# ENG (SERC): 1-4772 – Mathematics for Engineers

#### Faculty - Incharge:

Shri. J. Suriya Praksh, Dr. Madan Kumar Lakshmanan.

#### Module I. Linear Algebra

Week 1Introduction to LA
Vector & Linear combination., Length & Dot Product, Determinant – Properties, Cofactor, Cramers Rule, Volume Linear Equation – Elimination, A=LU form, Transpose & Permutation
Week 2Vector Spaces
Spaces & Subspaces, Null space Ax=0; Rank & Row Reduced Form, Solution to Ax=b; Independence, basis and Dimension, Dimension of four spaces
Week 3Orthogonality, Projection Gram- Schmidt
Orthogonality in four spaces, Projection & Least square Approximation, Gram- Schmidt Process
Week 4Eigen Values and Vectors, Differential Eqn solution, Matrix Diagonalization
Week 5Singular Value Decomposition, Other Matrices
Introduction to Singular Value Decomposition, Other important Matrices – Symetric, Positive Matrices,
Evaluation I

#### Module II. Differential Equations

Week 6Fundamentals of differential equations:
How differential equations have been applied in diverse fields to understand physical/biological phenomenon; Examples such as Maxwell's equations, heat equations, progress of Avian influenza virus where DE has found applicability to solve practical problems will be highlighted; Types of differential equations - ODE/PDE, linear/non-linear, constant/variable co-efficient; Solution of a differential equation - general solutions vs actual/particular solutions, initial value problems, existence and uniqueness of a solution; What are direction fields and how they can be used to obtain a geometric view of the differential equations.
Week 7First order differential equations
General form of expression; Methods to solve 1st order differential equations of the form (all cases handled with examples);
Problems which are linear in nature
Problems where the differentials are separable
Bernoulli's differential equations
Homogeneous differential equations
Week 8Numerical methods to solve first order differential equations
An introduction to numerical techniques for DEs; Euler's method to solve ODE's numerically (uniform and non-uniform steps); Runge Kutta method and backward Euler methods as extensions to the Euler's method to solving DEs numerically
Week 9Second Order differential equations
General form of expression; Types of second order ODEs (homogeneous/non-homogeneous, constant/variable coefficients, linear/non-linear); Solving 2nd order differential equations which are linear, homogenous and have constant coefficients (all cases handled with examples):
I General form of solution
What are characteristic equations
Solution for the cases when (a)the roots are real and distinct, (b)the roots are Complex and (c) double roots.
I Reduction of order of differential equations to solve cases when the coefficients are not constants.
I Wronskian condition and fundamental sets of solutions - the necessary conditions between the roots under which a valid solution can exist
Week 10Partial differential equations
An introduction to PDE + general form of expression; Types of PDEs - elliptic, parabolic and hyperbolic; The procedure to solve PDEs illustrated with 2 examples, namely, solving the 1st order heat equation and solving the 1st order wave equation
I Solving the heat equation (1st order)
General form of expression
Expatiation of the initial value and boundary conditions
Explanation of terminologies such as specific heat, mass density and heat flux
Simplification of the heat equation (moving from 2 unknowns to 1 unknown and considering the case of a uniform surface)
Moving from PDE to ODE through separation of variables
Finding out the general solutions to the heat equation.
I Solving the wave equation (1st order)
General form of expression
Initial value and boundary conditions
Separation of variables to simplify problem
Derivation of the general solution to the wave equation
The relationship between the solutions to the heat and wave equations and the connection to the derivation of the Fourier series representation
A brief introduction to solving 2nd and 3rd order heat and wave equations.
Evaluation II

#### Module III. Mathematical transforms

Week 11What are mathematical transforms, their significance and why they are useful; An introduction to Fourier series and Fourier transform (FFT, DFT, DTFT)
Week 12Gabor (Short time fourier) transform ; Wavelet (discrete and continuous) transforms; Karhunen-Loeve transform
Week 13Laplace and Z-transform (with examples)
Evaluation III

#### Module IV. Mathematical methods for data analysis

Week 14Review of Probability
Probability function; Conditional Probability; Independence, Random Variable; Expectation, Variance, Standard Deviation; Distributions – Binomial, Poisson, Exponential, Normal
Week 15Statistics Basics
Research methods, Histograms, Summary statistics; Correlation analysis, Sampling and standard Error,
Week 16Statistics Methods
Regression Analysis; Central Limit Theorem; Multiple regression; T test and ANOVA; Chi-Square Test; logistic Regression; Non parametric Tests
Week 17Data analysis Methods
Descriptive, Exploratory, Inferential, predictive, Causal Data Analysis; Clustering – Hierarchical, K-mean; Dimensionality reduction – SVD, PCA
Week 18Prediction and Data classification
Data collection, feature creation, training and test sets for Data Analysis; Methods – Naive Bayes, Random Forest, Classification trees
Final Evaluation

#### Books

 Calculus Text Book Thomas’ Calculus, Pearson education 11th ed, 2007, by G.B. Thomas, M.D. Weir, J. Hass and F.R. Giordano. Reference Books 1. James Stewart – Calculus, 5e, Cengage learning, 2003. 2. Erwin Kreyszig – Advanced Engineering. Mathematics, 8th Edition Wiley-India,1999. Linear algebra Text Book Introductory Linear Algebra with Applications by B. Kolman and D.R. Hill, 8th edition, 2005, Pearson Education, Inc Reference Books 1. Linear Algebra and its Applications by D.C. Lay, 3rd edition, Pearson Education, Inc. 2. Introduction to Linear Algebra, by Gilbert Strang, Wellesley-Cambridge Press (1993). 3. Linear Algebra: A Modern Introduction, by David Poole. 4. Linear Algebra and Its Applications, by Gilbert Strang. Sequences and Series Text Book Complex Variables and applications by R.V. Churchill and J.W. Brown, 8th edition, 2008, McGraw-Hill. Reference Books Complex Variables with Applications by A.D. Wunsch, 3rd edition, Pearson Education, Inc. Differential Equations Text Book Simmons G.F., Differential Equations with Applications and Historical Notes, TMH, 2nd  ed.,1991. Reference Books 1. Edwards & Penney: Differential Equations and Boundary value problems, Pearson Education, 3rd ed. 2. Shepley L. Ross: Differential Equations, Willy India Pvt. Ltd, 3rd ed. 3. Birkhoff & Rota: Ordinary Differential Equations, Wiley India Pvt. Ltd., 4th ed. 4. Zill, Differential Equation, Thomson Learning, 5th ed., 2004 5. R.K. Patnaik: Differential Equation, PHI, 2009. Transforms Text Book Oppenheim, Alan V., and A. S. Willsky. Signals and Systems. Prentice Hall, 1982.ISBN: 9780138097318. Reference Books B.Burke, The World According to Wavelets: The Story of a Mathematical Technique in the Making. Upper Saddle River,New Jersey: A K Peters, May 1998.