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engineering-syllabus · 2 years ago

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A Comprehensive Guide to the GATE Electrical Engineering Syllabus

Preparing for the Graduate Aptitude Test in Engineering (GATE) in the field of Electrical Engineering requires a solid understanding of the syllabus and a well-structured study plan. The GATE syllabus is designed to cover a wide range of topics within the field to assess your knowledge and skills. Here's a comprehensive guide to the GATE Electrical Engineering syllabus:

1. Engineering Mathematics:

Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors.

Calculus: Limits, continuity and differentiability, partial derivatives, maxima and minima, sequences and series, Taylor series.

Differential Equations: First-order equations (linear and nonlinear), higher-order linear differential equations, Laplace transforms.

2. Electric Circuits:

Circuit Elements: Resistors, inductors, capacitors, ideal independent and dependent voltage and current sources.

Kirchhoff's Laws: Analysis of resistive circuits, nodal and mesh analysis, superposition, Thevenin and Norton theorems.

AC Circuits: Phasors, sinusoidal steady-state analysis, power factor, three-phase circuits.

3. Electromagnetic Fields:

Electrostatics and Magnetostatics: Coulomb's law, Gauss's law, Biot-Savart law, Ampere's law.

Maxwell's Equations: Differential and integral forms, electromagnetic wave propagation, Poynting vector.

4. Signals and Systems:

Signal Classification: Continuous-time and discrete-time signals, periodic and aperiodic signals.

System Analysis: Linearity, time-invariance, causality, stability, impulse response, convolution.

5. Electrical Machines:

Transformers: Single-phase and three-phase transformers, efficiency, regulation.

DC Machines: Construction, characteristics, starting and speed control.

AC Machines: Synchronous and induction machines, principles of operation, characteristics, power factor correction.

6. Power Systems:

Power Generation: Thermal, hydro, nuclear, and renewable sources.

Transmission and Distribution: Line parameters, load flow analysis, economic operation, fault analysis, protection.

7. Control Systems:

Mathematical Modeling: Transfer function, block diagram, signal flow graph.

Time Response Analysis: Standard test signals, steady-state errors, stability.

Frequency Response Analysis: Bode plots, Nyquist plots, root locus.

8. Electrical and Electronic Measurements:

Measurement Basics: Accuracy, precision, errors, standards.

Measurement Devices: Ammeters, voltmeters, bridges, oscilloscopes, transducers.

9. Analog and Digital Electronics:

Semiconductor Devices: Diodes, transistors, operational amplifiers.

Analog Circuits: Amplifiers, oscillators, filters, voltage regulators.

Digital Circuits: Logic gates, combinational and sequential circuits, ADCs and DACs.

10. Power Electronics:

Semiconductor Switches: Diodes, thyristors, MOSFETs, IGBTs.

Converter Topologies: Rectifiers, inverters, choppers, voltage regulators.

11. Electric and Magnetic Fields:

Electrostatics and Magnetostatics: Gauss's and Ampere's laws, dielectric and magnetic materials.

Maxwell's Equations: Integral and differential forms, electromagnetic wave propagation.

12. Signals and Systems:

Continuous and Discrete Signals: Fourier series and transform, Laplace transform, Z-transform.

System Analysis: Linear time-invariant systems, convolution, stability, causality.

13. Control Systems:

Time Domain Analysis: Stability, transient and steady-state response.

Frequency Domain Analysis: Bode plots, Nyquist plots, root locus, compensation techniques.

14. Power Systems:

Power Generation: Thermal, hydro, nuclear, and renewable sources.

Transmission and Distribution: Fault analysis, voltage and frequency control, load flow studies.

15. Analog and Digital Electronics:

Diodes, Transistors, and Amplifiers: Diode circuits, small signal analysis of BJT and FET, feedback amplifiers.

Digital Electronics: Logic gates, combinational and sequential circuits, ADCs and DACs.

16. Electric and Magnetic Fields:

Electrostatics: Gauss's law, boundary conditions, Poisson's and Laplace's equations.

Magnetostatics: Ampere's law, Biot-Savart law, magnetic materials.

17. Power Systems:

Power Generation: Types of power plants, load characteristics, economics of power generation.

Protection and Switchgear: Relays, circuit breakers, fuses, protection schemes.

18. Power Electronics:

Power Semiconductor Devices: Diodes, thyristors, MOSFETs, IGBTs.

Converters: AC-DC converters, DC-DC converters, inverters.

19. Electrical Machines:

Transformers: Construction, regulation, efficiency.

Synchronous Machines: Characteristics, voltage regulation, parallel operation.

Induction Machines: Construction, characteristics, starting and speed control.

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mathematicianadda · 5 years ago

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Derivative and calculus over the set of rational numbers

I am interested in the derivative of a function defined on a subset $S$ of $[0, 1]$. The subset in question is dense in $[0, 1]$ but has Lebesgue measure zero. My actual question can be found at the bottom of this post.

There has been a few questions on the subject, but none leading to something interesting as far as I am concerned. See here, here (concept of arithmetic derivative) and also here (Minkowski's question mark function, related to the material discussed here.)

Generally these discussions lead to some kind of non-sense math. Here it is the opposite. I have a framework that does work as far as applications and computations are concerned, but I have a hard time putting it into some sound mathematical framework. It would have to be some kind of non-standard calculus.

Perhaps the simplest example (though I have plenty of other similar cases) is as follows. Let $Z$ be a random variable defined as follows: $$Z = \sum_{k=1}^\infty \frac{X_k}{2^k}$$

where the $X_k$'s are identically and independently distributed with a Bernouilli$(p)$ distribution. Thus $P(X_k = 1) = p$ and $P(X_k = 0) = 1-p$. Here $0 < p < 1$. In short, the $X_k$'s are the binary digits of the random number $Z$.

There are two cases.

Case $p=\frac{1}{2}$

In this case, $Z$ has a uniform distribution on $S$, where $S$ is the set of normal numbers in $[0, 1]$. It is known that $S$ has Lebesgue measure $1$, and that $S$ is dense in $[0, 1]$. Yet it is full of holes (no rational number is is a normal number due to their periodicity, thus the $X_k$'s are not independent for rational numbers.)

This is the simplest case. One might wonder if the density $f_Z$ (the derivative of the distribution $F_Z$) exists. Yet $f_Z(z) = 1$ if $z \in S$ works perfectly well for all purposes. It can easily be extended to $f_Z(z) = 1$ if $z \in [0, 1]$. Let us denote the extended function as $\tilde{f}_Z$. You can compute all the moments using the extended $\tilde{f}_Z$ and get the right answer. If $s$ is a positive real number, then $$E(Z^s) = \int_0^1 z^s \tilde{f}_Z(z) dz = \frac{1}{s+1}.$$

You could argue that $\tilde{f}_Z$ (and thus $f_Z$) can be obtained by inverting the above functional equation, using some kind of Laplace transform. So we can bypass the concept of derivative entirely, it seems.

Case $p\neq \frac{1}{2}$

Now we are dealing with a hard nut to crack, and a wildly chaotic system: $Z$'s support domain is a set $S'$ that is a subset of non-normal numbers in $[0, 1]$. This set $S'$ has now Lebesgue measure zero, yet it is dense in $[0, 1]$. For the distribution, it is not a problem: even discrete random variables have a distribution $F_Z$ defined for all positive real numbers: $F_Z(z) = P(Z \leq z)$.

The issue is with the density $f_Z = dF_Z/dz$. It sounds either it should be zero everywhere or not exist. My guess is that you might be able to define a new, workable concept of density. In the neighborhood of every point $z \in S'$, it looks like $g(z,h) = (F_Z(z+h) - F_Z(z))/h$ oscillates infinitely many times with no limit as $h\rightarrow 0$, yet these oscillations are bounded most of the time, perhaps leading to the fact that averaging $g(z, h)$ around $h = 0$, using smaller and smaller values of $h$, could provide a sound definition for the density $f_Z$.

Again, despite the chaotic nature of the system (see how the the would-be density could potentially look like in the picture below) all the following quantities exist and can be computed exactly and then confirmed by empirical evidence, even though the integrals below may not make sense:

$$E(Z) = \int_{0}^{1} z f_Z(z) dz = p \\ E(Z^2) = \int_{0}^{1} z^2 f_Z(z) dz =\frac{p}{3}(1+2p)\\ E(Z^3) = \int_{0}^{1} z^3 f_Z(z) dz =\frac{p}{7}(1+4p+2p^2)\\ E(Z^4) = \int_{0}^{1} z^4 f_Z(z) dz =\frac{p}{105}(7+46p + 44p^2+8p^3) $$ Indeed, a general formula for $E(Z^s) = \int_0^1 z^s f_Z(z)dz$ is available for $s \geq 0$, defined by the following functional equation (see here): $$E(Z^s) = \frac{p}{2^s-1+p}\cdot E((1+Z)^s) .$$

In other words, we would have, under some appropriate calculus theory with a sound definition of integral and derivative: $$\int_{S'}z^s f_Z(z)dz = \frac{p}{2^s-1+p}\cdot\int_{S'}(1+z)^s f_Z(z) dz .$$

Here is how the density $f_Z$, if properly defined, could look like for $p=0.75$ (see here and here):

My question

Is there an existing theory to handle this type of density-like-substance?

from Hot Weekly Questions - Mathematics Stack Exchange from Blogger https://ift.tt/34Vs6CM

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coursesforallacademynoida · 6 years ago

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Engineering Mathematics - II Tuition Classes In Greater Noida Near Alpha Commercial

Engineering Mathematics – II Tuition Classes In Greater Noida Near Alpha Commercial

Differential Equations Linear differential equations of nth order with constant coefficients, Complementary function and Particular integral, Simultaneous linear differential equations, Solution of second order differential equations by changing dependent & independent variables, Normal form, Method of variation of

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quanrel · 6 years ago

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Symbolab Calculator

https://www.quanrel.com/symbolab-calculator/ Symbolab Calculator - https://www.quanrel.com/symbolab-calculator/ Your private math tutor, solves any math problem with steps! Integrals, Derivatives, Equations, Limits and much more. (Steps require a subscription.) Symbolab Math Solver is composed of over a hundred of Symbolab most powerful calculators: Integral Calculator Derivative Calculator Limit calculator Equation Calculator Inequality Calculator Trigonometry Calculator Matrix Calculator Functions Calculator Series Calculator ODE Calculator Laplace Transform Calculator Download the app to experience the full set of Symbolab calculators. Symbolab Math Solver solves any math problem including Pre- Algebra, Algebra, Pre-Calculus, Calculus, Trigonometry, Functions, Matrix, Vectors, Geometry and Statistics. Symbolab solves any problem for free. In order to view step-by-step solutions, you can subscribe weekly ($0.99), monthly ($2.49), or yearly ($14.99). • Payment will be charged to iTunes Account at confirmation of purchase  • Subscription automatically renews unless auto-renew is turned off at least 24-hours before the end of the current period  • Account will be charged for renewal within 24-hours prior to the end of the current period, and identify the cost of the renewal  • Subscriptions may be managed by the user and auto-renewal may be turned off by going to the user’s Account Settings after purchase Terms of use: https://www.symbolab.com/public/terms.pdf Privacy policy: https://www.symbolab.com/public/privacy.pdf By Symbolab Find out more

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digitalcontentmarketorg-blog · 8 years ago

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Download Full Solution Manual for Differential Equations an Introduction to Modern Methods and Applications 3rd Edition By

Solution Manual for Differential Equations An Introduction to Modern Methods and Applications 3rd Edition by Brannan and Boyce

Link download full: https://getbooksolutions.com/download/solution-manual-for-differential-equations-an-introduction-to-modern-methods-and-applications-3rd-edition-by-brannan/

CLICK HERE TO VIEW SAMPLE OF Differential Equations An Introduction to Modern Methods and Applications 3rd Edition Solution manual by James R. Brannan and Boyce

Brannan/Boyce’s Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition is consistent with the way engineers and scientists use mathematics in their daily work. The text emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. The focus on fundamental skills, careful application of technology, and practice in modeling complex systems prepares students for the realities of the new millennium, providing the building blocks to be successful problem-solvers in today’s workplace. Section exercises throughout the text provide hands-on experience in modeling, analysis, and computer experimentation. Projects at the end of each chapter provide additional opportunities for students to explore the role played by differential equations in the sciences and engineering.

Table of Contents

Chapter 1: Introduction

1.1 Mathematical Models and Solutions

1.2 Qualitative Methods: Phase Lines and Direction Fields

1.3 Definitions, Classification, and Terminology

Chapter 2: First Order Differential Equations

2.1 Separable Equations

2.2 Linear Equations: Method of Integrating Factors

2.3 Modeling with First Order Equations

2.4 Differences Between Linear and Nonlinear Equations

2.5 Autonomous Equations and Population Dynamics

2.6 Exact Equations and Integrating Factors

2.7 Substitution Methods

Projects

2.P.1 Harvesting a Renewable Resource

2.P.2 A Mathematical Model of a Groundwater Contaminant Source

2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

Chapter 3: Systems of Two First Order Equations

3.1 Systems of Two Linear Algebraic Equations

3.2 Systems of Two First Order Linear Differential Equations

3.3 Homogeneous Linear Systems with Constant Coefficients

3.4 Complex Eigenvalues

3.5 Repeated Eigenvalues

3.6 A Brief Introduction to Nonlinear Systems

Projects

3.P.1 Estimating Rate Constants for an Open Two-Compartment Model

3.P.2 A Blood-Brain Pharmacokinetic Model

Chapter 4: Second Order Linear Equations

4.1 Definitions and Examples

4.2 Theory of Second Order Linear Homogeneous Equations

4.3 Linear Homogeneous Equations with Constant Coefficients

4.4 Mechanical and Electrical Vibrations

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

4.6 Forced Vibrations, Frequency Response, and Resonance

4.7 Variation of Parameters

Projects

4.P.1 A Vibration Insulation Problem

4.P.2 Linearization of a Nonlinear Mechanical System

4.P.3 A Spring-Mass Event Problem

4.P.4 Euler-Lagrange Equations

Chapter 5: The Laplace Transform

5.1 Definition of the Laplace Transform

5.2 Properties of the Laplace Transform

5.3 The Inverse Laplace Transform

5.4 Solving Differential Equations with Laplace Transforms

5.5 Discontinuous Functions and Periodic Functions

5.6 Differential Equations with Discontinuous Forcing Functions

5.7 Impulse Functions

5.8 Convolution Integrals and Their Applications

5.9 Linear Systems and Feedback Control

Projects

5.P.1 An Electric Circuit Problem

5.P.2 The Watt Governor, Feedback Control, and Stability

Chapter 6: Systems of First Order Linear Equations

6.1 Definitions and Examples

6.2 Basic Theory of First Order Linear Systems

6.3 Homogeneous Linear Systems with Constant Coefficients

6.4 Nondefective Matrices with Complex Eigenvalues

6.5 Fundamental Matrices and the Exponential of a Matrix

6.6 Nonhomogeneous Linear Systems

6.7 Defective Matrices

Projects

6.P.1 Earthquakes and Tall Buildings

6.P.2 Controlling a Spring-Mass System to Equilibrium

Chapter 7: Nonlinear Differential Equations and Stability

7.1 Autonomous Systems and Stability

7.2 Almost Linear Systems

7.3 Competing Species

7.4 Predator-Prey Equations

7.5 Periodic Solutions and Limit Cycles

7.6 Chaos and Strange Attractors: The Lorenz Equations

Projects

7.P.1 Modeling of Epidemics

7.P.2 Harvesting in a Competitive Environment

7.P.3 The Rossler System

Chapter 8: Numerical Methods

8.1 Numerical Approximations: Euler’s Method

8.2 Accuracy of Numerical Methods

8.3 Improved Euler and Runge-Kutta Methods

8.4 Numerical Methods for Systems of First Order Equations

Projects

8.P.1 Designing a Drip Dispenser for a Hydrology Experiment

8.P.2 Monte Carlo Option Pricing: Pricing Financial Option by Flipping a Coin

Chapter 9: Series Solutions of Second order Equations

9.1 Review of Power Series

9.2 Series Solutions Near an Ordinary Point, Part I

9.3 Series Solutions Near an Ordinary Point, Part II

9.4 Regular Singular Points

9.5 Series Solutions Near a Regular Singular Point, Part I

9.6 Series Solutions Near a Regular Singular Point, Part II

9.7 Bessel’s Equation

Projects

9.P.1 Diffraction Through a Circular Aperature

9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator

9.P.3 Perturbation Methods

Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems

10.1 Orthogonal Families in the Space PC [a,b]

10.2 Fourier Series

10.3 Elementary Two-Point Boundary Value Problems

10.4 General Sturm-Liouville Boundary Value Problems

10.5 Generalized Fourier Series and Eigenfunction Expansions

10.6 Singular Boundary Value Problems

10.7 Convergence Issues

Chapter 11: Elementary Partial Differential Equations

11.1 Terminology

11.2 Heat Conduction in a Rod—Homogeneous Case

11.3 Heat Conduction in a Rod—Nonhomogeneous Case

11.4 Wave Equation—Vibrations of an Elastic String

11.5 Wave Equation—Vibrations of a Circular Membrane

11.6 Laplace Equation

Projects

11.P.1 Estimating the Diffusion Coefficient in the Heat Equation

11.P.2 The Transmission Line Problem

11.P.3 Solving Poisson’s Equation by Finite Differences

11.P.4 Dynamic Behavior of a Hanging Cable

11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media

11.P.6 Fisher’s Equation for Population Growth and Dispersion

Appendices

11.A Derivation of the Heat Equation

11.B Derivation of the Wave Equation

A: Matrices and Linear Algebra

A.1 Matrices

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank

A.3 Determinates and Inverses

A.4 The Eigenvalue Problem

B: Complex Variables

ISBN-13: 978-1118531778 ISBN-10: 1118531779

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