University of Massachusetts, Amherst
Fall 2017
MATH 425.1: Advanced Multivariate (=multivariable) Calculus
(schedule number 35750)

Description:
This is a course in the differential and integral calculus of several variables from a more advanced perspective than Math 233. We will study geometry(curves, surfaces, ...) and topology
(properties of thesegeometric objects)
of the ndimensional Euclidean space forn=1,2,3,... .
The main focus will be on
integration over regions, paths and surfaces,
the change of variables formula, and
thefundamental theorem of calculus (the theorems of Green, Gauss, and Stokes).

The required knowledge.
This includes the Linear Algebra 235 course and the
Chapters 1,2,3,4 of the Marsden and Tromba book.
Thismaterial will be only reviewed in a very fast fashion and it will be assumed throughout.
 Some of the Basic Notions in the course:
differentiability,
directional and partial derivatives and gradient of functions;
critical points without or with constraints
(Lagrangemultipliers/tangentialgradient) 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).

Possible additional topics:
from physics
(fluids and electromagnetism)
and from
differential geometry (curves and surfaces in space).

The basic organizational information:

Office :
1235I Lederle Graduate Tower.

Email :
mirkovic@math.umass.edu

Phone :
(413) 5456023.

Meet :
TuTh at
11:3012:45
in LGRT 219.

Office hours :
In my office: Tuesday 1:002:00, Wednesday 2:003:00.
[[Check here for changes  temporary or permanent!]]

Text:
Vector Calculus by Marsden and Tromba, Ed., W. H. Freeman,
5th edition.
(For those who have 6th edition I will post the
homework problems below.)

SYLLABUS:
Syllabus contains general information on
topics, exams, grade, course structure
and policies.

HOMEWORKS
There will be weekly homeworks. They will be due in class
a week after they appear on this web page.

The scanned problems from the 5th edition
(for those who have the 6th edition book)

Notes

The old notes
The notes will evolve during the semester,
The current notes are from a previous year.

The LAST UPDATE was on ...
It covered ...

The CONTENT of the notes:
The notes mainly indicate how lectures differ from
the book.
Here are some differences:

The book concentrates on functions of up to
three variables. This means that the geometry is that of spaces
R, R^2 and R^3. I will try to give some
indication of how these ideas extend to
any number of variables (the corresponding geometric
spaces are all R^n).

Another aspect is that the ideas behind some procedures
may be explained in more details.
In mathematics a clear explanation is called a proof.
The homeworks and exams will not require complete proofs,
but it will be necessary to express
certain levl of conceptual understanding.

EXAMS.
There will be two midterm exams and a final exam or a final project.

EXAM 1. Thursday October 12. At 7:00.
The place to be announced.
The exam covers chapters ... in the book
and also some questions on how the change of variable formula
works (for these read the web page notes on chapter 6).
For the style of answering these questions enquire in class.

Sample Exam 1
The sample exam is supposed to be Long and some of it should be Hard.
The idea is that once you can do this you should be OK at the exam.

The actual exam should be much shorter, the length of the exam will be
appropriate so that you can do the work in the scheduled time.

Review session: ...

EXAM 2.
Thursday November 17. At 7:00.
The place to be announced.

What is covered on the exam is ....

Sample Exam 2

Review session:
...

!! MAKE UP ??

The FINAL PROJECT
The project covers sections ...

The FINAL Exam or PROJECT

Review session: ...

HOW TO LEARN abstract MATHEMATICS.
The following is what I see as the {\em basic} approach
towards learning mathematics at the conceptual level.
The procedure is

(0)
You start by hearing (or reading) of a new idea, new procedure, new trick.

(1)
To make sense of it you check what it means in sufficiently many
examples. You discuss it with teachers and friends.

(2)
After you see enough examples you get to the point where you
think that you more or less get it. Now you attempt
the last (and critical)
step:

(3)
Retell this idea or procedure, theorem, proof or
whatever it is, to yourself in YOUR OWN words.

More on step (3).

Trying to memorize someone else's formulation,
is a beginning but it is far from what you really need.

You should get to the stage where you can
tell it as a story,
as if you are teaching someone else.

When you can do this, and your story makes sense
to you,
you are done. You own it now.

However, if at some point you find
a piece that does not make sense, then you have to
return to one of the earlier steps (13) above.
Repeat this process as many times as necessary.