SE EXTC SEM 3 – ENGINEERING MATHEMATICS III

Module 1 – Laplace Transform
14 Topics
1.1.a – Definition of Laplace Transforms
1.1.b – Condition of Existence of Laplace transform
1.2 – Laplace Transform (L) of Standard Functions like ? ?? , ???(??), ???(??), ???ℎ(??), ???ℎ(??) and ? ? , ?ℎ??? ? ≥ 0
1.3.a – Linearity
1.3.d – change of scale Property
1.3.b – First Shifting Theorem
1.3.e – multiplication by t
1.3.c – Second Shifting Theorem
1.3.f – Division by t
1.3.g – Laplace Transform of derivatives and integrals (Properties without proof)
1.4 – Evaluation of real improper integrals by using Laplace Transformation
1.5.a – Heaviside’s Unit Step function
1.5.b – Laplace Transform. of Periodic functions
1.5.c – Dirac Delta Function
Module 2 – Inverse Laplace Transform
7 Topics
2.1.a – Inverse Laplace Transform
2.1.b – Linearity property
2.1.c – use of standard formulae to find inverse Laplace Transform
2.1.d – finding Inverse Laplace transform using derivatives.
2.2 – Partial fractions method to find inverse Laplace transform.
2.3 – Inverse Laplace transform using Convolution theorem (without proof).
2.4.a – Applications to solve initial and boundary value problems involving ordinary differential equations.
Module 3 – Fourier Series
7 Topics
3.1 – Dirichlet’s conditions, Definition of Fourier series and Parseval’s Identity (without proof).
3.2 – Fourier series of periodic function with period 2? and 2l.
3.3 – Fourier series of even and odd functions
3.4 – Half range Sine and Cosine Series
3.5.a – Complex form of Fourier Series
3.5.b – Orthogonal and orthonormal set of functions
3.5.c – Fourier Transform.
Module 4 – Complex Variables
13 Topics
4.1.a – Function f(z) of complex variable
4.1.b – limit
4.1.c – Continuity and Differentiability of f(z)Analytic function
4.1.d – Necessary and sufficient conditions for f(z) to be analytic (without proof)
4.2 – Cauchy-Riemann equations in cartesian coordinates (without proof).
4.3 – Milne-Thomson method to determine analytic function f(z)when real part (u) or Imaginary part (v) or its combination (u+v or u-v) is given.
4.4 – Harmonic function, Harmonic conjugate and orthogonal trajectories
4.5.a – Conformal mapping
4.5.b – Linear
4.5.c – Bilinear mapping
4.5.d – Cross ratio
4.5.e – Fixed points
4.5.f – Standard transformations.
Module 5 – Linear Algebra: Matrix Theory
10 Topics
5.1.a – Characteristic equation
5.1.b – Eigen values and Eigen vectors
5.1.c – Example based on properties of Eigen values and Eigen vectors.(Without Proof).
5.2.a – Cayley-Hamilton theorem (Without proof)
5.2.b – Examples based on verification of Cayley- Hamilton theorem and compute inverse of Matrix
5.3.a – Similarity of matrices
5.3.b – Diagonalization of matrices
5.3.c – Functions of square matrix
5.4.a – Application of Matrix Theory in machine learning and google page rank algorithms
5.4.b – Derogatory and non-derogatory matrices.
Module 6 – Vector Differentiation and Integral
9 Topics
6.1.a – Basics of Gradient
6.1.b – Divergence
6.1.c – Curl (Without Proof).
6.2.a – Solenoidal and irrotational (conservative) vector fields
6.3.a – Line Integral
6.3.b – Green’s theorem in a plane (Without Proof)
6.3.c – Stokes’ theorem (Without Proof) only evaluation.
6.4.a – Gauss’ divergence Theorem
6.4.b – Applications of Vector calculus
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1.1.a – Definition of Laplace Transforms

SE EXTC SEM 3 – ENGINEERING MATHEMATICS III Module 1 – Laplace Transform 1.1.a – Definition of Laplace Transforms
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