| Unit 1: Partial Differentiation | Functions of two variables Neighborhood of a point(a, b) Continuity of a Function of two
variables Continuity at a point Limit of a Function of two variables Partial Derivatives - Geometrical representation of a Function of two Variables - Homogeneous Function Theorem on Total Differentials Composite Functions Differentiation of Composite Functions Implicit Functions Equality of fxy(a,b) and fyz(a,b) Taylor’s theorem for function of two variables Maxima and Minima of functions of two variables Lagrange’s Method of undetermined
multipliers
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| Unit 2: Curvature and Lengths of Plane Curves | Definition of Curvature Radius of Curvature Length of Arc as a Function Derivative of arc Radius of Curvature Cartesian Equations Cartesian Equations Newtonian Method Centre of Curvature Chord of Curvature Evolutes and Involutes Properties of the evolutes One Parameter Family of Curves, Consider the family of straight lines. Determination of Envelope. Lengths of Plane Curves: Expression for the lengths of curves y = f (x), Expressions for the length of arcs x =
f(y); x = f(t), y = φ(t); r=f(θ)
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| Unit 3: Differential Equations of First Order | Variables and Separable method Homogeneous Differential Equations DifferentialmEquations Reducible to Homogeneous Form Linear Differential Equations Differential Equations Reducible to Linear Form Exact differential equations Integrating Factors Change in variables. Differential Equations of first order but not first degree: Equations Solvable for p Equations solvable for y Equations Solvable for x Equations that do not contain x or y Equations Homogeneous in x and y Equations of the First degree in x and y Clairaut’s equation.
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| Unit 4: Higher order Linear Differential Equations and PDE | Higher order Linear Differential Equations: Solution of homogeneous linear Differential equations with constant coefficients Solution of non-homogeneous differential equations P (D)y = Q(x) with constant coefficients by means of polynomial operators when Q(x) = eୟ୶, b sin ax, b cos ax, bx୩, Veୟ୶ Method of undetermined coefficients Method of variation of parameters Linear differential equations with non-constant coefficients the Cauchy-Euler Equation Legendre’s Linear EquationsPartial Differential Equations: Formation and solution,
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| Unit 5: Sequences and Series | Sequences: Limits of Sequences Limit Theorems for Sequences Monotone Sequences and Cauchy Sequences Subsequences, Lim sup and Lim infimum, Series, Alternating Series and Integral Tests.
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| Unit 6: Continuity, Differentiation and Riemann Integral | Continuity: Continuous Functions Properties of Continuous Functions Uniform Continuity Limits of Functions Basic properties of the derivative The mean value theorem L-Hospital Rule Taylor’s theorem. The Riemann Integral, Properties of Riemann Integral, Fundamental Theorem of Calculus.
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| Unit 7: Groups | Definition and Examples of Groups Elementary Properties of Groups Finite Groups, Subgroups, Subgroup Tests, Cosets and Lagrange’s Theorem Properties of Cosets, Cyclic Groups Properties of Cyclic Groups Normal Subgroups and Factor Groups Group Homomorphism’s Properties of Homomorphism’s The First Isomorphism Theorem, Automorphisms Permutation Groups: Definition and Properties of Permutations Isomorphisms, Cayley’s Theorem
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| Unit 8: Rings | Rings Examples of Rings Properties of Rings Subrings Integral Domains Fields Characteristics of a Ring Ideals, Factor Rings, Prime Ideals and Maximal Ideals Ring Homomorphisms and Isomorphisms.
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| Unit 9: Vector Spaces | Vector Spaces and Subspaces -Null Spaces, Column Spaces,, and Linear Transformations Linearly Independent Sets, Bases, Coordinate Systems The Dimension of a Vector spaces Rank Change of Basis
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| Unit 19: Diagonalization and Orthogonality | Eigen values and Eigenvectors The Characteristic Equation Diagonalization, Eigenvectors of Linear Transformations Inner Product spaces Length, and Orthogonality Orthogonal Sets Orthogonal Projections The Gram-Schmidt Process
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