University of Massachusetts, Amherst                                                                                                                                                                                     Fall 2018

## MATH 425.1: Advanced Multivariate (=multivariable) Calculus

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• ##### 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 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: Office :            1235I Lederle Graduate Tower. Email :           mirkovic@math.umass.edu Phone :           (413) 545-6023. Meet :            TuTh at 11:30-12:45 in Goessman Lab, Addition, room 152. Office hours : In my office: Tuesday 1:00-2:00, Wednesday 2:00-3:00.       [[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 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 6.1-6.2 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: 5-6 in my office, 6-7 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) Re-tell 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 (1--3) above. Repeat this process as many times as necessary.