Vector Calculus 20E, Winter 2017

Justin Roberts (Lecture B)

Update: March 19

Added solutions for practice finals and this term's MT2 at the bottom of the page.

My office hours this week will be on Wednesday, 1-4pm.

Lu Yu will have normal office hours Tuesday 4-6pm and extra on Friday from 9-11am in AP&M 2402A.

Note: there won't be any physics on the final!

Who, what, where

Lectures.  MWF 1:00-1:50 in Peterson Hall 110.

Office hours. Mondays and Wednesdays 4-5pm in room 7210, AP&M. My email is and my phone is 534-2649. Questions about grading should go to the TAs, who maintain the spreadsheet. 

Discussion sections/TAs. All sections are on Wednesdays:

B01 - 5pm - AP&M 2301 - Yu Lu
B02 - 6pm - AP&M 2301 - Yu Lu
B03 - 7pm - AP&M 2301 - Yu Lu
B04 - 8pm - AP&M 2301 - Yu Lu
B05 - 6pm - Centre 207 - Amir Babaeian
B06 - 7pm - Centre 207 - Amir Babaeian

Lu Yu ( has office hours
in the calc lab (AP&M B204) on Mondays, 2-4pm and then in AP&M 2402A on Tuesdays, 4-6pm.
Amir Babaeian ( has office hours in AP&M 5720 on Wednesdays, 4-6pm.

Course calendar (as of Jan 9)

Week Monday Tuesday Wednesday Thursday Friday
Jan 9

Jan 11
Section (HW 0)

Jan 13
Jan 16

Jan 18
Section (HW 1)

Jan 20
   5.1, 5.2, 5.3
Jan 23 
   5.4, 5.5

Jan 25
   6.1, 6.2
Section (HW 2)

Jan 27
Jan 30
   4.3, 4.4

Feb 1
Section (HW 3)

Feb 3
   Midterm 1
Feb 6

Feb 8
Section (HW 4)

Feb 10
Feb 13

Feb 15
   7.5, 7.6
Section (HW 5)

Feb 17
Feb 20 
   Presidents Day

Feb 22
Section (HW 6)

Feb 24
   8.1, 8.2
Feb 27

March 1
Section (HW 7)

March 3
   Midterm 2
March 6

March 8
Section (review)

March 10
10 March 13

March 15
Maxwell's Equs
Section (HW 8)

March 17

March 24


Homework. The problems for each week are listed at the bottom of this page. Homework should be delivered to the dropbox in the basement of the AP&M building by 4pm on Thursday - the day after the Wednesday section at which it has been discussed. Late homework will not be accepted. A representative sample of the exercises will be graded; your homework grade will be based on your best five of the eight graded homework assignments.

Exams. There will be two midterms given during the lecture hour, and a final on Friday of exam week - see the calendar below. Please bring blue books and IDs for all exams! You may bring one 8.5x11 inch handwritten sheet of notes 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. There's no "curve" as such, but the grade bands will be adjusted at the end of the term.

    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?!) Last year Jeremy Semko compiled a list of these, and I'll add his comments 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 Jan 19)

    • 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 Jan 26)

    • 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 Feb 2)

    Homework 4 (due Feb 9)


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

    Homework 5 (due Feb 16)

    Homework 6 (due Feb 23)

    Notes (sigh... here we go again...):

    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 March 2)

    Homework 8 (due March 16)

    Homework 9

    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'll upload additional practice exams when we get closer to the midterm and final dates.

    First midterms:

    Spring 2013 first midterm

    Fall 2014 first midterm

    Winter 2016 first midterm

    Solutions for the above three midterms

    Second midterms:





    Review sheet about integrating on curves and surfaces (revised, with fourth page added)

    Solutions for the above exams
    Thanks to Kakeru Imanaka for finding a couple of typos in these solutions:
    1. on 2012 #1 in the calculation using the first parametrisation the "-t^2.t^{3/2}" should be "-t. t^{3/2}". This doesn't affect the answer because I actually evaluated it using the second parametrisation.
    2. On 2013 #2 I appear to have written "pi" for the upper limit of integration in both places where it occurs, when it should be "pi/2", corresponding to integrating over the hemisphere, not the whole sphere. But I appear to have been _thinking_ about it correctly, because I've evaluated the bit in square brackets as if the upper limit were pi/2, and consequently the answer is correct as given.

    Old finals:





    2012 solutions
    2013 solutions
    2014 solutions
    2016 solutions
    2017 MT2 solutions

    Some comments about notation and terminology...

    The notation and terminology of vector calculus is not particularly standardised: different books may use different names and symbols for the same concepts. This is of course a bit annoying, but it's just something you have to get used to. I'll try to put some notes here about this...