Robert B. Kusner

Director of the Center

(This page is like a cathedral: always under reconstruction ;-)

Advanced Multivariate Calculus (Math 425.2) Fall 2018

(While under construction, please be patient: more details will follow during the semester.)


After our first class on Wednesday 5 September, as Rob biked into town (to testify on the need for restoring some recently cut PVTA routes serving students and non-students alike), he and his bike collided with a truck. His bike is OK, but Rob suffered (lots of bruises and abrasions, a fractured left tibia, and a deep laceration to his left shin requiring 6 sutures); he's back (not missing a single class, but on crutches with a knee brace for a while)....

Time and Place:

Mondays & Wednesdays 2:30-3:45 in LGRT 121 (just south of Lederle Tower entrance, near the Physical Sciences Library)

Office Hours:

After class (3:45-5:00) Mondays, starting in LGRT 121 and migrating up to LGRT 1435G or 1535 (and other times TBA), but in any event, please make an appointment via email:


Multivariable Calculus and Linear Algebra

Recommended Text:

J. Marsden and A. Tromba, Vector Calculus [an earlier edition is fine and may be less expensive: W. H. Freeman, Fifth Edition edition (2003) ISBN-10: 0716749920, ISBN-13: 978-0716749929; or Sixth Edition (2012) ISBN-10: 1-4292-9411-6, ISBN-13: 978-1-4292-9411-9]

H. M. Schey, div, grad, curl and all that [ISBN-13: 978-0393925166; ISBN-10: 039 3925161]

M. Spivak, Calculus on Manifolds [ISBN-13: 978-0805390216; ISBN-10: 0805390219]


This course covers the basics of differential and integral calculus in many variables: differentiability, directional and partial derivatives and gradient of functions; critical points without or with constraints (Lagrange-multipliers/tangential-gradient) and the Hessian; vector fields and differential forms; divergence, curl and exterior derivative; line- and surface-integrals; the fundamental theorem of calculus (Gauss/Green/Stokes/Thomson). If time and taste permit, topics from physics (fluids and electromagnetism) and differential geometry (curves and surfaces in space) may also be explored.


Course grades will be based in roughly equal parts on the homework, on in-class midterm/quizzes, and on the final exam.

Informal Notes: These are for the first part of the course - still under construction with plenty of little errors (e.g. on page 12, the integrand should involve the 3/4-power of (1-y^2), not the 3/2-power; if you find more, please email me :-).

Topic and Homework Schedule (please finish and submit within 9 days of the date at the left; if you work together in small groups (2 to 4 is optimal), which is strongly encouraged, be good scholars and name your collaborators):

09/03: Locate and try to classify (as maxima, minima, saddles...) all the critical points on the topographic map of the Norwottuck region that I shared in class. A more interesting problem is to locate some (rather long) contour that forms a simple loop (closed curve which does not self-intersect) on the map, and count the numbers of maxima, minima and saddles on the region inside; do this for several such loops (using at least one other map like this one of the Pelham hills) and see if you can discover a relation for these numbers. Of course, if the weather is good, you may also use these maps to ... Go Take a Hike (and watch out for trucks if you bike)!

Maxwell (of the eponymous electromagnetic equations and the thermodynamics daemon) was one of the earlier folks (Moebius and Morse being others) to study this critical points count carefully: see On Hills and Dales.
Much more on this can be found in Milnor's monograph Morse Theory and in the (rather quaint: the math starts about 24 minutes in, after a lengthy tour of 1960s Princeton and the Institute for Advanced Study) film Pits, Peaks, and Passes featuring the (elderly) Marston Morse himself! (It may be more fun to start with Part II.)

09/10: Derivatives, Critical Points and Convexity.
Solutions by Riya Patel (page 1), (page 2), (page 3), (page 4); and note that in Problem 2, there are actually infinitely many critical points of the same types as those indicated, but translated in a doubly-periodic way (like a checkerboard) by integer multiples of 2π in the x and y directions.

09/17: Integrals, Volumes and Centers of Mass.
Solutions by Jeffrey Liang (page 1), (page 2), Jessica Lam, and Julie Doty.

09/24: Gradient, Divergence and Laplacian in n-space
[For Problem 2, the behavior at the origin - the minimum of r - is singular: while the "average" of grad(r) may be 0 there, this vectorfield is not well-defined because r is not differentiable (it fails to have a linear approximation - its graph has a tangent "cone" rather than a tangent plane) at the origin; the set of limit values for grad(r), as r tends to 0, is the whole (n-1)-sphere of unit vectors in R^n. This singularity at r=0 may explain (in case n=2) why cooked spaghetti is often soft on the outside, but al dente in the middle!]
Solutions by KD (page 1), (page 2), (page 3), (page 4)

10/01: Homework Hiatus! The HW above is now due Wednesday 10 October!!!
The Open University has made some charming videos on grad, div, and curl. (The Laplace and Poisson equations, describing steady-state heat distribution, make a cameo appearance in the second video.)

10/08: Curling Vectorfields on R^3
[For Problem 4, it is nice that the algebraic formula for iterated cross product A × (B × C)=B(A•C)-C(A•B) works, but one must use care to justify this, and it can be misleading when the Leibniz/product rule comes into play, e.g. for expanding curl(v × w).]
Solutions by DE, HP (page 1), (page 2), (page 3), AH (page 1), (page 2), (page 3) and KD(page 1), (page 2).

10/15: Practice with Path Integrals
Solutions by TB, (page 2) and KD

10/22: Mini-Midterm Homework Hiatus!!!!!
How about a midterm exam sometime soon?

We've have converged to holding a Mini-Midterm in class on Wednesday 24 October.
No make-ups, but please let me know by email before Monday 15 October if you have a special situation that we might accommodate.

Alert: Material & schedule below may still be revised quite a bit!

[Please begin the reading below and get started with the next problem sets, because things are about to accelerate into the wild blue yonder!!!]

The last part of the course concerns differential and integral calculus with forms. We'll soon do a bit of this with 1-forms and path integration, but please also study Spivak chapter 4, especially pages 86-108. In particular, try to understand "pullback" - which is the forms version of the "change of variables" formulas we briefly discussed this month which you may have seen in M&T 6. Also, try to understand the notion of a "singular k-chain" over which we'll integrate a k-form - this generalizes a path integral to higher dimensions and is the natural way to set up the Fundamental "Stokes" Theorem of Calculus (FToC).]

10/29: Fun with Forms
Reminder: as usual, this is due 9 days later, on Wednesday 7 November!
[Please see M&T section 8.6 or Spivak for more 'fun' with forms; you may also try these older Fun with Forms problems, but no need to turn them in - there are a few "write-os" which I hope you can figure out.]
Solutions by KD, (page 2); by WYW, (page 2), (page 3); and by AH, (page 2), (page 3) .

11/05: Practice with Differential Forms
Domains and Forms in n-Space
[These sets (along with the following hand-written material) are from my course last year: please try them, especially Problems 3 & 4 in the second set, and you may turn them in for extra credit before or right after the Thanksgiving Break, the latest being in class in Monday 11/26!!!]

11/12: Div, Grad, Curl via Hodge and Music; Pullbacks and Naturality, (page 2), (page 3).
[Problems 1, 2, 3 of this set are due Wednesday 11/28. The remaining problems we'll discuss in class the week of 11/26 and are due Monday 12/10.]
On page 3 the goal is to deal with a k-form β on R^n, and its pullback F*β to the m-cube Q^m via a smooth map F:Q^m -> R^n. [Sorry for the write-os there (I meant to keep k and m distinct), but I hope you can sort that out. In particular, as a warm-up to the general case, try finishing the special case (k=1, m=2, n=3) of the "naturality" formula d(F*β)=F*(dβ) we began in class where β=xdx+ydy+zdz and F(s,t)=(s,t,sin(s)sin(t)); then try more general cases to see what's going on.]
[Technically, the last problem should have been done with a nonconstant function f in front of a basic form....]
Solutions by TB and by KD, (page 2).

We talked about this briefly in class the week of 11/26, and my notes above touch on it too, but here's more about Hodge-star. Cf. also Spivak (page 96, problem 4-19) for an alternative to "music" and Hodge-star for treating div, grad & curl in terms of "d" in \R^3. [Hint for the last problem: it may help to first observe that F*(α∧β)=F*(α)∧ F*(β), which one proves first for basic forms.... In class we have considered a warm-up problem: using the polar coordinates map F:\R^3 -> \R^3: F(r,θ,φ)=(r cosθ cosφ, r sinθ cosφ, r sinφ), compute F*(dx∧dy∧dz): it will be a function times dr∧dθ∧dφ.]

11/19: Thanksgiving Break!!!

[We're about a week behind on the syllabus, but we have left some extra time at the end of the semester to deal with this.]

11/26: Integration of k-forms over k-cubes and k-chains [These are due Monday 12/10, and Tetsuya will help you work through them in class next week. Note: In problems 3 and 4, the domain Q is a stretched/translated version of the 2-cube: one is [0,2π]×[-π/2,π/2] and the other is [0,2π]×[0,2π]. And sorry: the scanner inserted a couple of "backsides" - there are only 5 pages, not 7! ;-] [[Further comments: In problem 1, technically we're restricting everything to S, which can also be thought of as pulling back by the inclusion of S into \R^3. In problem 4, there's a "writing" error: it should be x(s,t)=(R + r cos(t))cos(s), y(s,t)=(R + r cos(t))sin(s) and z(s,t)=r sin(t).]]
Solution sketches by Rob.

12/03: The Big FToC [Rob will be away at math meeting in Korea this week, but Tetsuya will meet with you! It is a good idea to try this final HW, but I will NOT collect these.... :-]
[Again, sorry: the scanner duplicated and "wiggled" a couple pages - I hope there's at least one clear version of each page! You should try some explicit choices of V=α^# or V=∗β^# in problems 3, 4 and 5 to be sure of what you're doing. My favorite examples are constant (translation) vector fields like V=(1,0,0), rotation vector fields like
V=(y,-x,0), and the position vector field V=(x,y,z)!]
Solution sketches by Rob.

12/10: [Final week of classes - we'll be finishing up (and updating) the above materials on integration!]

Exam details:

Students who qualify for DS accommodations should email me at least 2 weeks before the midterm and final to let know me if they are planning to take the exam at the DSC, and they should then immediately contact* the DSC to arrange taking the exam there if desired:

Trisha Link
DSC Exam Proctoring Coordinator
169A Whitmore

The final exam is on 12/19 (Wednesday) 3:30-5:30PM in Hasbrouck Lab Add room 124 (please check SPIRE for any updates).
[You may bring a two-sided "help sheet" prepared by yourself in your own handwriting. Most of the exam topics (differentiation and integration of form, mainly) will be from after the midterm, but earlier topics (div, grad, curl and their interpretation using forms) naturally feed into the latter topics.]

Have a great winter!!!!!!