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3.1: Continuity and Intermediate value theorem-lVT: first and second, version, 3.2: Increasing and decreasing function - Increasing and decreasing test,, critical point test, first derivative test, 3.3: Second derivative and concavity- second derivative test for local, maxima, minima and concavity, inflection points, 3.4: Drawing of Graphs- graphing procedure, asymptotic behaviour, 3.5: Maximum- Minimum Problems- maximum and minimum values on, intervals, extreme value theorem, closed interval test, word problems, 3.6: The Mean Value Theorem- The MVT, consequences of MVT-Rolles, , Theorem, horserace theorem, , 11.2: L'Hospital rule- Preliminary version, strengthened version, 4.1: Summation- summation, distance and velocity, properties of summation,, telescoping sum (quick introduction- relevant ideas only), 4.2: Sums and Areas-step functions, area under graph and its counterpart in, , distance-velocity problem, 4.3: The definition of Integral- signed area (The counterpart ofsigned area for, our distance-velocity problem), The integral, Riemann sums, 4.4: The Fundamental Theorem of Calculus-Arriving at FTC intuitively using, distance velocity problem, Fundamental integration Method, proof ofFTC, Area, under graph, displacements and velocity, 4.5: Definite and Indefinite integral-indefinite integral test, properties of, definite integral, fundamental theorem of calculus: alternative version, , (interpretation and explanation in terms ofareas), 4.6: Applications of the Integral- Area between graphs, area between, intersecting graphs, total changes from rates of change,, 9.1: Volume by slice method- the slice method, volume of solid of revolution, by Disk met hod, , 82, , 9.3: Average Values and the Mean Value Theorem for Integrals - motivation, , and definition ofaverage value, illustration, geometric and physical interpretation,, the Mean Value Theorem for Integrals, , References: I, 1 Soo T Tan: Calculus Brooks/ Cole, Cengage learning(2010 ) ISBN 0-534·, 46579-X, , 2, , Gilbert Strang: Calculus Wellesley Cambridge Press(J991JISBN:0-96140882-0, , 3, , Ron La rso n. Bru ce Edwards: Calculus( ll /e) Cengage learning(2018}, ISBN: 978-1-337-27534-7, , 4, , Robert A Adams & Christopher Essex : Calculus Single Variable (8/e), Pearson Education Canada (20131 ISBN: 032 1877403, , s, 6, , Joel Hass, Christopher Heil & Maurice D. Weir : Thomas', Calculus[14/e1 Pearson (2018) ISBN 0134438981, Jon Rogawski & Colin Adams: Calculus Early Transcendentals (3/e) W, fl., , Freeman and Company(2015) ISBN: 1319116450

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SECOND SEMESTER, MTS2 C0Z:MATHEMATICS-2, 4 hours/week, , 3 Credits, , 75 Marks[lnt.15 + Ext. 60), , Text ( 1), , Calculus I (2/e) : Jerrold Marsden & Ala n Wei nstein Springer-Verlag, , Text (2), , Calculus II (2/e) : Jerrold Marsden & Alan Weinstein Springer-Verlag, , Text[3), , Advanced Engineering Mathematics(6/e) : Dennis G Zill Jones &, , New York lnc(1985) ISBN 0-387-90974-5, New York lnc{1985) ISBN 0-387-90975-3, Bartlett learninQ, LLC(2018J/SBN: 978-1 -284-10590-2, , 5.1: Polar coordinates and Trigonometry - Cartesian and polar coordinates, (Ono, representation of points in polar coordinates, relationship between Cartesian, and polar coordinates, converting from one system to another and regions, represented by inequalities in polar system ore required), , 5.3 : Jnverse functions-inverse function test, inverse function rule, 5.6: Graphing in polar coordinates- Checking symmetry ofgraphs given in polar, equation, drawings, tangents to graph in polar coordinates, 8.3: Hyperbolic functions- hyperbolic sine, cosine, tan etc., derivatives, anti, differentiation formulas, 8.4: Inverse hyperbolic functions- inverse hyperbolic functions {their, , derivatives and anti derivatives), , 10.3: Arc length and surface area- Length of curves, Area of surface of, revolution about x and y axes, 10.4: Parametric curves- parametric equations of lines and circles, tangents, to parametric curves, length of a parametric curve, speed, 10.5: Length and area in polar coordinates- arc length and area in polar, coordinates , Area between two curves in polar coordinates, , 84, , 11.3: Improper integrals- integrals over unbounded intervals, comparison, test, integrals of unbounded functions, 11.4: Limit of sequences and Newton's method- E - N definition, limit of, powers, comparison test, Newton's method, 11.5: Numerical Integration· Riemann Sum, Trapezoidal Rule, Simpson's, Rule, 12.1: The sum of an infinite series- convergence of series, properties of, limit of sequences (statements only), geometric series, algebraic rules for, series, the i t h term test, 12.2: The comparison test and alternating series- comparison test ratio, comparison test, alternating series, alternating series test, absolute and, conditional convergence, 12.3: The integral and ratio test-integral test, p-series, ratio test, root test, 12.4: Power series - ratio test for power series, root test, differentiation, and integration of power series, algebr aic operation on power series, 12.5: Taylor's formula- Taylor and Maclaurian series, Taylor's formula with, remainder in integral form, Taylor's formula with remainder in derivative form,, convergence of Taylor series, Taylor series test, some important Taylor and, Maclaurian series

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7.6: Vector spaces - definition, examples, subspaces, basis, dimension, span, 7.7: Gram-Schmidt Orthogonalization Process- orthonormal bases for llln,, construction of orthonomal basis of !Rn, 8.2: Systems of Linear Algebraic Equations- General form, solving systems,, augmented matrix, Elementary row operations, Elimination MethodsGaussian elimination, Gauss-Jordan elimination, row echelon form, reduced row, echelon form, inconsistent system , networks, homogeneous system, over and, underdetermined system, 8.3: Rank of a Matrix- definition, row space, rank by row reduction, rank and, linear system, consistency oflinear system, , 85, , 8.4: Determinants- definition, cofactor (quick introduction), 8.5: Properties of determinant- properties, evaluation of determinant by row, reducing to triangular form, , 8.6: Inverse of a Matrix - finding inverse, properties of inverse, adjoint, method, row operations method, using inverse to solve a linear system, 8.8: The eig,envalue problem- Definition, finding eigenvalues and, eigenvectors, complex eigenvalues, eigenvalues and singular matrices,, eigenvalues of inverse, 8.9: Powers of Matrices- Cayley Hamilton theorem, finding the inverse, 8.10: Orthogonal Matrices- symmetric matrices and eigenvalues, inner, product, criterion for orthogonal matrix, construction of orthogonal matrix, 8.12Diagonalization- diagonalizable matrix -sufficient conditions, orthogonal, diagonalizability ofsymmetric matrix, Quadratic Forms, 8.13: LU Factorization- definition, Finding an LU- factorization, Doolittlle method,, solving linear systems ( by LU factorization), relationship to determinants, , References: I, 1, 2, 3, , Soo T Tan: Calculus Brooks/Cole, Cenaaae Learnina{ZOJO J/SBN 0-534-46579-X, Gilbert Stram!: Calculus Wellesley Cambridae Pressf199/ )ISBN:0-9614088-Z-O, Ron Larson. Bruce Edwa rds: Calculus( 11/e) Cengage l eamir1g(Z018) ISBN: 978-1-, , 337-27534-7, 4, , Robert A Adams & Christopher Essex : Calculus Single Vc,riab/e (8/e) Pearson, , liducation Canodo (2013) ISBN: 0321877403, 5, , Joel Hass, Christopher Heil & Maurice D. Weir : Thomas' Calculus(14/e) Pearson, , (ZOJBJ ISBN 0134438981, 6, , Peter, , V, , O'Neil:, , Advanced, , Engineering, , Leamina/201ZJ/SBN: 978-l-111-4Z741 -2, 7, , Mathematics(7 / e), , Cengage, , Erwin Kreyszig : Advanced Engineering Mathe matlcs(l0/e) Jahn Wiley &, , Sons(Z011) ISBN: 978-0-470-45836-5, 8, , Glyn J.imes: Advanced Modern Engineering Mathemalics(4/c) Pearson E'ducat1011, , limited(ZOJ 1) ISBN: 978-0-Z73-719Z3-6, , 86

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THIRD SEMESTER, MTS3 C03:MATHEMATICS-3, 5 hours/week, , 3 Credits, , 75 Marks[lnt.15 + Ext. 60), , Text Advanced Engineer ing Mathematics(6/c) : Dennis G Zill Jones & Bartlett, Leornin, , LLC 2018 ISBN: 978-1-284 -10590-2, , 9.1: Vector Functions - Vector-Valued Functions, Limits, Continuity, and Derivatives,, Geometric Interpretation of r '(t), Higher-Order Derivatives, Integrals of Vector, Functions, Length of a Space Curve, Arc Length as a Parameter, 9.2: Motion on a Curve•Velocity and Acceleration, Centripetal Acceleration,, Curvlltnear Motlon In the Plane, 9.3: Curvature and components of Acceleration· definition, Curvature ofa Circle,, Tangen·llal and Normal Components of Acceleration, The Blnormal, Radius of Curvature, 9.4: Partial Derivatives-Functions of Two Variables, Level Curves, Level Surfaces,, Higher-Order and Mixed Derivatives, Functions of Three or More Variables, Chain Rule,, Generalizations, 9.5: Directional Derivative-The Gradient of a Function, A Generalization of Partial, Differentiation, Method for Computing the Directional Derivative, Functions of Three, Variables, Maximum Value of the Directional Derivative, Gradient Points In Direction of, Most Rapid Increase of/, 9.6: Tangent planes and Normal Lines-Geometric Interpretation of the Gradient,, Tangent Plane, Surfaces Given by z = f(x,y), Normal Line, 9.7: Curl and Divergence-Vector Fields, definition of curl and divergence, Physical, Interpretations, 9.8: Line Integrals-definition of smooth.closed and simple closed curves, Line, Integrals In the Plane, Method of Evaluation-curve as explicit function and curve glven, parametrically, Line Integrals In Space, Method of Evaluation, Work, Circulation, 9.9: Independenc e of Path· Conservative Vector Fields, Path Independence, A, Fundamental Theorem, definition of connected,slmply connected and multlconnected, , 87, , regions, Integrals Around Closed Paths, Test for a Conservative Field, Conservative, Vector Fields In 3-Space, Conservation of Energy, 9.10: Double Integral- lntegrablltty, Area, Volume, Properties, Regions of Type I and, II, Iterated Integrals, Evaluation of Double Integrals (Fubinl theorem), Reversing the, Order of Integration, Lamlnas with Variable Density- Center of Mass, Moments of, Inertia, Radius of Gyration, 9.11: Double Integrals in Polar Coordinates· Polar Rectangles, Change of, Variables: Rectangular to Polar Coordinates,, 9.12: Green's Theorem· Line Integrals Along Simple Closed Curves, Green's theorem In, plane, Region with Holes,, , 9.13: Surface Integral· Surface Area, Differential of Surface Area, Surface Integral,, Method of Evaluation, Projection of S Into Other Planes, Mass of a Surface, Orientable, Surfaces, Integrals of Vector Fields-Flux,, 9.14: Stokes's Theorem- Vector Form of Green's Theorem, Green's Theorem in 3Space-Scoke's Theorem, Physical Interpretation of Curl

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9.15:Triple Integral· defin i tion, Evaluation by Iterated Integrals,, Applications, Cylindrical Coordinates, Conversion of Cylindrical, Coordinates to Rectangular Coordinates, Conversion of Rectangular, Coordinates to Cylindrical Coordinates, Triple Integrals in Cylindrical, Coordinates, Spherical Coordinates, Conversion of Spherical Coordinates to, Rectangular and Cylindrical Coordinates, Conversion of Rectangular, Coordinates to Spherical Coordinates, Triple Integrals in Spherical, Coordinates, 9.16: Divergence Theorem- Another Vector Form of Green's Theorem ,, divergence or Gauss' theorem, ( proof omitted }, Physical Interpretation of, Divergence, 9.17: Change of Variable in Multiple Integral - Double Integrals, Triple, Integrals, 17.1: Complex Numbers- definition, arithmetic operations, conjugate,, Geometric Interpretation, 17.2: Powers and roots-Polar Form, Multiplication and Division, Integer Powers, of z, DeMoivre's Formula, Roots, , 88, , 17.3: Sets in the Complex Plane- neighbourhood, open sets, domain, region etc., 17.4: Functions of a Complex Variable- complex functions, Complex Functions, as Flows, Limits and Continuity, Derivative, Analytic Functions· entire functions, 17.5: Cauchy Riemann Equation- A Necessary Condition for Analyticity,, Criteria for analyticity, Harmonic Functions, Harmonic Conjugate Functions,, 17.6:Exponential and Logarithmic function- (Complex)Exponential Function,, Properties, Periodicity, ('Circuits' omitted), Complex Logarithm-principal value,, properties, Analyticity, 17.7: Trigonometric and Hyperbolic functions- Trigonometric Functions,, Hyperbolic Functions, Properties -Analyticity, periodicity, zeros etc., 18.1: Contour integral- definition, Method of Evaluation, Properties,, inequality. Circulation and Net, , ML-, , 18.2: Cauchy-Goursat Theorem- Simply and Multiply Connected Domains, Cauchy's, Theorem, Cauchy-Goursat theorem, Cauchy-Goursat Theorem for Multiply Connected, Domains,, 18.3: Independence of Path· Analyticity and path independence, fundamental, theorem for contour integral, Existence ofAntiderivative, 18.4: Cauchy's Integral Formula- First Formula, Second Formula-C./.F., derivatives. Liouville's Theorem, Fundamental Theorem of Algebra, References:, , 1, , for, , I, , Soo T Tan: Calculus, , Brooks/Cole, Cengage Learning{2010 )ISBN 0-534-, , 46579-X, , 2, , Gilbert Strang: Calculus, , Wellesley Cambridge Press{199l}ISBN:0-9614088-2-, , 0, , 3, , Ron Larson. Bruce Edwards: Calculus(l 1/e), , Cengage Learning(2018) ISBN:, , 978-1-337-27534-7, , 4, , s, 6, 7, 8, , Robert A Adams & Christopher Essex : Calculus several Variable (7/e), l'eorson Educat,011 Canada (20101 ISBN: 978-0-32 1-54929-7, Jerrold Marsden & Anthony Tromba : Vector Calculus (6/e) W. H. Freeman, and Comoanv ISBN 978-1 -4292-1508-4, Peter V O'Neil: Advanced Engineering Mathematics(7 /e) Cengage, Learnina(2012J/SBN: 978-1- 111 -42741 -2, Erwin Kreyszig : Advanced Enginee1ing Mathematics(lO/e) John Wiley &, Sons(2011) ISBN: 978-0-470-45836-5, Glyn James: Advanced Modern Engineering Mathematics(4/e) Pearson, Education Li mited[2011) ISBN: 978-0-273-71923-6, , 89

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', , FOURTH SEMESTER, , -, , MTS4 C04:MATHEMATICS-4, 5 hours/week, , 3 Credits, , 75 Marks[lnt.15 + Ext. 60], , Text Advanced Engineering Mathematics(6/e) : Dennis G Zill Jones & Bartlett, Learnin<J, LLC(2018}/SBN: 978-1-284-10590·2, , Ordinary Differential Equations, , 1.1: Definitions and Terminology- definition, Classification by Type,, Classification by Order, Classification by Linearity, Solution, Interval of, Definition, Solution Curve, Explicit and Implicit Solutions, Families of, Solutions, Singular Solution, Systems of Differential Equations, 1.2: Initial Value Problems-First- and Second-Order IVPs, Existence ofsolution, 1.3: Differential Equations as Mathematical Models- some specific differentialequation models in biology, physics and chemistry., 2.1: Solution Curves without Solution-Direction Fields {'Autonomous FirstOrder DEs' omitted], , 2.2: Separable Equations- definition. Method of solution, losing a solution, An, Integral-Defined Function, 2.3: Linear Equations-definition, standard form, homogeneous and non, homogeneous DE, variation of parameter technique, Method of Solution, General, Solution, Singular Points, Piecewise-Linear Differential Equation, Error Function, 2.4: Exact Equations- Differential of a Function of Two Variables, Criteria, for an exact differential, Method of Solution, Integrating Factors,, 2.5: Solutions by Substitution-Homogeneous Equations, Bernoulli's Equation,, Reduction to Separation of Variables, 2.6: A Numerical Method- Using the Tangent Line, Euler's Method [upto and, including Example 2; rest omitted], , 90

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Higher Order Differential Equations, 3.1: Theory of Linear Equations- Initial -Value and Boundary-Value, Problems [Existence and Uniqueness (of solutions), Boundary-Value Problem), , Homogeneous Equations [Differential Operators, Superposition Principle,, Linear Dependence and Linear Independence, Wronskian], Nonhomogeneous, Equations, Superposition Principle], , [Complementary, , Function,, , Another, , 3.2: Reduction of Order- a general method to find a second solution of linear, second order equation by reducing to linear first order equation, , 3.3: Homogeneous Linear Equations with Constant Coefficients- Auxiliary, Equation, Distinct Real Roots , Repeated Real Roots , Conjugate Complex Roots,, Higher-Order Equations, RationaJ Roots ['Use of computer' part omitted], 3.4: Undetermined Coefficients- Method of Undetermined Coefficients for finding, , out particular solution, 3.5: Variation of parameter- General solution using Variation of parameter, technique, 3.6: Cauchy-Euler Equations- Method of solution, Distinct Real Roots,, Repeated Real Roots, Conjugate Complex Roots, 3.9: Linear Models & Boundary Value Problems- Deflection of a Beam,, Eigenvalues and Eigenfunctions [upto and including Example 3: the rest is, omitted], Laplace Transforms, , 4.1: Definition of Laplace Transform- definition, examples, linearity,, Transforms of some basic functions, Sufficient Conditions for Existence of, transform,, 4.2: Inverse Transform and Transforms of Derivative- Inverse Transforms:A few important inverse transforms, Linearity, Partial Fractions, Transforms of, Derivatives, Solving Linear OD Es, , 91

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4.3: Translation Theorems- Translation on the s-axis, first translation, theorem, its inverse form, Translation on the t-axis, Unit step function, second, translation theorem. Its Inverse form, Alternative Form of second translation, theorem Beams, 4.4: Additional Operational Properties- Derivatives of Transforms,, Transforms of Integrals -convolution, convolution theorem (without proof] and, its inverse form, Volterra Integral Equation, Series Circuits {'Post ScriptGreen's Funclion Redux' omitted},Transform of a Periodic Function, 4.5: The Dirac delta Function- Unit Impulse, The Dirac Delta Function and its, transform,, , 12.1:, , Orthogonal Functions- Inner Product, Orthogonal Functions,, Orthonormal Sets, Vector Analogy, Orthogonal Series Expansion, Complete Sets,, 12.2: Fourier Series-Trigonometric Series, Fourier Series, Convergence of a, Fourier Series, Periodic Extension, Seql.llence of Partial Sums,, 12.3: Fourier Cosine and Sine Series- Even and Odd Functions., Properties,, Cosine and Sine Series, Gibbs Phenomenon, Half-Range Expansions, Periodic, Driving Force,, 13.1: Separable Partial Differential Equations- Linear Partial Differential, Equation, Solution of a PDE, Separation of Variables ( Method ), Superposition, Principle, Classification of Equations (• hyperbolic, parabolic, elliptic), 13.2: Classical PDE's and BVP's- Heat Equation, Wave Equation, Laplace's, Equation, Initial Conditions, Boundary Conditions, Boundary-Value Problems, ('Variations' omitted), 13.3: Heat Equation- Solution of the BVP ( method ofSeparation of Variables), References:, , 1, , 2, 3, 4, , I, , Peter V O'Neil: Advanced Engineering Mathematics(7 /e) Cengage, learnina(2012HSBN: 978-1-111 -42741 -2, Erwin Kreyszig : Advanced Engineering Mathematics(l0/e) John Wiley &, Sons(2011 I ISBN: 978-0-470-45836-S, Alan Jeffrey: Advanced Engineering Mathematics Harcourt/Academic, Press(2002) ISBN: 0-12-382592-X, Glyn James: Advanced Modern Engineering Mathematics( 4 /e) Pearson, Education Limitedf201 l) ISBN: 978-0-273 -71923-6, , 92