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COURSE TITLE: Topics in Applied Mathematics

INSTRUCTOR: Doug Hensley, 311 Milner. Email Doug.Hensley@math.tamu.edu, phone 409-8453654 (on campus, 5-3654). Office hours Tues 1:30-3:30, Thurs 1-2, or other mutually convenient time (best arranged by email or after class.)

TEXTBOOK: 
Multivariable Mathematics, $3^{\hbox{rd}}$ Ed. by Williamson \& Trotter

CATALOG DESCRIPTION:
Matrices, determinants, systems of linear equations, eigenvalues,
eigenvectors, diagonalization of symmetric matrices. Vector analysis,
including normal derivative, gradient, divergence, curl, line and surface
integrals, Gauss', Green's and Stokes' theorems. Prerequisite: Math 221 or
253 and Math 308 or current enrollment therein.

The first 60\% of the course is devoted to linear algebra, with a strong bias
toward applications to analysis. The remaining 40\% is devoted to the
material traditionally called vector calculus (or vector analysis), along
with some other topics in advanced calculus which provide nice applications
of the linear algebra covered in the first part. The unifying theme is that both are concerned with mathematics involving several variables, or variables with ``dimension''.

Linear algebra deals with matrices and with associated situations in which objects add and scalar multiply like vectors. For instance, the set of polynomials of the form $a+bx+cx^2$ is in a sense three-dimensional, and the subset of these polynomials which take the value $0$ at $x=1$ is a two-dimensional subset. The operation which takes as input the polynomial $f(x)$, and returns as output the new polynomial $\int_{x-1}^xf(t)dt$, is a linear operator on this three-dimensional set. The whole situation can be described and analyzed using matrices. The set of solutions to the differential equation $y'''-3y'+2y=0$ is likewise three-dimensional, and the subset of solutions obeying the initial condition $y'(0)=0$, two dimensional. In both cases, much of the situation can be described and analyzed using matrices.

We shall also be concerned with the eigenvalues and eigenvectors of a matrix, and with various notions of an `inner product'. This is a generalization of the dot product from calculus. 


The main goals in the calculus part of 311 are a
firm understanding of the Gauss-Green-Stokes theorems, techniques for
evaluating surface integrals, Jacobi's theorem for a general change of
variables, other aspects of curvilinear coordinates, and some idea of how
linear algebra helps us to understand partial derivatives, the chain rule,
the implicit function rule, etc. Just to get a sense of where this is going, here is an example: If $u$ is a function of $x$ and $x$ is a function of $t$, then the (one variable) chain rule says that $\frac{du}{dt}=\frac{du}{dx}\cdot\frac{dx}{dt}$. If now $u$ is a function of $x_1$ and $x_2$, themselves functions of $t_1$ and $t_2$, then the matrix 
$$\pmatrix{\frac{dx_1}{dt_1}&\frac{dx_1}{dt_2}\cr \frac{dx_2}{dt_1}&\frac{dx_2}{dt_2}\cr}$$ takes the place of $dx/dt$, the vector $\pmatrix{\frac{du}{dx_1}&\frac{du}{dx_2}\cr}$ takes the place of $du/dx$, and the chain rule reads$$\frac{d{\bf u}}{d\bf {t}}=\left(\frac{d{\bf u}}{d{\bf x}}\right)\left(\frac{d{\bf x}}{d{\bf t}}\right)$$For instance, if $x_1=2t_1+3t_2$ and $x_2=4t_1+t_2^2$, and $u=x_1^2+5x_2$, then $$\frac{d{\bf u}}{d\bf {t}}=\pmatrix{2x_1&5}\pmatrix{2&3\cr 4&2t_2\cr}$$so that if $t_1=7,t_2=11$, (so $x_1=47$ and $x_2=149$) and $t_1$ is incremented by $0.004$ and $t_2$ by $0.002$ then the resulting change in $u$ will be approximately $$(94,5)\pmatrix{2&3\cr 4&22}\pmatrix{0.004\cr 0.002\cr}$$(which works out to a single number).\vfill\eject

Week

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  1. 1.1--1.3 Jan 20, 22
  
  2. 2.1--2.2, Jan 27, 29
  
  3. 3.1--3.2 Feb 3, 5
  
  4. Review,  Feb 10, Test I, Feb 12
  
  5. 3.3. 3.4 Feb 17, 19
  
  6. 3.4, 3.5 Feb 24, 26
  
  7. 4.1, 4.2, 4.3 Mar 3, 5
  
  8. 4.4, Review, Test II Mar 10, 12
  
  9. 5.1, 5.2, 5.3 Mar 24, 26

 10. 6.1, 6.2, 6.3 Mar 31, Apr 2
 
 11. 7.1, 7.2, 7.3 item 3.7 Apr 7, 9
 
 12. 8.1, 8.4, Apr 14, Test III Apr 16
 
 13. 9.1, 9.2 Apr 21, 23
 
 14. 9.3, 9.4, 9.5 Apr 28, 30. 
 
 Grading: There will be a short quiz each Thursday except for those weeks on which an exam is scheduled. The best ten quizzes will count for 3\% of the grade each. Each exam will count for 15\% of the grade, and the final will count 25\%. Scale is $85=$A, $75=$B, $60=$C, $50=$D. 

Suggested Homework Problems

1.1     1, 7, 11

1.3     1c, d, 5a, d 12a, b

1Rev    4, 7, 8, 13, 15 a, b



2.1     5, 7, 8, 12, 18

2.2     3, 6, 10, 15a,  

2C      1a, e, f, 13, 15

2Rev    3, 5, 8.

3.1     1d, e, f, 2d, e, f, 4, 6, 7

3.2     1b, d, 2c, 3, 4, 6

3.3     1c, d, f, 2a, 3, 8a, 11, 12b, 17

3.4     1b, 5, 7, 8c, 9d, 10a, d

3.5     1, 2, 3, 4, 6, 7, 10

4.1     1b, d, 2, 4a, 7a, e, 8, 10, 14

4.2     1a, 4, 6 p 116, and on p 119, 4, 9.

4.3     1c, e, 3 b, c, 6, 10, 11, 14, 15, 17 a,b,f,h, 18

4.4     1a, b, 3, 6, 7, 9

5.1     2b, 6, 8, 11, 15

5.2     1a, 2a, 3, 8,

5.3     1a, b, 2

6.1     2b, d, 3b, d, 4, 5, 6, 7

6.2     1, 2,  3a,  4a,  9, 11, 12

6.3     1, 8, 11

7.1     1, 2, 4, 8, 16

7.2     1, 7, 10

(homework for chapters 8 and 9 to be announced)





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