**Announcements****Course Outline**- Convex Analysis: convex sets, convex functions, calculus of convex functions.
- Optimality of Convex Programs: 1st order necessary and sufficient conditions, KKT conditions.
- Duality: Lagrange and conic duality.
- Linear and Quadratic Programs.
- Conic Programs: QCQPs, SOCPs, SDPs.
- Smooth Problems: gradient descent, Nesterov's accelerated method, Newton's methods.
- Non-smooth Problems: sub-gradient descent.
- Special topics: active set and cutting planes methods, proximal point method.
**Text Books**- S. Boyd and L.Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
**Reference Books**- R. T. Rockafellar. Convex Analysis. Princeton University Press, 1996.
- G. C. Calafiore and L. El Ghaoui, Optimization Models, Cambridge University Press, 2014.
**Question Papers and Answer Keys**- Quiz I: Question Paper Marking Scheme
- Mid-Sem: Question Paper Marking Scheme
**Marks****Announcements**- Problem set 11 is uploaded.
- The syllabus for the Mid Semester Examination will be from Lecture 5 to Lecture 12-13. Sequence, Series and Fundamental theorem of calculus are not included.
- Grading Policy: Quiz I: 20 points, Mid-sem Exam: 35 points, Quiz II: 20 points, End-sem Exam: 75 points, Total: 150 points. Note that marks will be rescaled (linearly) depending on the full marks of the examination.
**Course Outline**- Univariate Calculus: Function of one variable, Limit, Continuity and Differentiability of functions, Rolle’s theorem, Mean value theorem, maxima, minima, Riemann integral, Fundamental theorem of calculus, applications to length, area, volume, surface area of revolution.
- Infinite Sequences and Series: Sequences, Infinite series, The Integral test, Comparison tests, The Ratio and Root tests, alternating series, absolute and conditional convergence, Power series. Taylor and Maclaurin series, Convergence of Taylor Series, Error Estimates, applications of Power series.
- Multivariate Calculus: Functions of several variables, Limit, Continuity and Partial derivatives, Chain rule, Gradient, Directional derivative, and Differentiation, Tangent planes and normals. maxima, minima, saddle points, Lagrange multipliers, Double and Triple integrals, change of variables.
- Calculus on Vector Field: Vector fields, Gradient, Curl and Divergence, Curves, Line integrals and their applications, Green’s theorem and applications, Divergence theorem, Stokes’ theorem and applications.
**Text Books**- G. B. Thomas, M. D. Weir, and J. Hass,
*Thomas' Calculus*, Pearson. **Reference Books**- T. M. Apostol,
*Calculus, Vol. 1*, Wiley. - T. M. Apostol,
*Calculus, Vol. 2*, Wiley. - J. Stewart,
*Calculus*, Thompson Press. **Lecture Notes by Prof. P. Shunmugaraj (Department of Mathematics, IIT Kanpur)**- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5
- Lecture 6
- Lecture 7
- Lecture 8
- Lecture 9
- Lecture 10
- Lecture 11
- Lecture 12-13
- Lecture 14
- Improper integrals: Lecture 15
- Area between two curves; Polar coordinates: Lecture 16
- Area in Polar coordinates; Volume of Solids: Lecture 17
- Washer and Shell Methods; Length of a plane curve: Lecture 18
- Areas of surfaces of revolution: Lecture 19
- Sequences in R^3: Lecture 20
- Vector valued functions: Lecture 21
- Functions of Several Variables: Lecture 22
- Directional Derivatives: Lecture 23
- MVT: Lecture 24
- Maxima and Minima for function of two variables: Lecture 25
- Double Integrals: Lecture 26
- Change of Variable in a Double Integral; Triple Integral: Lecture 27
- Change of Variable in a Triple Integral; Parametric Surface: Lecture 28
- Surface Area, Surface Integral: Lecture 29
- Line Integrals: Lecture 30
- Line Integrals: Lecture 30
**Problems**- The Real Number System: Problem Set 01
- Convergence of Sequences, Monotone Sequences: Problem Set 02
- Cauchy criterion, Subsequence: Problem Set 03
- Series: Necessary and sufficient conditions, absolute convergence: Problem Set 04
- Comparison, Limit comparison and Cauchy condensation tests: Problem Set 05
- Ratio, Root and Leibniz’s tests: Problem Set 06
- Limit and Continuity: Problem Set 07
- IVP, Existence of maxima/minima: Problem Set 08
- Differentiability and Rolle's theorem: Problem Set 09
- Mean Value Theorem: Problem Set 10
- Test for maxima and minima: Problem Set 11
**Question Papers and Answer Keys**- Quiz I: Question Paper Marking Scheme
- Mid-Sem: Question Paper Marking Scheme
- Quiz II: Question Paper Marking Scheme Figures
- End-Sem: Question Paper Marking Scheme Figures
**Course Outline**- Probability: Axiomatic definition, Properties, Conditional probability, Bayes rule and independence of events.
- Random Variable: Random Variables, Distribution function, Discrete and Continuous random variables, Probability mass and density functions, Expectation, Function of random variable, Moments, Moment generating function, Chebyshev's inequality.
- Special discrete distributions: Bernoulli, Binomial, Geometric, Negative binomial, Hypergeometric, Poisson, Uniform.
- Special continuous distributions: Uniform, Exponential, Gamma, Normal.
- Random vector: Joint distributions, Marginal and conditional distributions, Moments, Independence of random variables, Covariance, Correlation, Functions of random variables.
- Law of Large Numbers: Weak law of large numbers, Levy's Central limit theorem (i.i.d. finite variance case), Normal and Poisson approximations to Binomial.
- Statistics: Introduction: Population, Sample, Parameters.
- Point Estimation: Method of moments, Maximum likelihood estimation, Unbiasedness, Consistency.
- Interval Estimation: Confidence interval.
- Tests of Hypotheses: Null and Alternative hypothesis, Type-I and Type-II errors, Level of significance, p-value, Likelihood ratio test, Chi-square goodness of fit tests.
- Regression Analysis: Scatter diagram, Simple linear regression, Least square estimation, Tests for slope, prediction problem, Graphical residual analysis, Q-Q plot to test for normality of residuals.
**Text Books**- Rohatgi, V. K. and Saleh, A. K. (2000),
*An Introduction to Probability and Statistics*, 2nd Edition, Wiley-interscience. - Montgomery, D. C., Peck, E. A. and Vining, G. G. (2012),
*An Introduction to Linear Regression Analysis*, 5th Edition, Wiley. **Reference Books**- Bertsekas, D. P. and Tsitsiklis, J. N. (2008),
*Introduction to Probability*, Athena Scientific, Massachusetts. - Seber, G. A. F. and Lee, A. J. (2003),
*Linear Regression Analysis*, 2nd Edition, Wiley-interscience. **Questions and Keys**- Back Paper Examination: Question Paper Marking Scheme
**Course Outline**- Laplace Transforms: Definition and properties, Sufficient condition of Existence, Transforms of derivatives and integrals, Derivatives and integrals of transforms, Inverse Laplace Transforms, Exponential shifts, Convolutions, Applications: Differential and Integral Equations.
- Fourier Series: Periodic functions, fundamental period, Trigonometric series, Fourier series, Bessel's inequality, Orthonormal and orthogonal set, Euler formulas, Functions with arbitrary periods, Even and odd functions , Half range expansions, Fourier coefficients without integration, Approximation by trigonometric polynomials, Application to differential equation.
- Fourier Transforms: Fourier integral theorem, Sine and Cosine Integrals, Inverse Transforms, Transforms of Elementary Functions, Properties, Convolution, Parsevals relation, Transform of Dirac Delta Function, Multiple Fourier transform, Finite Fourier transform.
- Z Transforms: Z-transforms, properties, Inverse Z- transforms, relationship with Fourier transforms.
- Complex Analysis: Complex numbers, Modulus, Argument, Curves and regions in complex plane, Functions, Limits, Derivatives, Analytic functions, Cauchy-Riemann equations, Complex exponential logarithms and trigonometric function, General powers, Line integrals, Cauchy's theorem, Cauchys integral theorem, Cauchys integral formula, Taylor and Laurent series , Zeros and singularities, Residues, Residues theorem, Evaluation of real improper integrals.
**Text Book**- E. Kreyszig,
*Advanced Engineering Mathematics*, Wiley. **Reference Books**- M. Braun,
*Differential Equations and Their Applications*, Springer-Verlag, New York. - W. Trench,
*Elementary Differential Equations*. - J. Schiff,
*The Laplace Transform: Theory and Applications*, Springer. - J. Brown and R. Churchill,
*Complex Variables and Application*, McGraw-Hill. - G. F. Simmons,
*Differential Equations*, Tata Mcgraw Hill. - R. Jain and S. Iyenger,
*Advanced Engineering Mathematics*, Narosa. **Questions and Keys**- Back Paper Examination: Question Paper Marking Scheme