*Director of the Center*

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

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

** Newsflash:**

*
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:* ** profkusner@gmail.com **

**Prerequisites:**

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]

**Description:**

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.

**Grading:**

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

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

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

**examsaccess@admin.umass.edu**

413-545-0892

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!!!!!!*