Vector Calculus 20E, Fall 2019

Justin Roberts (Lecture A)

Update: Dec 2

- Uploaded midterm 2 solutions

- Uploaded practice finals and solutions. (If you need even more practice (!), there are extra revision problems from the book listed after the last homework.)

- Added an extra page about Stokes/Gauss to the review sheet.

- Uploaded some solutions/comments for Marsden and Tromba problem 8.2 #15

Who, what, where

Lectures.  MWF 2:00-2:50 in Ledden Hall.

Office hours. Wed 10-12 in room 7210, AP&M. My email is and my phone is 534-2649. Questions about grading should go to your TAs, who maintain the spreadsheet. 

Discussion sections/TAs. All sections are on Thursdays. (Unfortunately the section numbers, times and rooms are not at all easy to remember!)

A01 - 12pm - AP&M 2301 - Xin
A02 - 1pm   - AP&M 2301 -
A03 - 8pm   - AP&M 7321 - Frank

A04 - 9pm   - AP&M 7321 - Frank
A05 - 6pm   - AP&M 2402 - Xin
A06 - 7pm   - AP&M 7321 - Qingyuan

Their emails and office hours are as follows:

Xin Tong                      Wed 4-6 in AP&M 5829
Shan (Frank) Jiang           Mon 3-5 in AP&M 2313
Qingyuan Chen           Tues 1-2 and Fri 12-1 in AP&M 6446

OASIS: additional help with this class is available in OASIS tutoring workshops. See their website for details.

Course calendar (as of Sept 26)

Week Monday Tuesday Wednesday Thursday Friday

Sept 27
Sept 30

Oct 2
Oct 3
Section: HW 1
Oct 4
Oct 7
   5.1, 5.2, 5.3

Oct 9
   5.4, 5.5
Oct 10
Section: HW 2
Oct 11
 6.1, 6.2
Oct 14

Oct 16
   4.3, 4.4
Oct 17
Section: HW 3
Oct 18
Oct 21

Oct 23
   Midterm 1
Oct 24
Section: HW 4
Oct 25
Oct 28

Oct 30
Oct 31
Section: HW 5
Nov 1
Nov 4

Nov 6
Nov 7
Section: HW 6
Nov 8
 8.1, 8.2
Nov 11
  Veterans' Day

Nov 13
Nov 14
Section: HW 7
Nov 15
Nov 18
   Midterm 2

Nov 20
Nov 21
Section: HW 8
Nov 22
Nov 25

Nov 27
Nov 28
Nov 29
10 Dec 2

Dec 4
Apps. to Physics
Dec 5
Section: HW 9
Dec 6

Dec 11
   Final 3-6pm


Homework. The problems for each week are listed at the bottom of this page. Homework should be delivered to the dropboxes in the basement of the AP&M building by
4pm on Friday after the Thursday discussion section listed above. (The blue boxes in the chart are labelled with which HW will be discussed on which day.) This gives you a bit of time to think further about the problems after the section before handing work in. Late homework will not be accepted. A representative sample of the exercises will be graded; your homework grade will be based on your best six of eight graded homework assignments.

Exams. There will be two midterms given during the lecture hour, and a final on Wednesday of exam week, as shown above. Please bring blue books and IDs for all exams! You may bring one 8.5x11 inch handwritten sheet of notes (front and back) with you to each exam; no other notes - and no calculators - will be allowed. There will be no makeup exams. It's your responsibility to avoid a conflict with other final exams; you should not enroll in this class unless you can take the final at its scheduled time. 

Grades. Your grade will be based on the better of the following two weighted averages (in particular, the second one will automatically apply if you miss one of the midterms):

  • 15% Homework,  20% First midterm,  20% Second midterm,  45% Final.
  • 15% Homework,  20% Best midterm,  65% Final.
  • In addition, you must pass the final in order to pass the course.

    "The curve". There is no running "curve", as such. I can only fix the grade bands at the end of the term because before that I don't have enough data to correctly compensate for the difficulty of the exams, strictness of marking, and so on. Since problems in vector calculus usually involve several stages of calculation, and the integrals which arise are often relatively tricky, it's harder to get a perfect score in this course than in other 20-series courses: in previous years it has generally turned out that about 85% was enough for an A and about 70% for a B.  

    Academic dishonesty. Cheating is considered a serious offense at UCSD. Students caught cheating may be suspended or expelled from the university.


    Textbook. Vector Calculus, sixth edition, by Jerrold E. Marsden and Anthony J. Tromba; published by W. H. Freeman, 2011. I have mixed feelings about this book - here are some notes and warnings:

    1. If you have a previous edition than the sixth, the actual text isn't much different, so as far as reading the book goes, everything is fine. Unfortunately though the problems get renumbered in every new edition, so you'll have to borrow a friend's sixth to get the correct homework questions.

    There are some rather infuriating typos in the book, especially in the answers at the back. (You'd think that by the sixth edition they'd have got it right, wouldn't you?!) Comments about these will be added to the list of HW problems below.

    3. In many of the problems in this book, once you've done the "20E" (vector calculus) part of the question, you end up with the "20B" part - having to evaluate an integral. This term's course is not meant to be all about doing integrals, but at the same time you mustn't forget how to do them! There are certain types which come up repeatedly in vector calculus (for example integrals of powers of sin and cos, which often appear when using spherical coordinates) and even if you find these frightening to start with, you'll be used to them by the end of the course. Nevertheless, occasionally this book leads you to a genuinely hard integral... if you really can't get the thing to come out by yourself, don't waste too much time on it - just use some software and move on. In my exams I'll make sure that only "reasonable" integrals will be needed. 

    4. Annoyingly, there are lots of different systems of terminology and notation used in vector calculus - if you look at different books you will find different names and symbols. That means that to read papers and references involving vector calculus you have to be comfortable, in principle, with all of them! Some of the terms and symbols used in this book strike me as confusing, and I won't necessarily use the same ones, but I'll try to maintain somewhere on this webpage a list of these alternatives. You can use whichever conventions you prefer. 

    Course outline

    In 20C you learned about functions of several variables and their partial derivatives. This course continues directly on from 20C (I have no idea why it's called 20E!) and concerns calculus primarily in three dimensions. The main ideas all originated from the 19th century study of electromagnetism, and the culmination of the course is seeing how to combine various simple experimental observations about EM fields to arrive at Maxwell's equations, the partial differential equations governing EM theory. The language of vector calculus gives us powerful methods for writing and working with these equations in a way that doesn't depend on what coordinate system we use and lets us understand the intrinsic geometrical meaning of the various terms. The same techniques are incredibly useful in all parts of the physical sciences.

    A. Revision of prerequisites from 20C. (I won't lecture on this, but check you understand it by doing HW 0 (not for credit) below.)

    1.1. Vectors in 2d and 3d
    1.2. Scalar product
    1.3. Cross product
    1.4. Cylindrical and spherical coordinates

    2.1. Scalar- and vector-valued multivariable functions, graphs and level sets
    2.2. Limits and continuity
    2.4. Paths
    2.6. Gradient and directional derivative
    3.1. Second and higher derivatives

    4.1. Acceleration
    4.2. Arc length of a curve

    B. Basic stuff about differentiation - mostly revision, but very important!

    2.3. The derivative of a function of several variables.
    2.5. Properties of the derivative

    3.2. Taylor series for multivariable functions

    C. Multiple integrals - partly revision

    5.1. Multiple integrals
    5.2. Double integrals over rectangles
    5.3. Double integrals over more complicated shapes. (Parametrising such shapes)
    5.4. The trick of switching the order of integration
    5.5. Triple integrals over funny-shaped regions.

    6.1. Maps between Euclidean spaces; changes of coordinate system
    6.2. The change of variables formula for integrals (multivariable "integration by substitution")

    D. Vector fields

    4.3. Vector fields and their flow lines
    4.4. Divergence and curl: in 3d space, the only two geometrically-meaningful kinds of derivative other than gradient

    E. Four new types of integrals

    7.1. Integral of F.ds (e.g. work done by force as object moves along a path)
    7.2. Integral of f ds (e.g. for computing average value of f over a curve)
    7.3. Parametrisation of surfaces
    7.4. Surface area
    7.5. Integral of f dA (e.g. for computing average value of f over a surface)
    7.6. Integral of  F.dA (e.g for computing flux of a vector field through a surface)

    F. The fundamental theorems of calculus

    8.1. Green's theorem (FTC for curl in 2d)
    8.2. Stokes's theorem (FTC for curl in 3d)
    8.3. Conservative fields (idea of scalar and vector potential)
    8.4. Gauss's theorem (FTC for divergence in 3d)
    x.x. Maxwell's equations and other physical applications (in the sixth edition of the book, this section has been removed, but it's really the point of the whole course!)

    Homework problems

    Homework 0 (review of 20C - not to be turned in!)

    • Section 1.1: (pg 18) 1, 4, 7, 11, 17
    • Section 1.2: (pg 29) 3, 7, 12, 22
    • Section 1.3: (pg 49) 2b, 5, 11, 15ad, 16b, 30
    • Section 1.4: (pg 58) 1, 3ab, 9, 10, 11
    • Section 2.1: (pg 85) 2, 9, 10b, 30, 40
    • Section 2.2: (pg 103) 2, 6, 9c, 10b, 16
    • Section 2.4: (pg 123) 1, 3, 9, 14, 17
    • Section 2.6: (pg 142) 2b, 3b, 9b, 10c, 20
    • Section 3.1: (pg 156) 2, 9, 10, 25
    • Section 4.1: (pg 227) 2, 5, 11, 13, 19
    • Section 4.2: (pg 234) 3, 6, 7, 9, 13

    Homework 1 (due Friday Oct 4)

    • Section 2.3: (pg 115) 5, 6, 9, 10, 12, 19, 20, 28 
    • Section 2.5: (pg 132) 6, 8, 11, 14, 20, 32, 34

    2.3 #20 This question doesn't make sense! It should say

    "Consider, for each function, the level set which passes through the point (1,0,1) - in other words, the surface in R^3 defined by the equation f(x,y,z)=c where c=f(1,0,1). Find the equation of the tangent plane to this surface at the point (1,0,1)."

    Hint: if we move in R^3 from (1,0,1) to the nearby point (1,0,1)+h (for some small vector h), the linear approximation tells us that f changes by Df (evaluated at (1,0,1)) multiplied by h. If we want f to _stay constant_ - that is, h is moving us _along_ (=in a direction tangent to) the surface, we need this change to be 0.

    2.3. #28 It's not clear when they say "f: R^n -> R^m is a linear map" whether they mean in the "geometrical sense" (having a flat graph, which means being of the form f(x) = Ax+b, where A is an mxn matrix and b a fixed m-vector) or in the "linear algebra sense" (which means satisfying the laws f(x_1+x_2) = f(x_1) + f(x_2) and f(lambda.x) = lambda.f(x), and means being of the form f(x) = Ax without the b). I think they actually intended the linear algebra sense, but you should be able to work out the derivative in both cases and then see what is special about the second...

    2.5 #8 Use the matrix form of the chain rule to do this. I know you can also do it by direct substitution, but it's important to learn how to apply the rule. (You have to do essentially the same calculations either way, but the matrix rule organises them more clearly and becomes more and more useful as the functions get more complicated or the numbers of variables increase.)

    2.5 #20 This is a ludicrously badly-written question - I mean that without the hint it makes absolutely no sense at all. It should probably say something like this:

    "Consider three variables x,y,z which are related by an implicit equation F(x,y,z) = 0. In principle you ought to be able to solve for each variable and write it as a function of the other two, obtaining functions x=f(y,z), y=g(x,z), z=h(x,y), though in practice it's probably impossible for us to do this using explicit formulae. (Consider for example, something nasty like F(x,y,z) = x^4  + y^4 + z^4 + xyz - 1! If you just consider that F(x,y,z)=0 describes a surface in R^3 , you ought to be able to see why functions f(y,z) etc ought to exist, whether or not we know how to write them down explicitly.)

    By differentiating the three formulae of the form F(f(y,z),y,z)=0, show that (dg/dx)(dh/dy)(df/dz)= -1."

    Homework 2 (due Friday Oct 11)

    • Section 3.2: (pg 165) 3, 4, 6
    • Section 5.1: (pg 269) 3ac, 7, 11, 14
    • Section 5.2: (pg 282) 1d, 2c, 7, 8, 9, 17
    • Section 5.3: (pg 288) 4ad, 7, 8, 11, 15
    • Section 5.4: (pg 293) 3ac, 4ac, 7, 10, 14, 15

    5.2 #7 The roof is just like any normal roof - it is part of a plane, but it is inclined at an angle. (Describing it as a "flat roof" is just stupid!)

    Homework 3 (due Friday Oct 18)

    6.2 #3 The answer given in the back of the book is wrong, should just be pi.(e-1) without the half (thanks to Kevin Nguyen for spotting this)!

    Homework 4 (due Friday Oct 25)

    4.3 #9 The answers given in the back of the book are swapped around!

    Homework 5 (due Friday Nov 1)

    7.1#7 The question has a typo: z=x^3 has to be replaced with z=x^2 (otherwise the points listed are not on the surface)!

    Homework 6 (due Friday Nov 8)

    7.4 #5 The answer to (a) is wrong; it has been used when doing (c), resulting in an easier integral and the wrong answer!

    7.5 # 19 Wrong answer at the back, should be 17/2

    7.6 # 3 Answers are wrong, should be 54pi and 108pi.

    Homework 7 (due Friday Nov 15)

    Homework 8 (due Friday Nov 22)

    Homework 9 (nominally due Friday Dec 6 - but won't be marked, so don't hand this in!)

    Revision problems (for practice before final)

    These are some suggested problems from the end-of-chapter review sets, if you need extra practice problems, but the questions from the old finals below will obviously be the most useful things to work on.

    Some old exams

    Here are some old exams for you to look at. They are meant to reassure you that even though the HW contains some "theoretical/explore this concept" type questions, the exams will only contain standard calculational problems.

    I recommend not looking at the solutions until you've done the exams, and especially saving the last couple for "real-time" practice exams.

    Midterm 1

    First midterms from S12, S13, F14, W16, W17, F17

    Solutions for those first midterms(and the original)

    - My answer on 2012#2 is wrong - the substitution should change the limits so that u goes from 4 to 9, not 0 to 5! D'oh!

    Midterm 2

    Second midterms from S12, S13, F14, W16, W17, F17

    - in S12 and F14 I was using the book's convention of writing "dS" instead of dA. I wish I hadn't!

    - in S12#4 the "dS" is a typo, it should be a lower-case "ds"!

    - F14#3 was too hard! I wrote a lot about it in the solutions.

    Solutions for those second midterms

    Solutions for this term's second midterm (and the exam itself)

    Review sheet

    Review sheet about integrating on curves and surfaces

    Marsden and Tromba problem 8.2 #15


    Final exams from S12, S13, F14, W16, W17

    Solutions for the finals

    - The TA who wrote the solutions did not use Green's theorem on 2012#1, but I would have done!

    - On 2012#6 the TA has made two mistakes! The stretch factor should only have sin(phi) in it, NOT sin^2(phi); and the upper limit for phi should be pi NOT pi/2. The correct answer is 12pi/5.