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SE MECHANICAL SEM 3 – ENGINEERING MATHEMATICS III

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Module 1 – Laplace Transform 12 Topics

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1.3 – Properties of Laplace Transform

1.1.a – Definition of Laplace Transform

1.1.b – Condition of Existence of Laplace Transform

1.2 – Laplace Transform (L) of Standard Functions like eat, sin(at), cos(at), sinh(at), cosh(at) and tn , where n>_0.

1.3.a – Linearity

1.3.b – First Shifting Theorem

1.3.c – Second Shifting Theorem

1.3.d – Change of Scale Property

1.3.e – Multiplication by t

1.3.f – Division by t

1.3.g – Laplace Transform of Derivatives and Integrals (Properties without proof)

1.4 – Evaluation of Integrals by using Laplace Transformation

Module 2 – Inverse Laplace Transform 6 Topics

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2.1.b – Linearity Property

2.1.a – Inverse Laplace Transform

2.1.c – Use of Standard Formulae to Find Inverse Laplace Transform

2.1.d – Finding Inverse Laplace Transform using Derivative

2.2 – Partial Fractions Method & First Shift Property to Find Inverse Laplace Transform

2.3 – Inverse Laplace Transform using Convolution Theorem ( without proof)

Module 3 – Fourier Series 5 Topics

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3.1.a – Dirichlet’s conditions

3.1.b – Definition of Fourier series and Parseval’s Identify (without proof)

3.2 – Fourier Series of Periodic Function with Period 2pi and 2l

3.3 – Fourier Series of Even and Odd Functions

3.4 – Half Range Sine and Cosine Series

Module 4 – Complex Variables 7 Topics

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4.1.a – Function f(z) of Complex Variable, Limit, Continuity and Differentiability of f(z)

4.1.b – Analytic Function

4.1.c – Necessary and Sufficient Conditions for f(z) to be Analytic (without proof)

4.2 – Cauchy-Riemann Equations in Cartesian Co-ordinates (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.a – Harmonic Function

4.4.b – Harmonic Conjugate and Orthogonal Trajectories

Module 5 – Matrices 8 Topics

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5.1.a – Characteristic Equation

5.1.b – Eigen values and Eigen vectors

5.1.c – Properties of Eigen values and Eigen vectors

5.2.a – Cayle-Hamilton Theorem (without proof)

5.2.b – Application to Find The Inverse of The given Square Matrix and to Determine The given Higher Degree Polynomial Matrix

5.3 – Functions of Square Matrix

5.4.a – Similarity of Matrices

5.4.b – Diagonalization of Matrices

Module 6 – Numerical methods for PDE 6 Topics

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6.1.a – Introduction of Partial Differential Equations

6.1.b – Method of Separation of Variables

6.1.c – Vibrations of String

6.1.d – Analytical Method for One Dimensional Heat and Wave Equations

6.2 – Crank Nicholson Method

6.3 – Bender Schmidt Method

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