MAIR21 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 e^{ax} , e^{ax} cos (bx), e^{ax} sin (bx) – Equation reducible to linear equations with constant coefficients using x e^t - 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.

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.

4. Apostol, T.M. ‘Calculus’, Volume I & II, 2ndEdition, John Wiley & Sons (Asia), 2005.

5. Greenberg, M.D. ‘Advanced Engineering Mathematics’, 2ndEdition, Pearson Education Inc. (First Indian reprint), 2002.

6. Strauss. M.J, Bradley, G.L. and Smith, K.J. ‘Calculus’, 3rdEdition, Prentice Hall, 2002.

7. Venkataraman, M. K. ‘Linear Algebra’, The National Publishing Co, 1999

Evaluation and Examinations: One has to write an assignment test (10%), two class tests (20% each) and a final examination (50%). Attendance requirement is as per Institute norms.

The best place for more information and problems on vector calculus is the Schaum's outline of Theory and Problems of VECTOR ANALYSIS and an introduction to tensor analysis by Murray R. Spiegel, McGraw-Hill, London, 1959.