Course Details
Subject {L-T-P / C} : MA1002 : Mathematics - II { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
Matrix Theory: Gauss elimination method, Gauss-Jordon method for finding inverse of a matrix, Vector space, subspace, linear span, linear dependence and independence, Basis and dimension of vector space, Row and column spaces, Rank and nullity of a matrix, Rank and Nullity Theorem, Inner product spaces, Gram-Schmidt Orthogonalization, Matrix representation of Linear Transformations, Solvability of systems of linear equations, Eigen values, Eigen vectors, Diagonalization of matrices, Reduction of a quadratic form to canonical form.
ODE of first order: Geometrical interpretations, Separable equations, Reduction to separable form, Exact equations, Integrating factors, Linear equations, Bernoulli equations, orthogonal trajectories, Existence and uniqueness of IVP (Picard’s Theorem), Applications to physical problems.
ODE of higher order: Fundamental system and general solutions of homogeneous equations of order two, Wronskian, reduction of order, Solution of non-homogeneous equations by method of undetermined coefficients and variation of parameters. Extension to higher order differential equations, Euler-Cauchy equation, Power series method, Applications to physical problems.
Laplace Transform: Laplace and inverse Laplace transforms, existence of Laplace transform, first shifting theorem, transforms of derivative and integral, second shifting theorem, differentiation and integration of transforms, Convolution theorem, Solution of ordinary differential equations by using Laplace transform.
Course Objectives
- To introduce the concept of vector spaces, basis, linear transformation and inner product spaces.
- To introduce rank of a matrix and its application for testing the invertibility of a square matrix and solvability of linear system of equations.
- To teach different methods of solving ordinary differential equations of one/higher order.
- To introduce Laplace transform, it’s properties and applications in solving differential equations.
Course Outcomes
CO1: Students will learn to solve systems of linear equations, and to find inverse of a matrix by using Gauss-Jordan elimination method. Students will be able to find rank/nullity and eigenvalues/eigenvectors of a matrix. They will also learn about the diagonalization of a matrix. <br /> <br />CO2: Students will be able to find basis of finite dimensional vector spaces. They will learn about inner product, and how to transform a set of non-zero vectors into an orthonormal set. <br /> <br />CO3: They will be familiar with the methods of solving ordinary differential equations of one/higher order, including the power series method, and how to find orthogonal trajectories for a given one parameter family of curves. <br /> <br />CO4: Students will learn about the Laplace transform and it’s application in solving differential equations.
Essential Reading
- E. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons
- G. F. Simmons and S. G. Krantz, Differential Equations: Theory, Technique and Practice, Tata McGraw-Hill
Supplementary Reading
- G. Strang, Linear Algebra and its applications, Cengage Learning
- N. Piskunuv, Differential and Integral Calculus, CBS Publishers & Distributors