- 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
Office hours. Wed 10-12 in room 7210, AP&M. My email is email@example.com 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!)
Section: HW 1
5.1, 5.2, 5.3
Section: HW 2
Section: HW 3
Section: HW 4
Section: HW 5
Section: HW 6
Section: HW 7
Section: HW 8
| Nov 28
Apps. to Physics
Section: HW 9
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):
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:
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
2. 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.
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.
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.)
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
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."
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!)
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)!
4.3 #9 The answers given in the back of the book are swapped around!
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)!
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.
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.
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.
First midterms from S12, S13, F14, W16, W17, F17Solutions for those first midterms(and the original)
Midterm 2Second 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 about integrating on curves and surfacesMarsden and Tromba problem 8.2 #15
Final exams from S12, S13, F14, W16, W17Solutions for the finals