multivariable calculus

## 课程介绍

This course covers vector and multi-variable calculus. Topics include vectors and matrices, parametric curves, partial derivatives, double and triple integrals, and vector calculus in 2 – and 3 -space. As its name suggests, multivariable calculus is the extension of calculus to more than one variable. That is, in single variable calculus you study functions of a single independent variable
$$y=f(x) .$$
In multivariable calculus we study functions of two or more independent variables, e.g.
$$z=f(x, y) \text { or } w=f(x, y, z)$$
These functions are interesting in their own right, but they are also essential for describing the physical world.

## 参考案例

\begin{aligned} \frac{d}{d x} r^{2} &=\frac{d}{d x}\left(x^{2}+y^{2}\right) \ 0 &=2 x+2 y y^{\prime} \ y^{\prime} &=\frac{-2 x}{2 y}=-\frac{x}{y} \end{aligned}

Problem 1.

Compute $\int e^{x} \cos x d x$

Proof .

With $d v=e^{x} d x$ and $u=\cos x, v=e^{x}$ and $d u=-\sin x d x .$ So,
$$I=\int e^{x} \cos x d x=e^{x} \cos x+\int e^{x} \sin x d x$$
Repeat with $d v=e^{x} d x$ and $u=\sin x .$ We get $v=e^{x}$ and $d u=\cos x d x$. Hence,
$$\int e^{x} \sin x d x=e^{x} \sin x-\int e^{x} \cos x d x$$
Now, insert the second equation into the first to yield
$$I=e^{x} \cos x+\left[e^{x} \sin x-I\right]$$
Solving for $I$ yields
$$\int e^{x} \cos x d x=\frac{1}{2}\left(e^{x} \cos x+e^{x} \sin x\right)$$

Problem 2.

Compute $\int e^{\arcsin x} d x$

Proof .

Let $u=\arcsin x .$ Then, $x=\sin u$, so, $d x=\cos u d u .$ So,
$$\int e^{\arcsin x} d x=\int e^{u} \cos u d u=\frac{1}{2} e^{u}(\sin u+\cos u)$$
by the previous exersice. Since $\cos u=\sqrt{1-x^{2}}$, we obtain
$$\int e^{\arcsin x} d x=\frac{1}{2} e^{\arcsin x}\left(x+\sqrt{1-x^{2}}\right)$$

Problem 3.

Compute $\int \frac{x+1}{\sqrt{1-x^{2}}} d x$.

Proof .

\begin{aligned} \int \frac{x+1}{\sqrt{1-x^{2}}} d x &=\int \frac{x}{\sqrt{1-x^{2}}} d x+\int \frac{d x}{\sqrt{1-x^{2}}} \ &=-\frac{1}{2} \int \frac{d\left(1-x^{2}\right)}{\sqrt{1-x^{2}}}+\arcsin x \ &=-\left(1-x^{2}\right)^{1 / 2}+\arcsin x=\arcsin x-\sqrt{1-x^{2}} . \end{aligned}

Problem 4.

Compute $\int x^{2}(\log x)^{2} d x$

Proof .

If $u=(\log x)^{2}$ and $d v=x^{2} d x$, then, $v=x^{3} / 3$ and $d u=2(\log x) d x / x$. Hence,
$$\int x^{2}(\log x)^{2} d x=\frac{1}{3} x^{3}(\log x)^{2}-\frac{2}{3} \int x^{2} \log x d x .$$
If $u=\log x$ and $d v=x^{2} d x$, then, $v=x^{3} / 3$ and $d u=d x / x$. Hence,
$$\int x^{2} \log x d x=\frac{1}{3} x^{3} \log x-\frac{1}{3} \int x^{2} d x=\frac{1}{3} x^{3} \log x-\frac{1}{9} x^{3} .$$
Now, insert the second integral into the first equation, and rearrange to obtain
$$\int x^{2}(\log x)^{2} d x=\frac{x^{3}}{27}\left(9 \log ^{2} x-6 \log x+2\right)$$

Problem 5.

Compute $\int \sqrt{1-e^{-2 x}} d x$.

Proof .

Take $u=\sqrt{1-e^{-2 x}}$. Then, $u^{2}=1-e^{-2 x}$, so, $2 u d u=2 e^{-2 x} d x=$
$2\left(1-u^{2}\right) d x$. Hence,
$$\int \sqrt{1-e^{-2 x}} d x=\int \frac{u^{2} d u}{1-u^{2}}=\int \frac{d u}{1-u^{2}}-u=\frac{1}{2} \log \left(\frac{1+u}{1-u}\right)-u$$
which simplifies to
$$\int \sqrt{1-e^{-2 x}} d x=\log \left(1+\sqrt{1-e^{-2 x}}\right)+x-\sqrt{1-e^{-2 x}} .$$

Problem 6.

Show that
$$\log 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots$$
following the procedure discussed above for the Leibnitz series.

Proof .

3.6.17 Let $s_{n}(x)$ denote the $n$ th partial sum in (3.6.3). Then, if $0<x<1$, by the Leibnitz test,
$$s_{2 n}(x) \leq \log (1+x) \leq s_{2 n-1}(x), \quad n \geq 1$$
In this last inequality, the number of terms in the partial sums is finite. Letting $x \nearrow 1$, we obtain
$$s_{2 n}(1) \leq \log 2 \leq s_{2 n-1}(1), \quad n \geq 1$$
Now, let $n \nearrow \infty$

# MAT201 Multivariable Calculus

Announcement:

This summer, Princeton  will be offering a small suite of  STEM courses this summer (MAT 103, 104, 201, 202 and PHY 110) for students who need these specific courses for degree or concentration progress. Students accepted to the program will not be charged tuition for the courses; students on financial aid will also receive an allotted living expense budget.Please visit: https://forms.gle/52iszwnhbMGLXmCe7 for more information and to apply. The deadline for application is April 7, 2021.

Schedule: MW at 8:30 AM, at 11 AM, at 1:30 PM, and at 3:00 PM (Fall only), Friday precepts; Fall and Spring. Optional review sessions are very helpful, but they are scheduled only after classes begin.

Brief Course Description: Third semester of the standard 3-semester calculus sequence. Gives a thorough introduction to multivariable calculus and mathematical methods needed to understand real world problems involving quantities changing over time in two and three dimensions. Topics include vectors, lines, planes, curves, and surfaces in 3-space; limits, continuity, and differentiation of multivariable functions; gradient, chain rule, linear approximation, optimization of multivariable functions; double and triple integrals in different coordinate systems; vector fields and vector calculus in 2- and 3-space, line integrals, flux integrals, and integration theorems generalizing the Fundamental Theorem of Calculus (Green’s theorem, Stokes’ theorem and Gauss’s theorem, also known as the divergence theorem).

Why take this course? It provides important mathematical foundations for advanced studies in life sciences, physical sciences, social sciences, computer science and engineering, building vocabulary and tools to describe and understand phenomena in the natural world, and improving analytic and problem-solving skills valuable in many disciplines.

Who takes this course? Most students in this course are first- or second-year students who consider majoring in one of the sciences or engineering.   More mathematically inclined economics majors will take this course along with 202 (instead of 175).  Although it is not a prerequisite, many students in the course will have had a more basic multivariable calculus course in high school.

Prerequisites:  104 or equivalent. A solid knowledge of single-variable calculus and precalculus is essential: how to analyze and graph functions, how to compute and interpret derivatives, how to interpret, set up, and calculate definite integrals with speed and accuracy. The very fast pace means that solid grasp of the prerequisite material is especially important.

### FAQ:

• You already took multivariable in high school or at a local college, so you want to place out of MAT201.  Most students in 201 have some multivariable calculus and/or linear algebra before, but very rarely with the same depth and thoroughness.  Most students will find that the sample problems are much more sophisticated than problems they have encountered in high school. You may want to consider 203, which requires an intense commitment and interest in deeper understanding of the subject and its applications.
• How much work is this course?  This course generalizes both 103 and 104 to higher dimensions. To accomplish this in a single semester, 201 keeps an extremely fast pace and requires that students be fluent in all the main ideas and techniques from single variable calculus. All math courses require a steady time commitment, but this one is particularly demanding. We expect that the weekly problem sets will take at least 3 hours to complete, although this can vary quite a lot for individual students, and this is only the beginning! The exams often include more challenging problems which require complex analytic skills. Learning to think independently and creatively in a mathematical setting takes time and lots of practice.  To do well on math exams, you need to work through a lot of extra problems from past exams and quizzes. All in all, you should be ready to spend at least 10 hours per week working outside of class.
• You want to take both MAT201 and MAT202 in the same semester to get your engineering prerequisites over with. Is this possible? It is not impossible but this is not a good idea for most students. The work load and pace of 201 is particularly overwhelming for many students, and adjusting to the abstraction in 202 is also a big step. Doing both these demanding courses in a single semester should not be undertaken lightly. We would not recommend it for anyone, but it is especially inadvisable for anyone who got less than an A in 104
• You think MAT201 is too hard after looking at the sample problems or attending the first couple classes.  MAT201 is a cumulative course — the topics build upon themselves throughout the semester.  If you are having difficulty at the beginning of the semester and you want to major in engineering, you should consider switching to 104 to get a thorough foundation for 201.  You should switch as soon as possible, as there is no overlap between 201 and 104 early in the semester, and it will be very difficult to catch up and do well in 104.   You may also consider a switch into 175 if you are absolutely sure that you don’t need to take any further math courses and your program will allow this substitution. (175 is not enough for BSE degrees.)
• You think MAT201 is not challenging enough.  Wait till you have had a quiz, which usually occurs in the 3rd week.  Homework problems are often quite routine compared to exam questions.  Try some old quiz problems, but don’t just read the questions and solutions. Instead, see if you can produce correct solutions to most of the problems in the allotted time.  If you can do well on old exams, then you may consider taking 203 or 215 instead.  201 and 203 cover essentially the same topics, so it is quite easy to switch between the two courses in the first few weeks.  There is no overlap between 201 and 215, so this course switch should take place as early as possible.
• What kind of calculator do I need for this class? We don’t use calculators in Princeton calculus classes. If you would normally rely on your calculator for graphing functions, solving equations and computing values of trigonometric functions, you should be very conservative in choosing your first math course.
• Are you serious about no calculators? Why? Calculators can be useful, but these courses want to teach students how to think independently in a quantitative setting and calculators can function as substitute for thinking at the beginning. Students need to learn the basic vocabulary and grammar of mathematics so that they can recognize patterns and common features by working through simple well-chosen examples. For instance, a program like ‘Google translate’ can be helpful to a person with basic knowledge of a language to decipher a complicated sentence or even to write a correct one, but without a good foundation to refine and direct its application, the results of blindly applying this useful technological tool can be wildly off the mark.

File AttachmentsMAT 201 Sample Problems

# MATH 210 Calculus III: Multivariable Calculus with Linear Algebra

This course includes topics in multi-variate and vector calculus, including vectors in a plane and in space, vector-valued functions, functions of several variables, partial derivatives, surfaces and space curves, multiple integrals, cylindrical and spherical coordinates, applications to area and volume, vector fields, line integrals, and Green’s Theorem. Topics in linear algebra include matrices, elementary row operations, systems of linear equations, augmented matrices, Gaussian and Gauss-Jordan elimination, inverse matrices, matrix algebra, eigenvalues and eigenvectors, determinants, vector spaces, subspaces, and basic vectors.
NOTE:A graphing calculator may be required and will be discussed in class.
Pre-requisite(s):MATH 190 Calculus II.
Terms Offered:Fall, Spring, Summer
Offered Distance Learning:Yes
Liberal Arts and Sciences Designation: Mathematics
SUNY General Education Designation(s): Mathematics
Credits:4
Contact Hours:
Lecture: 4

# Courses

## General Information:

01:640:251 Multivariable Calculus (4 Credits)

This course covers multi-variable and vector calculus. Topics include analytic geometry of three dimensions, partial derivatives, optimization techniques, multiple integrals, vectors in Euclidean space, and vector analysis.

Prerequisite: CALC2 (Math 152, 154, or 192).

## Textbook:

Thomas’ CALCULUS Early Transcendentals, 14/e,  by Joel Hass, Christopher Heil, and Maurice Weir. Pearson Education. ISBN: 9780134639543

MyMathLab access with etext: ISBN: 9780135901403

MyMathLab access can be purchased directly from Pearson.

## Information for Instructors

Information for instructors

### Lecture Topics & textbook homework for Math 251

This is a very rapid plan of study. A great deal of energy and determination will be needed to keep up with it. Modifications may be necessary. Periodic assignments (Maple labs, workshops, etc.) may be due at times, and additional problems may be suggested.

The text is the 14th edition of Thomas’ CALCULUS Early Transcendentals, by Joel Hass, Christopher Heil, and Maurice Weir. Pearson Education. ISBN: 9780134639543

## Exam Protocols

The exams will be administered on MyLab  on Mondays. Each midterm will be 50 minutes long, but the time window for accessing and completing the exam will be 9:00pm-11pm. The exams are open notes. Tablets, calculators, phones, or other electronic devices is not allowed during the exams.

Midterm 1:  Monday, February 15

Midterm 2:  Monday, March 8

Midterm 3:  Monday, April 5

Midterm 4:  Monday, April 26

Final Exam: Thursday, May 6, 4-7pm

## MyLab – Online Homework

Students are required to purchase access to MyLab Math to complete the online homework, and possibly quizzes and exams. The MyLab assignments are similar to the exercises in the official list of HW exercises. (The official HW exercises are not handed in for grading but instead form a significant, but not exhaustive, portion of your study guide for the course.) Each assignment will have a specific due date set by the professor, and these assignments must be completed online.

How to use MyLab properly:
If you take shortcuts like trying to find answers to MyLab problems from various “homework help” web services without solving all of the problems yourself in their entirety, then your performance on exams will suffer. Instead, use the built-in help tools within MyLab. This online homework exists primarily to give you feedback on your ability to calculate correct answers at early stages of the learning process. The homework is not intended to measure your mastery of the material; only the midterm exams and final exam measure mastery. Without doing well on the exams, it is impossible to pass the course, even with a perfect score on the homework. So be sure to take full advantage of MyLab to get as much feedback as possible on your problem-solving skills.

#### Getting started with MyLab:

• You will be able to access your MyLab course directly through your Canvas site for Math 251.
• In your Canvas site, navigate to MyLab and Mastering and follow the on-screen instructions to create a Pearson account (or link an existing Pearson account) to your Canvas account.
• You will automatically be enrolled in the MyLab course.
• If you switch to a different section of Math 251, you can enroll in your new section’s MyLab course by following these same instructions.

#### Student support for MyLab:

• System Requirements
MyLab works on a series of pop-up screens. You MUST enable pop-ups when working in MyLab. For help on how to do this, as well as make sure your browser is up to date, use the link above.
• How to Use MyLab on a Mobile Device
This video shows you how to set up your mobile device with any necessary browser add-ons and apps to use MyLab properly.
• Pearson Support Database
Use the above link to search Pearson’s database for support topics (e.g., resetting password).
• Contact Support
Use the above link to contact technical support. Fill out the required form and you will be immediately connected to a support agent based on your issue.
• Pearson sales representative: [email protected]
Melissa Blum is our Pearson Sales representative. If you are having technical issues, please first contact Technical Support. If you are still having issues after contacting Technical Support, please email Melissa Blum with the Incident Number you received from working with Technical Support. You must have an Incident Number for Melissa to be able to help.

• MyLab is an interactive, online homework system. The assignments follow the lecture topics.
• Questions are algorithmically generated to give each student their own random versions of the questions.
• After entering an incorrect answer, students are given helpful feedback and hints. Most exercises will also include learning aids, such as guided solutions and sample problems.
• You have three attempts to get an answer correct. If you use all three attempts, you will be told the correct answer and given a new, random version of the same problem. There is no limit to the number of versions of a particular problem you can be given. So you are strongly encouraged to work on a problem until you get the correct answer. There is no penalty for the number of attempts taken.

## Matlab Computer Labs

We will have some computer labs to show how technology facilitates the learning of certain calculus concepts. These were primarily designed for Matlab, since you can use the Rutgers license to download it for free. However, you may also use Maple or Mathematica to work on these assignmets. More information about them will be given during at the beginning of the Spring Semester.

## Contacts

[email protected]

[email protected]

Professor Michael Beals

Full Departmental Directory

## Disclaimer: Posted for informational purposes only

This material is posted by the faculty of the Mathematics Department at Rutgers New Brunswick for informational purposes. While we try to maintain it, information may not be current or may not apply to individual sections. The authority for content, textbook, syllabus, and grading policy lies with the current instructor.Information posted prior to the beginning of the semester is frequently tentative, or based on previous semesters. Textbooks should not be purchased until confirmed with the instructor. For generally reliable textbook information—with the exception of sections with an alphabetic code like H1 or T1, and topics courses (197,395,495)—see the textbook list.