*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:**

Mon & Wed 2:30-3:45 in LGRC A301 (3rd floor of the Lederle lowrise, near the Physical Sciences Library)

**Office Hours:**

After class Mon (3:45-5:00 starting in LGRC A301 and migrating up to
LGRT 1435G or 1535); before class Wed (noonish-2:15 in LGRT 1435G or
1535); and 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.

**Topic and Homework Schedule **(*please finish and submit within
14 days of the date at the left*; if you work together, be good
scholars and *name your collaborators*)**:**

09/04: 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 Wednesday. 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) and see if you can discover a relation for these numbers. Of course, the weather is splendid, so you may also use this map to ... Go Take a Hike!

09/11:
Derivatives, Critical Points and Convexity.

Solutions by Cass Kornhiser (page 1) and (page 2).

09/18:
Integrals, Volumes and Centers of Mass.

Solutions by Thom Barron (page 1)
and (page
2); a
different solution
to Problem 2 by Dan Weber;
more solutions by Alex Hargrove (page
1), (page
2), (page
3) and (page
4).

09/25: Holiday Homework Hiatus: the next set is due 10/16!

10/02:
Gradient, Divergence and Laplacian in n-space

Solutions by Kate Donoghue (page
1),
(page
2),
(page
3),
(page 4). [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.]

10/09: Curling Vectorfields on R^3

Solutions by Derek
Enlow,
Henry Phan (page
1), (page
2), (page
3),
Alex Hargrove (page
1), (page
2), (page
3)
and
Kate Donaghue (page
1), (page
2). [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).]

10/16: Another HW hiatus [you guys had a lot of other midterms, so I figure the HW and self-assessed quizzes have been enough for now].

10/23: Practice with Path Integrals

Solutions by Thom Barron, (page 2) and Kate Donaghue.

10/30: Fun
with Forms [This set is due Wed 15 Nov!!! 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 Kate Donaghue,
(page 2);
by Wilson Y. Wang,
(page
2), (page
3);
and
by Alex Hargrove, (page
2), (page 3).

11/06:
Domains and Forms in n-Space [This (along with the following
hand-written material) is from my course last year and is to be worked
on over the long Thanksgiving Break! Please try them, especially
Problems 3 & 4, and **turn them in for extra credit**.]

11/13:
Practice with Differential Forms [This is also from my course last
year and is also to be worked on over the long Thanksgiving Break!
These ** really are for practice**, so it will be smart if you try
them, but **no need to turn them in**!]

11/20: Even more for the long Thanksgiving Break! Div, Grad, Curl via Hodge and Music; Pullbacks and Naturality, (page 2), (page 3) - **these WILL be to turn in** (in 2 weeks, i.e. 12/04)!

On page 3 the goal is to deal with a k-form \beta on R^n, and its
pullback F*\beta 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*\beta)=F*(d\beta) we began in class where
\beta=xdx+ydy+zdz and F(s,t)=(s,t,sin(s)sin(t)); then try more general
cases to see what's going on.]

Solutions by Thom Barron and by Kate Donaghue, (page 2). [Technically, the last problem should have been done with a * nonconstant function f* in front of a basic form, but we gave full credit for checking 0=0.... ;-]

We talked about this briefly in class, 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*(\alpha∧\beta)=F*(\alpha)∧
F*(\beta), which one proves first for basic forms.... In class on 11/29 we
considered a warm-up problem: using the polar coordinates map
F:\R^3 -> \R^3: F(r,\theta,\phi)=(r cos\theta cos\phi, r sin\theta
cos\phi, r sin\phi), compute F*(dx∧dy∧dz): it will be a
function times dr∧d\theta∧d\phi.]

The last part of the course concerns integral calculus with forms.
We've already done a bit of this with 1-forms and path integration,
but **over the Thanksgiving Break, please study Spivak chapter 4,
especially pages 86-108**. Please try to understand "pullback" -
which is the forms version of the "change of variables" formulas you
may have seen in M&T 6. Also, please 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).]

11/27: 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 [0,2\pi]×[-\pi/2,\pi/2] and the other is [0,2\pi]×[0,2\pi]. 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).]]

Solutions by Rob.

12/04: 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=\alpha^# or V=\star\beta^# 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)!]

Solutions by Rob.

FINAL EXAM:
The SPIRE schedule indicates 3:30-5:30 Wednesday 20 December in ILC
S140, which is the very last exam session. By popular demand, the
alternative plan is to hold the exam in our classroom LGRC A301
2:30-4:30 Wednesday 13 December, and also to hold homework/review
sessions in our classroom 2:30-4:00 Friday 8 and Monday 11
December (our room IS available and reserved for us).

** Anyone who needs more time for the exam, or for whom this alternative time
poses a problem, MUST email me BEFORE Friday 8 December to arrange an
accommodation.
To be clear, the default time and place for the exam is the original: 3:30-5:30 Wednesday 20 December in ILC
S140.**

[You may bring a two-sided "help sheet" prepared by yourself in your own handwriting. About half the exam topics (integration, mainly) will be from after the midterm, but earlier topics (

*Have a wonderfully warm winter solstice and a perfectly pleasant perihelion!*