# Math2

MAIR 21 MATHEMATICS II

Objectives: To learn mathematical concepts and methods.

Course Content

Vector space – Subspaces – Linear dependence and independence – Spanning of a subspace – Basis and Dimension. Inner product – Inner product spaces – Orthogonal and orthonormal basis – Gram- Schmidt orthogonalization process.

Basic review of first order differential equation - Higher order linear differential equations with constant coefficients –Particular integrals for x n eax , eax cos (bx), eax sin (bx) – Equation reducible to linear equations with constant coefficients using x et - Simultaneous linear equations with constant coefficients – Method of variation of parameters – Applications – Electric circuit problems.

Gradient, Divergence and Curl – Directional Derivative – Tangent Plane and normal to surfaces – Angle between surfaces –Solenoidal and irrotational fields – Line, surface and volume integrals – Green’s Theorem, Stokes’ Theorem and Gauss Divergence Theorem (all without proof) – Verification and applications of these theorems.

Analytic functions – Cauchy – Riemann equations (Cartesian and polar) –Properties of analytic functions – Construction of analytic functions given real or imaginary part – Conformal mapping of standard elementary functions (z^2, e^z, sin z, cos z, z+k^2/z ) and bilinear transformation.

Cauchy’s integral theorem, Cauchy’s integral formula and for derivatives– Taylor’s and Laurent’s expansions (without proof) – Singularities – Residues – Cauchy’s residue theorem – Contour integration involving unit circle.

Outcome: After the completion of the course, students are able to solve industrially applicable problems.

Text Books

1. Kreyszig, E., Advanced Engineering Mathematics, 9th edition, John Wiley Sons, 2006.
2. Grewal, B.S., ‘Higher Engineering Mathematics’, 42ndedition, Khanna Publications, Delhi, 2012.
3. Hsiung, C.Y. and Mao, G. Y. ‘Linear Algebra’, World Scientific Pub Co Inc., 1999.

Reference Books

1. Apostol, T.M. ‘Calculus’, Volume I & II, 2ndEdition, John Wiley & Sons (Asia), 2005.
2. Greenberg, M.D. ‘Advanced Engineering Mathematics’, 2ndEdition, Pearson Education Inc. (First Indian reprint), 2002.
3. Strauss. M.J, Bradley, G.L. and Smith, K.J. ‘Calculus’, 3rdEdition, Prentice Hall, 2002.
4. Venkataraman, M. K. ‘Linear Algebra’, The National Publishing Co, 1999

Evaluation and Examinations: For FA, one has to write four assignemnt tests. For ReDo, one has to write four assignments (20%), a class test (30%) and a final examination (50%). Attendance requirement is as per Institute norms.