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 please turn in during class on the Monday two weeks after it's assigned; if you work together in small groups (2 to 4 is optimal), which is strongly encouraged, be good scholars and name your collaborators):
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.
09/16:
Integrals, Volumes and Centers of Mass.
09/23:
Gradient, Divergence and Laplacian in n-space
09/30: Homework Hiatus! The HW above is now due Tuesday 15 October!!!
10/07: Curling Vectorfields on R^3
10/14: Practice with Path Integrals
10/21: Mini-Midterm Homework Hiatus!!!!!
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
11/04: Practice
with Differential Forms
11/11:
Domains and Forms in n-Space
11/18:
Div, Grad, Curl via Hodge and Music; Pullbacks and
Naturality, (page
2), (page
3).
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!!!
12/02: Integration
of k-forms over k-cubes and k-chains
12/09: The
Big FToC
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
Have a great winter!!!!!!
Advanced Multivariate Calculus (Math 425.2) Fall 2019
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!]
Solutions
by Liwen O.
[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).
[The Open University has made some charming videos on grad, div, and curl. Please take some time to watch these during the homework hiatus (the Laplace and Poisson equations, describing steady-state heat distribution, make a cameo appearance in the second video). (There also this on div and curl from YouTube, but please skip their $@#%&*$ ads!)]
[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.
TeXed
solutions by Liwen O.
Handwritten solutions by TB, (page 2) and KD
The HW above is now due Wednesday 30 October!!!
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 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)
.
This one is due Wednesday 20 November!
[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.]
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).
[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 in class on Monday 12/09!!!]
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.
[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.
DSC Exam Proctoring
Coordinator
examsaccess@admin.umass.edu
413-545-0892
169A Whitmore