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SE EXTC SEM 3 – ENGINEERING MATHEMATICS III

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Module 1 – Laplace Transform 14 Topics

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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

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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

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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

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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

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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

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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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