Advanced Linear Algebra, Complex Analysis and Partial Differential Equations

(Text 1: Relevant topics from sections 2.1, 2.2, 2.3, 2.4, 2.5) Vector Spaces, Subspaces -Definition and Examples. Linear independence of vectors, Linear span, Basis and dimension, Co-ordinate representation of vectors, Row space and Column space

(Text 1: Relevant topics from sections 5.1, 5.2, 5.5) Inner Product: Inner product spaces, properties of inner product, length and distance, Orthogonality, Cauchy-Schwarz inequality, Orthogonal projection, orthogonal compliment, Orthonormal basis, Gram Schmidt orthogonalization process.

(Text 2: Relevant topics from sections 13.3,13.4) Circles and disks half planes, complex functions, limit, continuity and derivatives, analytic functions, Cauchy-Riemann equations, Laplace equation, Harmonic functions, harmonic conjugate functions

(Text 2- Relevant topics from sections 14.1,14.2,14.3,14.4,15.4,16.1,16.2,16.3) Cauchy’s integral theorem for simply connected domains (without proof), Cauchy’s Integral formula for simply connected domains (without proof), Cauchy’s Integral formula for derivatives of an analytic function, Taylor's series, Maclaurin series and Laurent's series, Poles and Residues, Evaluation of residues, Cauchy's residue theorem.

(Text 3: Relevant topics from sections 17.1,17.2,17.3,17.4,17.5) Introduction, Formation of partial differential equations -elimination of arbitrary constants-elimination of arbitrary functions, Solutions of partial differential equations, Equations solvable by direct integration, Linear equations of the first order, Lagrange's linear equation