Linear Algebra :
Rn as a vector space over R, subspaces of Rn, linear independence and linear span, standard basis for Rn and examples of different bases in R2 and R3. Linear transformations from Rn to Rm, null space, range space, statement and illustration of the rank-nullity theorem, matrix of a linear transformation with respect to standard basis, matrices as linear transformations.Hermitian, unitary and normal matrices. Rn as a real inner product space (dot product), orthogonality, length of a vector, examples of orthonormal basis, Pythagoras’ theorem, Cauchy- Schwartz Inequality.
Real valued functions of one variable:
Limit and continuity of real valued functions of one variable, sum and product of continuous functions, sign preserving property for continuous functions, intermediate value theorem, extreme value theorem for continuous functions. Derivability of real valued functions of one variable, Rolle’s theorem, mean value theorem.
Calculus of several variables:
Definition and examples of sequences, open sets, closed sets, compact sets, connected subsets of R1 and R2 Limit and continuity for real valued functions on R2, differentiability of real valued functions on R2, directional derivatives and gradients for these functions. Statement of Taylor’s theorem for functions of two variables, Maxima and Minima of functions of two variables.
1. T. M. Apostol: Calculus, Volume 1, John Wiley and Sons (Asia) Pvt. Ltd., 2002.
2. R. G. Bartle and D. R. Sherbert: Introduction to real analysis, John Wiley and Sons (Asia) Pte. Ltd. 2000.
3. H. Anton, I Bivens and S. Davis: Calculus, John Wiley and Sons (Asia) Pvt. Ltd., 2002.
4. C. P. Simon and L. Blume: Mathematics for Economists, W W Norton and Company, 1994.