Differential calculus: Functions of real variable and their plots, limit. Continuity and derivatives, physical meaning of derivative of a function. Successive derivatives: Leibniz Theorem; Roll’s theorem, new value theorem, and Taylor’s theorem. Taylor’s and Maclaurin’s series and expansion inunctions. Maximum and minimum values of function: Functions of two or three variables partial and total derivatives. Euler’s theorem. Tangent and normal. Subtangent and subnormal in Cartesian and polar coordinates. Curvature, asymptotes and curve tracing. Integral Calculus: Different techniques of integration. Definite integral as the limit of a sum and as an area. Definition of remain integral, fundamental theorem of integral calculus and its applications to definite integrals, reduction formulae, Walli’s formulae. Improper integrals. Improper integrals: Beta and gamma functions.

# Course Catalogue

Differential Calculus:

Functional Analysis and Graphical Information: function, properties of functions, graphs of functions, new function from old, lines and family of functions, Limit: Limits( an informal view), one sided limits, the relation between one sided and two sided limits, computing limits, Continuity: continuity and discontinuity, some properties of continuity, the intermediated value theorem, Derivatives: slop and rate of change, tangent and normal, derivative of a function, physical meaning of derivative of a function, techniques of differentiation, chain rule, successive derivatives, Derivative in graphing and applications: analysis of functions, maximum and minimum, Expansion functions: Taylor’s series, Maclaurian’s series, Leibniz; Rolle’s and Mean Value theorems, Partials and total derivatives of a function of two or three variables.

Integral Calculus:

Different technique of integration: integration, fundamental integrals, methods of substitutions, integration of rational functions, integration by parts, integrals of special trigonometric functions, reduction formulae for trigonometric functions, Definite integrals: general properties of definite integral, definite integral as the limit of sum and as an area, definition of Riemann integral, Fundamental theorem of integral calculus and its applications to definite integrals, determination of arc length, Improper integrals, Double integrals, Evaluation of Areas and Volumes.

Differential Calculus:

Functional Analysis and Graphical Information: function, properties of functions, graphs of functions, new function from old, lines and family of functions, Limit: Limits( an informal view), one sided limits, the relation between one sided and two sided limits, computing limits, Continuity: continuity and discontinuity, some properties of continuity, the intermediated value theorem, Derivatives: slop and rate of change, tangent and normal, derivative of a function, physical meaning of derivative of a function, techniques of differentiation, chain rule, successive derivatives, Derivative in graphing and applications: analysis of functions, maximum and minimum, Expansion functions: Taylor’s series, Maclaurian’s series, Leibniz; Rolle’s and Mean Value theorems, Partials and total derivatives of a function of two or three variables.

Integral Calculus:

Different technique of integration: integration, fundamental integrals, methods of substitutions, integration of rational functions, integration by parts, integrals of special trigonometric functions, reduction formulae for trigonometric functions, Definite integrals: general properties of definite integral, definite integral as the limit of sum and as an area, definition of Riemann integral, Fundamental theorem of integral calculus and its applications to definite integrals, determination of arc length, Improper integrals, Double integrals, Evaluation of Areas and Volumes.

Coordinate Geometry:

Coordinates, polar coordinates, straight lines, changes of axes, pair of straight lines, circle parabola, ellipse, hyperbola, rectangular coordinate, plane.

Vector Analysis: vector components, vector components in spherical and cylindrical system, vector operators, scalar and dot product, application of vector geometry, derivative of vector, del, gradient, divergence and curl, physical significance, integration of vector, line surface and volume integration, Theorems (Green’s, Gauss’s, Liouville’s, Stoke’s) and their applications.

Linear Algebra:

Systems of Linear Equations (SLE): introduction to SLE, solution of a SLE, solution of a system of homogeneous LE, Gaussian and Gauss-Jordan elimination, Determinants: factorization, determinant, fundamental properties of determinants, minors and cofactors, Cramer’s rules for solving a SLE, Algebra of Matrices: Matrix, some special types of matrices, transpose, adjoint and inverse of a matrix, algebraic operation on matrices, quadratic forms solution of a LE by applying matrices, Vector Space: space and subspace, Euclidean n-space, basis and dimension, rank and nullity, Linear Transformations (LT) and its Matrix Representations: LT from to , properties of LT, matrix representation of a LT, diagonalization of LT, Eigen Values and Eigen Vectors: polynomials of matrices and linear operators, eigen values and vectors, diagonalizability, Cayley-Hamilton theorem, characteristic and minimum polynomial Inner Product Spaces: inner product spaces, Cauchy-Schwarz inequality, orthogonality, Gram-Schmidt orthogonalization process, linear functional and adjoint operators, Some Applications of LA.

Geometry: Double integration: Evaluation of surface areas and volumes by integration. Area under a plane curve in Cartesian and polar coordinates. Area of the region enclosed by two curves in Cartesian and polar coordinates, Trapezoidal rule, Simpson’s rule, Arc lengths of curves, in Cartesian and polar coordinates, parametric and pedal equations, Intrinsic equation. Volumes of solids of revolution. Volumes of hollow solids of revolutions by shell method. Area of surface of revolution. Linear Algebra: Introduction to systems of linear equations. Gaussian elimination. Definition of matrices. Algebra of matrices. Transpose of a matrix and inverse of matrix. Factorization. Determinants. Quadratic forms. Matrix polynomials. Euclidean n-space. Linear transformation from IRn to IRm. Properties of linear transformation from IRn to IRm. Real vector spaces and subspaces. Basis and dimension. Rank and nullity. Inner product spaces. Gram-Schmidt process and QR-decomposition. Eigenvalues and eigenvectors. Diagonalization. Linear transformations. Kernel and Range. Application of linear algebra to electric networks.