review session through both classes.  
August 13  Final exam from 7pmpm in Herrin hall room T (for both classes). 
.
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