What you will learn
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Important Formulae and Procedures
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Derivations
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Engineering relevance of critical concepts
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Model Question Paper Problems
21MAT31: Mathematics-III (Common for all Branches) of Visvesvaraya Technological University (VTU). The course incepts with, detailed ideas on Laplace Transform which is central to analyzing the damped systems, control engineering, system response, stability etc in engineering. This is followed by Fourier, a series which is a power tool for many engineering applications to get information in frequency domain. Further Fourier Transform and Z transforms are perused which are equivalent to of Laplace transform tool for undamped systems. This is followed by solution of partial differential equations both explicit and implicit methods. Finally, we end with calculus of variation perhaps finds widest application in engineering from optimization.
Learning Map & Key Takeaways
Understanding Laplace Transform - Symbolism
Difference between Laplace & Fourier Transforms
Understanding the Role of the exponential function in the laplace integral - Part 1
Understanding the Role of the exponential function in the laplace integral - Part 2
Understanding Transfer Function
Engineering Applications of Laplace Transform: (Application - Pitching of a Satellite)
Engineering Applications of Laplace Transform: (Practical Example - Second Order System)
Important formula and standard results to remember
Important Properties of Laplace Transforms
Laplace Transforms - Problems: (Problem 1 - sin(at) and cos(at))
Laplace Transforms - Problems: (Problem 1 - sin(at) and cos(at): Physical Interpretation)
Laplace Transforms - Problems: (Problem 2 - sinh(at) and cosh(at))
Laplace Transforms - Problems: (Problem 3)
Laplace Transforms - Problems: (Problem 4)
Laplace Transforms - Problems: (Problem 5)
Laplace Transforms - Problems: (Problem 6)
Laplace Transforms - Problems: (Problem 7)
Laplace Transforms - Problems: (Problem 8)
Introduction to Gamma functions
Finding the Laplace Transform of
Relevance of Gaussian Function
Gamma Function - Problem
Laplace Transforms - Problems: (Problem 9)
Laplace Transforms - Problems: (Problem 10)
Laplace Transforms - Problems: (Problem 11)
Laplace Transforms - Problems: (Problem 12)
Laplace Transforms - Problems: (Problem 13)
Laplace Transforms - Problems: (Problem 14)
Laplace Transforms - Problems: (Problem 15)
Laplace Transforms - Problems: (Problem 16)
Introduction to Laplace Transform of Periodic Functions
Deriving Formula for Laplace Transform for Periodic Functions
Laplace Transforms of Periodic Functions - Problems: (Problem 1 (By Euler Definition) -
Laplace Transforms of Periodic Functions - Problems: (Problem 1 (By Integration by parts) -
Laplace Transforms of Periodic Functions - Problems: (Problem 1 (By time shift) -
Laplace Transforms of Periodic Functions - Problems: (Problem 2 (Triangular wave) -
Laplace Transforms of Periodic Functions - Problems: (Problem 3)
Laplace Transforms of Periodic Functions - Problems: (Problem 4 (Half Rectified Wave)
Introduction to Heavy Side Function
Computing Standard Results - Laplace Transforms of Heavy Side Function
Computing the Laplace Transform of a function multiplied by a Heavy Side Function
Construction of a Pulse using Heavy Side Function
Heavy Side Function - Problems: (Problem 1)
Heavy Side Function - Problems: (Problem 2)
Heavy Side Function - Problems: (Problem 3)
Introduction to Inverse Laplace Transforms
Inverse Laplace Transforms - Understanding Method by Inspection
Inverse laplace Transforms - Problems: (Problem 1 - By completing the Square)
Rule for finding Inverse Laplace Transform for "Log and Inverse trignometric functions"
Inverse laplace Transforms - Problems: (Problem 2 - By Differentiation)
Inverse laplace Transforms - Problems: (Problem 3 - By Differentiation & Completing the squares)
Inverse laplace Transforms - Problems: (Problem 4 - By Partial Fraction)
Inverse laplace Transforms - Problems: (Problem 5 - By Partial Fraction)
Introduction to Convolution Theorem
Physical Interpretation of Convolution Theorem
Convolution Theorem - Problems: (Problem 1)
Convolution Theorem - Problems: (Problem 2)
Convolution Theorem - Problems: (Problem 3)
Introduction of solution of Ordinary Differential Equations using Laplace Transforms
Differential Equations - Laplace Transforms Problems: (Problem 1)
Differential Equations - Laplace Transforms Problems: (Problem 2)
Differential Equations - Laplace Transforms Problems: (Problem 3)
Introduction to Fourier Series
Introducing the Learning Metrix
Introduction to Infinite Series (Convergence & Divergence)
Fourier Series and Nature of Signals
Infinite Series Examples - (Example 1)
Infinite Series Examples - (Example 2)
Introduction to Periodic Functions
Periodic Functions Examples - (Example 1)
Periodic Functions Examples - (Example 2)
Periodic Functions Examples - (Example 3)
Periodic Functions Examples - (Example 4)
Periodic Functions Examples - (Example 5)
Periodic Functions - Quick Solving Trick
Understanding Orthogonal Functions
Periodic Functions - Quick Recap
Understanding Dirichlet's Conditions for Fourier Series
Fourier Series for periodic functions with period 2pi and arbitrary period - (Introduction)
Fourier Series - Inroduction and Derivation
Understanding Odd and Even Functions
Odd and Even functions - (Example 1 - Even Function)
Odd and Even functions - (Example 2 - Even Function)
Odd and Even functions - (Example 3 - Even Function)
Odd and Even functions - (Example 4 - Odd Function)
Odd and Even functions - (Example 5 - Even Function)
Odd and Even functions - (Example 6 - Odd Function)
Odd and Even functions - (Example 7 - Even Function)
Odd and Even functions - (Example 8 - Odd Function)
Odd and Even functions - (Example 9 - Odd Function)
Odd and Even functions - (Example 10)
Odd and Even Functions - Quick Recap
Fourier Series - Problems: (Problem 1)
Fourier Series - Problems: (Problem 2)
Fourier Series - Problems: (Problem 3)
Fourier Series - Problems: (Problem 4)
Fourier Series - Problems: (Problem 5)
Fourier Series - Problems: (Problem 6)
Fourier Series - Problems: (Problem 7)
Introduction to Half range Fourier Series
Half range Fourier Series - Problems: (Problem 1)
Half range Fourier Series - Problems: (Problem 2)
Half range Fourier Series - Problems: (Problem 3)
Introducing Practical Harmonic Analysis
Practical Harmonic Analysis - Problems: (Problem 1)
Practical Harmonic Analysis - Problems: (Problem 2)
Practical Harmonic Analysis - Problems: (Problem 3)
Introducing Fourier Transforms
Order of Learning - Curriculum
Developing Equations for Fourier Transforms, and Inverse Fourier Transforms
Fourier & Inverse Fourier Transforms - Problems: (Problem 1)
Fourier & Inverse Fourier Transforms - Problems: (Problem 2)
Fourier & Inverse Fourier Transforms - Problems: (Problem 3)
Fourier & Inverse Fourier Transforms - Problems: (Problem 4)
Fourier & Inverse Fourier Transforms - Problems: (Problem 5)
Introducing Fourier & Inverse Fourier sine & cosine Transforms (Understanding Negative Frequency)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 1)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 2)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 3)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 4)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 5)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 6)
sine & cosine Fourier & Inverse Fourier Transforms - Problems: (Problem 7)
Introduction of the Concept of Z-Transforms
Introducing Standard Z-Transforms - Examples
Introducing Damping and Shifting Rules
Z-Transforms - Problems: (Problem 1)
Z-Transforms - Problems: (Problem 2)
Z-Transforms - Problems: (Problem 3)
Z-Transforms - Problems: (Problem 4)
Z-Transforms - Problems: (Problem 5)
Z-Transforms - Problems: (Problem 6)
Introducing Inverse Z-Transforms
Inverse Z-Transforms - Problems: (Problem 1)
Inverse Z-Transforms - Problems: (Problem 2)
Inverse Z-Transforms - Problems: (Problem 3)
Inverse Z-Transforms - Problems: (Problem 4)
Understanding Finite Differences for Difference Equations
Understanding the Idea of Difference Equations
Difference Equations - Problems: (Problem 1)
Difference Equations - Problems: (Problem 2)
Difference Equations - Problems: (Problem 3)
Difference Equations - Problems: (Problem 4)
Introduction to Numerical Solutions of PDE's
Holistic Understanding of the Classification of 2nd order PDE's
Classification of PDE's - Example 1
Classification of PDE's - Example 2
Classification of PDE's - Example 3
Introducing Finite Difference Approximations to Derivatives
Idea of Finite Differences
Introducing Solution of Laplace Equations - Standard 5-point Formula
Understanding the 5-point and Diagonal 5-point Formulae
Example 1 - 16 Cell Grid
Example 2 - 9 Cell Grid
Solution to Laplace Equation (5-point forumla) - Problems: (Problem 1)
Solution to Laplace Equation (5-point forumla) - Problems: (Problem 2)
Solution to Laplace Equation (5-point forumla) - Problems: (Problem 3)
Solution to 1-D Heat Equations: (Understanding Schmidt Explicit Method)
Solution to 1-D Heat Equations: (Understanding Bendre Schmidt Explicit Method)
Solution to 1-D Heat Equations: (Understanding Crank - Nicolson Method)
Solution to 1-D Wave Equations: (Understanding)
Solution to 1-D Heat Equations - Problems: (Problem 1 - Bendre Schmidt Method)
Solution to 1-D Heat Equations - Problems: (Problem 2 - Schmidt Method)
Solution to 1-D Heat Equations - Problems: (Problem 3 - Bendre Schmidt Method)
Solution to 1-D Heat Equations - Problems: (Problem 4 - Crank-Nicloson Method)
Solution to 1-D Heat Equations - Problems: (Problem 5 - Crank-Nicloson Method)
Solution to 1-D Wave Equations - Problems: (Problem 1)
Solution to 1-D Wave Equations - Problems: (Problem 2)
Introduction to Order of Learning - Curriculum
Understanding the Range-Kutta Method
Runge-Kutta Method - Problems: (Problem 2)
Understanding Milne's Predictor-Corrector Method
Milne's Predictor-Corrector Method - Problem: (Problem 1)
Milne's Predictor-Corrector Method - Problem: (Problem 2)
Idea to Calculus of Variations - Physical Meaning
Order of Learning: Curriculum
Understanding Functionals and the Euler-Lagrange Equation - Derivation
Euler-Lagrange Equation - Problems: (Problem 1)
Euler-Lagrange Equation - Problems: (Problem 2)
Euler-Lagrange Equation - Problems: (Problem 3)
Euler-Lagrange Equation - Problems: (Problem 4)
Euler-Lagrange Equation - Problems: (Problem 5)
Understanding Geodesic on a Plane
Variational Problems - Brachistochrone
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LanguageEnglish
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