## Tentative syllabus:

### I. Laplace transform

3.1 Def'n of Laplace transform and basic properties 3, 5, 7, 13, 15, 17, 27, 29.
3.2 Solution of initial value problems using the Laplace transform 3, 5, 7, 9.

3.3 Shifting theorems and the Heaviside function 1, 3, 7, 9, 13, 15, 17, 19, 21, 25, 31, 37, 47, 49.

3.4 Convolution 1, 3, 7, 9, 11, 13, 17, 19, 23.

3.5 Unit impulses and the Dirac delta function 1, 3, 7, 9, 13, 19.

3.6 Laplace transform solution of systems 1, 3, 5, 7, 9, 11, 15.

3.7 Differential equations with polynomial coefficients 1, 3, 7, 11.

5.4 The vector space R^n. 1, 7, 9, 14, 15, 19, 26.
5.5 Linear independence, spanning sets, and dimension in R^n
1, 2, 5, 17, 19, 21
6.1 Matrices 10, 21, 27, 28, 29

6.3 Row echelon form of a matrix 1, 3, 9.
6.4 Row & column spaces of a matrix and rank 1, 3, 9.

6.5 Solution of a homogeneous system of linear equations 1, 5, 13.

6.7 Nonhomogeneous systems of linear equations 1, 5, 9.

6.9 Matrix inverses 1, 7, 13.

7.1 Permutations
7.2 Definition of determinant
7.3 Properties of determinant
7.4 Evaluation of determinant by elementary row & column ops. 1, 7

7.5 Cofactor expansions 5, 13, 19 a, b.
7.6 Determinants of triangular matrices 1

7.7 Determinant formula for matrix inverse 1, 7.
7.8 Cramer's Rule 1, 5, 9.

8.1 Eigenvalues and eigenvectors 1 a, b, 7 a, b, 19 a, b.

8.2 Diagonalization of matrices 1, 7

### III. Fourier transform/series and solution of heat and wave partial differential equations

13.1 Why Fourier series? 1
13.2 The Fourier series of a function 1, 5, 7, 9, 11.
13.4 Fourier cosine and sine series 1, 3, 5, 7.

13.5 Integration and differentiation of Fourier series 3, 5.

16.1 The wave equation and initial and boundary conditions 1, 3.
16.2 Fourier series solution of the wave equation 1, 3, 5, 9. 16.2

16.4 Fourier series and d'Alembert solutions of the wave equation
1, 3, 5, 11, 13, 17.

16.7 Vibrations of a rectangular membrane 1, 3.

17.1 The heat equation and initial and boundary conditions. 1, 2, 3.
17.2 Fourier series solution of the heat equation 1, 3, 7, 9, 11.

17.5 Heat conduction on a rectangular plate 7, 13, 15.