University of Massachusetts, Amherst
Fall 2018
MATH 425.1: Advanced Multivariate (=multivariable) Calculus
(schedule number )

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, solids, 4...) and topology
(properties of these geometric objects)
of the ndimensional Euclidean space for $n=1,2,3,...$ .
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 information:

Office :
1235I Lederle Graduate Tower.

Email :
mirkovic@math.umass.edu

Phone :
(413) 5456023.

Meet :
TuTh at
11:3012:45
in Goessman Lab, Addition, room 152.

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.
(If you have 6th edition let me know and I will post the
homework problems below. Hopefully everybody has the 5th
as announced on the department web page.)

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).
Chapter 5.16.2

Homeworks

H1.
H2.
H3.
H4.
H5. This is also the sample exam.
H6.
H7.

H8.
This is also the sample exam.

H9.
We have not covered new material
so the this homework is based on reading
the textbook.

The FINAL PROJECT
It is due Friday the 15th.

Review sessions: both in our classroom,
 Friday at 5:00.
 Tuesday at 7:00.
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.
The notes and consequently also homeworks and exams cover
more: how these ideas extend to
any number of variables (the corresponding geometric
spaces are all R^n).

Another aspect is that the in the notes 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 level of conceptual understanding.
EXAMS.
There will be two midterm exams and a final exam or a final project.
Exams and reviews are at 7:00 in the evening.
Places are TBA (hopefully in our classroom).
The Revised Schedule of Exams:

Midterm 1 (October):
Tuesday the 16th at 7:00 in our classroom.
Review: Thursday the 11th at 7:00 in our classroom.

Midterm 2 (November):
Thursday the 15th.
Review: Tuesday the 13th.

EXAM 1.
It covers chapters 5 and 6.16.2 in the book.
Dimensions beyond 3 will not be on this exam.
For the style of answering these questions enquire in class.

EXAM 2.
We will have EXTRA OFFICE HOURS before EXAM2:
56 in my office, 67 in the exam room LGRC A301.
!!!!!!!! EXAM room CHANGE: !!!!!!!
The exam will actually be in
LGRC A301.

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.