Program of the 131 Course

Linear Algebra: Vectors in 2D and 3D spaces Ref(1)

Week 1
Lecture  1.  Vector Spaces, Inner Product, Cross Product. and
Scalar Triple Product,
Lecture  2   Linear Independence and Orthonormal vectors
Lecture  3.  Gram-Schmidt Process
Week 2
Lecture  4. Examples
Lecture  5  Examples
Lecture  6. Examples

Matrices and Systems of Linear Equations. Ref(2)

Week 3
Lecture  7. Definition of  a Matrix, Square Matrices,
Lecture  8. Scalar and Matrice Multiplication
Lecture  9. Matrices as vector spaces
Week 4
Lecture 10. Matrix of a system of Linear equation
Lecture 11. Augmented Matrix, Elementary operations,
Lecture 12. Gaussian Elimination process, reduced row-echelon form
Week 5
Lecture 13. Gauss-Jordan method of Inverse matrix.
Lecture 14. Examples
Lecture 15. Examples

Determinants. Ref(3)

Week 6
Lecture 16.  Second order Determinants, Existence, Permutations
Lecture 17.  Elementary Product, signed elementary product,
Lecture 18.  Examples
Week 7
Lecture 19.
Properties of Determinants, upper, lower triangular matrices
Lecture 20.  Properties of Determinants, Invertibility of Matrices and determinants
Lecture 21.
Examples

Linear Algebra: Eigenvectors and Eigenvalues. Ref(3)

Week 8
Lecture 22.  Prerequisites
Lecture 23.  Eigenvalues & Eigenvectors
Lecture 24.  Examples
Week 9
Lecture 25.  Orthogonal Matrices,  Diagonalization
Lecture 26.  Orthogonal Diagonalization
Lecture 27.  Examples
Week 10
Lecture 28.  New Lecturer
Lecture 29.  Exercises
Lecture 30.  Exercises

Multivariable Functions and Partial Derivatives
Ref(5)

Week  11
Lecture  31. Index Notation Kronecker Delta & Epsilon of Levi-Civita

Lecture  32. Generalised Delta of Kronecker
Lecture  33. Determinants and Matrices, index notation
Week  12
Lecture  34.
Review of calculus of a single variable,  Level curves & Graphing + Excercises
Lecture  35. Examples on second order Partial Derivatives
Lecture  36.
Limits and continuity, Definition and Notation and examples of Partial Derivatives
Week  13
Lecture  37. Differentiability and the Chain Rule.

Lecture  38. Exercises
Lecture  39. Standard Linear Approximation, Differentials and the Sensitivity to change.
Week  14
Lecture  40.
Lecture  41. Exercises
Lecture  42. Directional Derivatives, Lagrange Multipliers
Week  15
Lecture  43.
Taylor's Formula and the second derivative test.
Lecture  44. Exercises
Lecture  45. Exercises

Multiple Integrals
. Ref(6)

Week 18
Lecture 46.  Double Integrals. Definitions.
Limits of Integration. Double Integrals in Polar form
Lecture 47.  Exercises
Lecture 48.
Areas and Center of Mass. First and Second Moment.
Week 19
Lecture 49.  Triple integrals in Rectangular Coordinates. Triple Integrals in Cylindrical and Spherical Coordinates.
Lecture 50.  Exercises
Lecture 51.
Substitution method. Masses and Moments in Three dimentions.

First Order Differential Equations. Ref(7)

Week 20
Lecture 52.  Classification of differential equations

Lecture 53.  Method of Direct Integration and Separable Equations.
Lecture 54.  Exercises.

Week 21
Lecture 55.  Homogeneous ordinary differential equations.
Lecture 56.  Exercises.
Lecture 57.  Method of Integrating Factor.

Week 22
Lecture 58.  Exercises.
Lecture 59.  Review
Lecture 60.  Review

References: 1. Anton & Rorres: ch on Vectors in 2-Space and 3-Space
2. Anton & Rorres: ch on Systems of Linear Equations and Matrices
3. Anton & Rorres: ch on Determinants
4. Anton & Rorres: ch on Eigenvalues & Eigenvectors
5.Thomas & Finney: ch. on Multivariable Calculus.
6.Thomas & Finney: ch. on Multiple Integrals.
7. Boyce & Di Prima: ch. one to four (up to first order diff equtations).