# Vector Calculus & Linear Algebra - [2110015]

#### Teaching Scheme

 Theory Tutorial Practical Total 3 2 0 5

#### Examination Scheme

 Theory Examination Practical Examination Total ESE (E) PA (M) ESE Viva (V) PA (I) 70 30 30 20 150

ESE = End Semester Examination, PA = Progressive Assessment

#### Systems of Linear Equations and Matrices

Systems of Linear Equations , Matrices and Elementary Row Operations , The Inverse of a Square Matrix , Matrix Equations , Applications of Systems of Linear Equations

#### Linear Combinations and Linear Independence

Vectors in R^n,Linear Combinations, Linear Independence,Definition of a Vector Space, Subspaces, Basis and Dimension, Coordinates and Change of Basis

#### Linear Transformations

Linear Transformations ,The Null Space and Range ,Isomorphisms ,Matrix Representation of Linear Transformations ,Similarity ,Eigen values and Eigen vectors Diagonalization

#### Inner Product Spaces

The Dot Product on R^n and Inner Product Spaces, Orthonormal Bases, Orthogonal Complements, Application: Least Squares Approximation, Diagonalization of Symmetric Matrices, Application: Quadratic Forms

#### Vector Functions

Vector & Scalar Functions and Fields, Derivatives , Curve, Arc length, Curvature & Torsion , Gradient of Scalar Field, Directional Derivative , Divergence of a Vector Field , Curl of a Vector Field

#### Vector Calculus

Line Integrals , Path Independence of Line Integrals , Green`s Theorem in the plane , Surface Integrals , Divergence Theorem of Gauss , Stokes`s Theorem

#### Reference Books

1Introduction to Linear Algebra with ApplicationJim Defranza, Daniel GagliardiTata McGraw-Hill
2Elementary Linear Algebra, Applications versionAnton and RorresWiley India Edition
4Elementary Linear AlgebraRon LarsonCengage Learning
5Calculus, Volumes 2T. M. ApostolWiley Eastern

#### Course Outcome

1. System of linear equations in solving the problems of electrical engineering, mechanical engineering, applied mechanics etc.

2. Use of matrix in graph theory, linear combinations of quantum state in physics, computer graphics and cryptography etc.

3. Students will be able to apply vectors in higher dimensional space in experimental data, storage and warehousing, electrical circuits, graphical images, economics, mechanical systems and in physics.

4. Students will able to use eigen values and eigen vector in Control theory, vibration analysis, electric circuits, advanced dynamics and quantum mechanics.

5. Students will be able to apply linear transformation in computer graphics, cryptography, thermodynamics etc.