University of Massachusetts, Amherst
Spring 2020
MATH 425.1: Advanced Multivariate (=multivariable) Calculus
(schedule number )
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Lectures and Office Hours
are both at the usual hours.
There may be extra Lectures or Office Hours, these are announced in
EMAILS.
LECTURES
FINAL PROJECT REVIEWS (handwritten notes):
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EXTRA REVIEW SESSions
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Wednesday the 29th at 6
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SATURDAY the 2nd at 4.
These are for questions on the Final Project.
You should in principle
first consult the above notes of Final Project Reviews!
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Office hours:
I will try to have them simultaneously on Piazza
at
piaza.com/umass/spring2020/math4251/home
(for questions that do not require extensive computation or drawing) and on zoom
at 573-107-9058.
TYPED NOTES 2
Theese are the notes on chaperts 7-8. In the online partof the course
we cover chapter 8. However, I have lumped together chapter 7 since it is the closest to
this new material and I felt that everything should be clearly explained for the online part.
There will be updates ``all the time''.
Typed Notes 1
Theese are the notes on chaperts 1-6 which have been covered in class.
GRADE:
As I understand I will use maximum of the following two calculations
(For other suggestions use piazza.)
(A) Exam 1 (30%), Final (30%), Homework (40%).
(B) Exam 1 (30%), Final (40%), Homework (30%).
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HOMEWORKS
There will be weekly homeworks. They will be due in class
a week after they appear on this web page.
If you have a wrong edition:
Scans1,
the 1st group of homeworks
Scans2,
the 2nd group of homeworks
HW1.
HW2.
HW3.
HW4.
HW5.
HW6.
HW7.
HW8. NEW!
Due Sunday April 26.
EXAMS.
Due to a change to online course
there will only be one midterm exam and the final exam will be a final project.
There will still be an online review session
for the final exam.
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EXAM1 Wednesday February 26.
Review session is Monday February 24.
The Sample Exam is just the Homework 5 above.
The FINAL PROJECT
The project is DUE MAY 5th.
Description of the course:
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 n-dimensional 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
(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).
Possible additional topics:
from physics
(fluids and electromagnetism)
and from
differential geometry (curves and surfaces in space).
The basic information:
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Office :
1235I Lederle Graduate Tower.
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Email :
mirkovic@math.umass.edu
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Phone :
(413) 545-6023.
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Meet :
TuTh at
2:30-3:45
in Goessman Lab, Addition, room 152.
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Office hours :
Thursday 11:00-12:00, Wednesday 3:00-4:00.
[[Check here for CHANGES -- temporary or permanent!]]
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Text:
Vector Calculus by Marsden and Tromba, Ed., W. H. Freeman,
5th edition.
SYLLABUS:
Syllabus contains general information on
topics, exams, grade, course structure
and policies.
(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.)
The notes mainly indicate how lectures differ from
the book.
Here are some differences:
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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).
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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.
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
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(0)
You start by hearing (or reading) of a new idea, new procedure, new trick.
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(1)
To make sense of it you check what it means in sufficiently many
examples. You discuss it with teachers and friends.
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(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:
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(3)
Re-tell this idea or procedure, theorem, proof or
whatever it is, to yourself in YOUR OWN words.
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More on step (3).
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Trying to memorize someone else's formulation,
is a beginning but it is far from what you really need.
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You should get to the stage where you can
tell it as a story,
as if you are teaching someone else.
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When you can do this, and your story makes sense
to you,
you are done. You own it now.
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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 (1--3) above.
Repeat this process as many times as necessary.