*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.)

**Time and Place:**

Mondays & Wednesdays 2:30-3:45 in Goessmann 151

**Office Hours:**

After class (3:45-4:00) Mondays and Wednesdays, starting in Goessmann
151 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
14 days of the date at the left*, which means

09/02: Locate and try to classify (as maxima, minima, saddles...) all
the critical points on the topographic map of northeast Amherst and the northwest Pelham hills
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.)

09/09:
Derivatives, Critical Points and Convexity.

Solutions
by Tommy
K (page 1),
(page 2),
(page 3),
(page 4),
(page 5). [Of course, he's kidding about the units being *nm/ly*: both numerator and denominator are units of length, so their ratio is actually a (very tiny!!!!!!!!!!!!!!!!!!!!!!!!!) number, with no units at all!]

09/16:
Integrals, Volumes and Centers of Mass.

Solutions
by Liwen O.

09/23:
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. Some physicists in Italy think 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!]

TeXed solutions by Liwen O.

Handwritten solutions by KD (page
1),
(page
2),
(page
3),
(page 4).

09/30: ** Homework Hiatus!** The HW above is now due Tuesday 15 October!!!

[The Open University has made some charming videos on

10/07: Curling Vectorfields on R^3

[For both Problems 2 and 3, you may assume the Theorem of Clairut, Schwarz, ... , Young.
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).]

TeXed
solutions by Liwen O.

10/14: Practice with Path Integrals

TeXed
solutions by Liwen O.

Handwritten solutions by TB, (page 2) and KD

10/21: *Mini-Midterm Homework Hiatus!!!!! *

The HW above is now due Wednesday 30 October!!!

**Exam details**:

How about our **midterm exam**?

We've have converged to holding a **Mini-Midterm** *in class
(Goessmann 151) on* **Wednesday 23 October.**

No make-ups, but **please let me know by email before Wednesday 9
October if you have a special situation** that we might accommodate.

**Alert:** 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 "smooth 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).]

10/28: Fun
with Forms

*Reminder: as usual, homework is normally due two weeks later, but
Veterans Day means on this one is due Wednesday 13 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) .

11/04: Practice
with Differential Forms

*This one is due Wednesday 20 November!*

11/11:
Domains and Forms in n-Space

[Thanks to the flu, we're about a week behind on the syllabus, but
hope we can find some extra time at the end of the semester to deal
with this.]

11/18:
Div, Grad, Curl via Hodge and Music; Pullbacks and
Naturality, (page
2), (page
3).

**Problems 1, 2, 3** of this set are about things we (un)covered
before the Thanksgiving break. The remaining problems concerning
pullbacks and naturality (a.k.a. change of variables) we'll discuss in
class the week of 12/02. 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.
[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.]
[[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 Hodge duality briefly in class the week of 11/18, and my notes above touch on it too, but here's more about Hodge-star. See 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.

11/25: __Thanksgiving Break!!!__

[The above problem sets (and hand-written material) for the weeks of
11/11 and 11/18 are from my earlier courses: please explore
them *for fun* over the Thanksgiving Break. You may **turn them
in for a modest amount of extra credit**, the latest
being

12/02: Integration
of k-forms over k-cubes and k-chains

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.

12/09: The
Big FToC

[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.

Finally: The **final exam** is still **3:30-5:30PM on Friday 13
December**, but it was just reclocated from our usual classroom to
Integrative Learning Center ILC S131 (please check SPIRE for any
further updates).

[For each exam you may bring a two-sided "help sheet" prepared by yourself in your own handwriting. Most of the final exam topics (differentiation and integration of forms, mainly) will be from after the midterm, but earlier topics (**div**, **grad**, **curl** and their interpretation using forms) naturally feed into the latter topics.]

* 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

*Have a great winter!!!!!!*