Linear Algebra,complex variable functions

Course Description:

This course introduces the topics involving: Linear Algebra, Complex variable functions, Ordinary

Differential Equations and their applications. The course starts with algebra of matrix, systems of

linear equations and with preliminary course on complex variable. It introduces the CR equation,

analytic function, Taylor and Laurent series expansions and determination of residues. Emphasis also

placed on the development of concepts and applications for first and second order ordinary differential

equations (ODE), systems of differential equations and Laplace transforms.

Course Objectives:

1. To solve the system of linear equations, and develop orthogonal transformation with emphasis on the role of eigen-values and eigen-vectors.

2. To analyze the function of complex variable and its analytic property with a review of elementary complex function.

3. To understand the Taylor and Laurent expansion with their use in finding out the residue and improper integral.

4. To identify important characteristics of ODE and develop appropriate method of obtaining solutions of ODE.

5. Explore the use of ODE as models in various applications to solve initial value problems by using

Laplace transform method.

UNIT I: MATRICES

Symmetric, Skew-symmetric and Orthogonal matrices, Determinants, System of linear equations,Inverse and rank of a matrix, Rank-nullity theorem, Eigen values and eigenvectors, Diagonalization of matrices, Cayley-Hamilton Theorem, and Orthogonal transformation.(12)

UNIT II: COMPLEX VARIABLE – DIFFERENTIATION

Differentiation, Cauchy-Riemann equations, Analytic function, Harmonic functions, finding  harmonicconjugate, Elementary analytic functions (exponential, trigonometric, logarithm) and their

properties.(12)

UNIT III: COMPLEX VARIABLE – INTEGRATION

Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy integral formula (without proof),Liouville’s and Maximum-Modulus theorem (without proof); Taylor’s series, Zeros of analyticfunctions, Singularities, Laurent’s expansion (without proof), Residues, Cauchy Residue theorem (without proof), Evaluation of definite integral involving sine and cosine, Evaluation of certain improper integrals using the Bromwich contour.(12)

UNIT IV: FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

Exact, Linear and Bernoulli’s, Equations not of first degree: equations solvable for p, equations solvable for x, equations solvable for y and Clairaut’s type.(12)

UNIT V: ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDERS

Second order linear differential equations with variable coefficients, Method of variation of parameters, Laplace Transform, Inverse Laplace transform, Bromwich contour method,and its applications to solve ordinary differential equations.(12)

Text books:

1. Higher Engineering Mathematics by Dr. B.S. Grewal, 42nd Edition, Khanna Publishers.

2. Complex variables and applications by R. V Churchill and J. W. Brown, 8th edition, 2008,

McGraw-Hill.

3. Differential Equations with applications and historical notes by G.F. Simmonssecond edition,

McGraw Hill, 2003.

References:

1. Elementary linear Algebra by Stephen Andrilli and David Hecker, 4th Edition, Elsevier, 2010

2. Ordinary and partial differential equations. By M.D. Raisinghania, 2013. S. Chand Publishing.

3. Linear Algebra and its Applications by D.C. Lay, 3rd edition, Pearson Education, Inc.

Course outcomes

Students are able to

1. Solve the systems of linear equations occurring in engineering system.

2. Determine harmonic function, velocity potential and stream lines in fluid flow systems.

3. Evaluate a contour integral and definite integral involving exponential, sine and cosine functions.

4. Find general solutions to first and second order homogeneous differential equations by algebraic

and computational methods.

5. Determine the solution of ODE of second and higher order.