Math 421: Applied Analysis

Fall 2011 (CRN 75305), UAF

Instructor: Ed Bueler
Chapman 301C  (office hours online)
Time & Place:
     MWF   9:15am-10:15am  GRUE 202
     T    9:45am--10:45am  CHAPMAN 106
     Schey, Div, Grad, Curl and All That, 4th ed., 2005
     Farlow, Partial Differential Equations, 1993

The course:     Math 421 is really two courses but they work well together; this is why it is 4.0 credits.

First we do a short course in vector calculus, completing multivariable calculus.  That is the subject which you started in Calculus III (Math 202).  This part of Math 421 gives you the tools to do classical electricity and magnetism, but just as directly helps with fluids and elastic deformation and any other "field theory" that uses vector fields.  The remaining two thirds of the course is an introduction to partial differential equations (PDEs) and their solution by separation of variables.  We cover the heat, wave, and potential equations (PDEs).  There will be boundary value problems and initial-and-boundary value problems for PDEs on nice domains.  We will develop the elementary theory of Fourier series and orthogonal functions, as this is the substance of the separation of variables technique.

Lectures and homework together form the core of the class.  There is a lot of homework, both calculations and arguments.  Formal proofs will not be expected.  Computer experimentation is encouraged but is not a part of the course.  You are expected to ask questions in class about recent lectures or homework assignments.

Prerequisites:    Math 302 (ordinary) Differential Equations is the only prerequisite, but Math 302 requires Math 202 Calculus III, which is equally a prerequisite for Math 421.

Policies and makeup exams:   The department has specific policies on incompletes, late withdrawals, and early final examinations, etc; see .  You are covered by the UAF Honor Code.  I will work with the Office of Disabilities Services (203 WHIT, 474-7043) to provide reasonable accommodation to student with disabilities.   I will create makeup versions of exams provided I have a convincing reason to do so at least two class days before the exam, and the makeup must occur no later than two class days after the exam date.

The grade:  Your grade will be determined by three exams (60%) and homework (40%):
20 %
10 %
30 %
40 %
Vector calculus Exam
Midterm Exam (on PDEs)
Final Exam (on PDEs)
Monday, Oct. 17 (one hour in class; REVISED DATE)
Friday, Nov. 11 (one hour in class)
Wednesday, Dec. 14  (take home; due 5pm this day)

The final grade will be determined
by the total of your scores using
the schedule at right.  This is a
guarantee; I reserve the right to
raise grades by a small amount
based on evidence of performance
improvement over time.
93 - 100 %
90 - 92 %
87 - 89 %
82 - 86 %

77 - 81 %

74 - 76 %
69 - 73 %
66 - 68 % 
63 - 65 %
56 - 62 %
53 - 55 % 
0 - 52 % 

Goals and related courses:
    In Math 421 you'll get a clear, detailed presentation of the ideas in vector calculus, which you just touched on in the first calculus sequence, and then you get an introduction to partial differential equations (PDEs) and Fourier series.
    Why do vector calculus?  It is the mathematical core of electricity & magnetism (e.g. PHYS 331/332 and EE 311).  It is the mathematical core of the study of fluids of all types (e.g. air, water, ice, plasma, earth's mantle; see ME 451, MSL 629, PHYS 614, PHYS 629, ...) and the deformation of materials.
    PDEs, and the Fourier series which arise in their solutions, applies in many places.  They occur in all of the contexts in which vector calculus is useful, but also in quantum mechanics and vibrations and other applications with a less explicit "vector" feel.  Our coverage of Fourier series is also useful in understanding signals; compare EE 451.  In the development of mathematics, understanding Fourier series was central to developing real analysis as the theory of calculus, as it is taught in Math 401 for example.
    There is a caviat about the PDEs part of the course, namely, that the methods we use only work (directly) for linear equations with constant coefficients in nice domains.  You will get a feeling for these restrictions, but you may not see at first how severe the restrictions are because we will not look at other kinds of PDE problems in this first course.  In future work you might approximate harder problems (nonlinear or non-constant coefficient or irregular domain) by simpler problems of the type we solve.