# 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**

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

**Reference Books**

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