Institute-wide Undergrad Courses

• Sequences and Series:
  • Limit of a sequence, monotone and Cauchy sequences and properties of convergent sequences, examples.
  • Infinite series, positive series, tests for convergence and divergence, integral test, alternating series, Leibnitz test.
• Differential Calculus:
  • Continuity and differentiability of a function of single variable, statement of Rolle’s Theorem, Lagrange’s mean value theorem and applications.

Integral Calculus:

• Definite Integrals as a limit of sums
• Applications of integration to area, volume, surface area
• Improper integrals

Functions of several variables:

• Continuity and differentiability
• Mixed partial derivatives
• Local maxima and minima for function of two variables
• Lagrange multipliers

Functional Series:

• Sequence of Functions
• Pointwise and uniform convergence
• Basic aspects of Power series
• Fourier series

Double and Triple Integrals:

• Calculations, Areas, Volumes
• Change of variables, Applications

Integrals of Vector Functions:

• Line integrals, Green’s formula
• Path independence
• Surface integral: definition, evaluation
• Stoke’s formula, Gauss-Ostrogradsky divergence theorem

Matrices, Linear equations and solvability:

• Vector spaces
• Basis and dimension
• Linear transforms
• Similarity of matrices

Rank-Nullity theorem and its applications:

• Eigenvalues and eigenvectors
• Cayley-Hamilton theorem and diagonalization
• Inner-product spaces
• Gram-Schmidt process

Ordinary Differential Equations:

• First order linear equations
• Bernoulli’s equations
• Exact equations and integrating factor
• Higher order linear differential equations with constant coefficients

Partial Differential Equations:

• First order linear PDE
• Quasi-linear PDE
• Method of characteristics
• Cauchy problem
• First order nonlinear PDE’s of special type

Sample space and events, definitions of probability:

• Properties of probability
• Conditional probability

Random variables:

• Distribution functions
• Discrete and continuous random variables
• Moments of random variables
• Conditional expectation
• Chebyshev inequality
• Functions of random variables

Special Distributions:

• Bernoulli, Binomial, Geometric, Pascal, Poisson, Exponential, Uniform, Normal distributions
• Limit Theorems: Law of large numbers

Laplace and Inverse Laplace transform:

• Linearity
• Laplace transforms of Derivatives and Integrals
• Partial fractions
• Unit step function
• Shifting on the t-axis
• Periodic functions
• Applications of Laplace transform for solving differential equations

Fourier integral, Fourier Sine and Cosine transform:

• Convolution
• Applications of Fourier transform for solving differential equations

Complex Functions:

• Limits
• Continuity
• Differentiability
• Analytic functions
• Cauchy-Riemann equations
• Laplace equations
• Harmonic functions
• Conformal mapping

Cauchy Integral Theorem and Cauchy Integral Formula:

• Derivations of an analytic function
• Power series
• Taylor series
• Laurent series
• Zeros
• Singularities
• Residues
• Evaluation of real integrals

Random sampling:

• Estimation of parameters
• Confidence Intervals
• Testing of Hypothesis
• Goodness of fit
• Nonparametric tests
• Correlation Analysis