MA6251 4.4 MATHEMATICS-II Construction of Analytic Functions 56 Problems 4.5 Conformal Mapping 58 Problems 4.6 Bilinear Transformation 61 Problems UNIT V- COMPLEX INTEGRATION 5.1 Prerequisite 62 5.2 Introduction 62 5.3 Cauchy’s Theorem 62 Problems 5.4 Taylor’s and Laurent’s Series Expansion. 64 Problems 5.5 Singularities 67 Problems 5.6 Residues 69 Problems 5.7 Evaluation of real definite Integrals as contour integrals 72 Problems 5.8 Applications 79 APPENDICES A Question Bank B University Questions
MA6251 MATHEMATICS – II REGULATION 2013 SYLLABUS MA6251 MATHEMATICS – II L T P C 3104 OBJECTIVES: • To make the student acquire sound knowledge of techniques in solving ordinary differential equations that model engineering problems. • To acquaint the student with the concepts of vector calculus, needed for problems in all engineering disciplines. • To develop an understanding of the standard techniques of complex variable theory so as to enable the student to apply them with confidence, in application areas such as heat conduction, elasticity, fluid dynamics and flow the of electric current. • To make the student appreciate the purpose of using transforms to create a new domain in which it is easier to handle the problem that is being investigated. UNIT I VECTOR CALCULUS 9+3 Gradient, divergence and curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green’s theorem in a plane, Gauss divergence theorem and Stokes’ theorem (excluding proofs) – Simple applications involving cubes and rectangular parallelopipeds. UNIT II ORDINARY DIFFERENTIAL EQUATIONS 9+3 Higher order linear differential equations with constant coefficients – Method of variation of parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations with constant coefficients. UNIT III LAPLACE TRANSFORM 9+3 Laplace transform – Sufficient condition for existence – Transform of elementary functions – Basic properties – Transforms of derivatives and integrals of functions - Derivatives and integrals of transforms - Transforms of unit step function and impulse functions – Transform of periodic functions. Inverse Laplace transform -Statement of Convolution theorem – Initial and final value theorems – Solution of linear ODE of second order with constant coefficients using Laplace transformation techniques. UNIT IV ANALYTIC FUNCTIONS 9+3 Functions of a complex variable – Analytic functions: Necessary conditions – Cauchy-Riemann equations and sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of analytic function – Harmonic conjugate – Construction of analytic functions – Conformal mapping: w = z+k, kz, 1/z, z2, ez and bilinear transformation. UNIT V COMPLEX INTEGRATION 9+3 Complex integration – Statement and applications of Cauchy’s integral theorem and Cauchy’s integral formula – Taylor’s and Laurent’s series expansions – Singular points – Residues – Cauchy’s residue theorem – Evaluation of real definite integrals as contour integrals around unit circle and semi-circle (excluding poles on the real axis). TOTAL: 60 PERIODS
TEXT BOOKS: 1. Bali N. P and Manish Goyal, “A Text book of Engineering Mathematics”, Eighth Edition, Laxmi Publications Pvt Ltd.,(2011). 2. Grewal. B.S, “Higher Engineering Mathematics”, 41 (2011). Edition, Khanna Publications, Delhi, REFERENCES: 1. Dass, H.K., and Er. Rajnish Verma,” Higher Engineering Mathematics”, S. Chand Private Ltd., (2011) 2. Glyn James, “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education, (2012). 3. Peter V. O’Neil,” Advanced Engineering Mathematics”, 7th Edition, Cengage learning, (2012). 4. Ramana B.V, “Higher Engineering Mathematics”, Tata McGraw Hill Publishing Company, New Delhi, (2008).
MA6251 MATHEMATICS-II UNIT-1 VECTOR CALCULUS 1.1Gradient-Directional Derivative 1.1.1. Gradient 1.1(a) The Vector Differential Operator The differential operator ∇ (read as del) is defined as ∇≡ , , + + where are unit vectors along the three rectangular axes OX, OY, OZ. 1.1(b) The Gradient (Or Slope Of A Scalar Point Function) Let ( , , ) be a scalar point function and is continuously differentiable then the vector ∇ = + + = + + is called the gradient of the scalar function =∇ . and is written as Note: 1.1.1 ∇ is a vector differential operator and also it is a vector. Note: 1.1.2 ∇≡ + + Note: 1.1.3 If is a constant ,then ∇ = 0. Note: 1.1.4 If ∇ is a vector whose three components are Note: 1.1.5 ∇( ∇ ) 1. Find ∇(r), ∇ Solution: We know that ,→ = r=→ = + + = = ; (i) ; ∇(r) = = + + ∇φ + , , , . , = + + = + + = + + = (ii) ∇ =∑ı =∑ ı ∑xı = → = . 2. Prove that ∇( )= ( Solution: ∇( )=∑ )=∑ =∑ = = SCE + + 1 Dept of S&H