# Mathematical Methods

## Sergey Cherkis

### Instructors

 Sergey Cherkis Office: 19.31 Hamilton Building Phone:  01 608 3453 Office Hours: Monday 2:00-3:00 E-mail: cherkis@maths.tcd.ie Grader: Ted FitzPatrick E-mail: fitzpamt@tcd.ie

### Course's Web Page:

http://www.maths.tcd.ie/~cherkis/131/

## Final Exam

### Class Meetings

 Tutorial: Friday 12:00-12:50 2041a Lecture Room in Arts Building 2041a Lectures: Monday 13:00-13:50 Maxwell Lecture Theatre in Hamilton Building 5 Tuesday 17:00-17:50 Geography Lecture Theatre in Museum Building MGLT

### Course Description

First part is devoted to
Linear Algebra:

R3 (vectors, dot product, cross product, scalar triple product, lines and planes, distance formulae (point to a plane?) systems of linear equations, matrices for them solution by row-reduction matrix inversion by row-reduction determinants, det nonzero iff invertible (and other results like this)
Diagonalisation of 3 x 3 symmetric matrices, orthogonal diagonalisation also
Notion of a linear operator on a general vector spaces as an introduction to ODEs

and ODEs:
1st order linear, Bernoulli eqn (y^n on right, reducible to linear via substitution u = y^{1-n}) 2nd order linear. Idea of particular solution + general solution of the homogeneous equation. Constant coefficient homogeneous cases (real or complex roots, repeated roots). Method of "undertermined coefficients" for finding particular solution.

Second and third parts continue with
Review of Basic Calculus:
Differentiation and Integration

Systems of Linear ODEs:
Diagonalizable and nondiagonalizable cases, functions of matrices, exponentiation of matrices,  initial value problem for higher order linear ODEs

Multivariable Calculus:
Review of calculus in 1-dimension, partial differentiation, gradient operator and its geometrical significance. Taylor polynomials, Taylor series. Maxima and minima (extreme  values), local and absolute. Critical points, 1st and 2nd derivative tests. Extreme values subject to constraints, Lagrange multipliers. Multiple and iterate integrals, line, surface and volume integrals, change of variable, Jacobians.

First order differential equations:
Separable and homogeneous equations. Integrating factors.

### Books

Required:

"Thomas' calculus," 11th ed / based on the original work by George B. Thomas, Jr., as revised by Maurice D. Weir, Joel Hass, Frank R. Giordano.

Recommended:
Mathematical Analysis," (Universitext) Springer-Verlag (2004)

### Exams

Three hour End of year Final examination.

 Homework 10% Final Exam 90%

### Homework

problems will be given on Friday and are due at the BEGINNING of the class on Friday the following week.
Students can collaborate on their homework, however, each student should hand in her or his own copy of solutions. Homework:

First in PS or in PDF (due 18 February 2005)
Second in PS or in PDF (due 4 March 2005)
Third in PS or in PDF (due 13 April 2005)
Fourth in PS or in PDF (due by noon 5 May 2005)
Fifth in PS or in PDF (due by noon 11 May 2005)