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.

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.

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. 

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.

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!)