Engineering Mathematics Online Course

Engineering Mathematics Online Course

 

The Objective Of The Paper Is To Facilitate The Student With The Basics Of Engineering Mathematics Applied Mathematics That Are Required For An Engineering Student.

 

The First Basic Unit Of Engineering Mathematics Online Course Is Partial Differentiation And Its Applications So That Students Can Understand Partial Derivatives Of First And Second Order. It Focus On Euler's Theorem For Homogeneous Functions (Without Proof).  We Also Lean Derivatives Of Implicit Functions, Total Derivatives. Change Of Variables. Jacobian. Taylor's Theorem For Function Of Two Variables(Without Proof). Error And Approximation. Extreme Values Of Function Of Several Variables(Maxima ,Minima, Saddle Points). Lagrange Method Of Undetermined Multipliers.

 

The Latest Course Of Mathematics

 

Academy Of Engineers Provide Engineering Mathematics Video Lectures As It Is Very Beneficial To Gate Mathematics Online Course. This Subjects Is Taught At M.Tech As Engineering Mathematics & Advanced Engineering Mathematics M.Tech. The Basic Mathematics For Engineering Students Is Available Through Online University Math Courses. The Best Online Calculus Course Is Available With Academy Of Engineers.

 

Our Next Topics Is Partial Differential Equations. It Is Very Important To Know Its Formulation, Solution Of First Order Equations, Lagranges Equations, Charpit's Method. Laplace Transformation: Definition, Laplace Transformation Of Basic Functions , Existence Condition For Laplace Transformation, Properties Of Laplace Transformation(Linearity, Scaling And Shifting). Unit Step Function, Impulse Function, Periodic Functions. Laplace Transformation Of Derivatives, Laplace Transformation Of Integrals, Differentiation Of Transforms, Integration Of Transforms, Convolution Theorem , Inverse Laplace Transformation. Solution Of Ordinary Differential Equations. Complex Function: Definition, Derivatives, Analytic Function, Cauchy's Riemann Equation (Without Proof). Conformal And Bilinear Mappings.

 

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The Classification of Engineering Materials

CLASSIFICATION OF MATERIALS

 

Solid materials have been conveniently grouped into three basic classifications - metals, ceramics, and polymers. This scheme is based primarily on chemical makeup and atomic structure, and most materials fall into one distinct grouping or another, although there are some intermediates. In addition, there are three other groups of important engineering materials—composites, semiconductors, and biomaterials.

 

A brief explanation of the material types and representative characteristics can be given as ---

 

METALS

Metallic materials are normally combinations of metallic elements. They have large numbers of non-localized electrons; that is, these electrons are not bound to particular atoms. Many properties of metals are directly attributable to these electrons. Metals are extremely good conductors of electricity and heat and are not transparent to visible light; a polished metal surface has a lustrous appearance. Furthermore, metals are quite strong, yet deformable, which accounts for their extensive use in structural applications.

 

CERAMICS

Ceramics are compounds between metallic and nonmetallic elements; they are most frequently oxides, nitrides, and carbides. The wide range of materials that falls within this classification includes ceramics that are composed of clay minerals, cement, and glass. These materials are typically insulative to the passage of electricity and heat, and are more resistant to high temperatures and harsh environments than metals and polymers. With regard to mechanical behavior, ceramics are hard but very brittle.

 

POLYMERS

Polymers include the familiar plastic and rubber materials. Many of them are organic compounds that are chemically based on carbon, hydrogen, and other nonmetallic elements; furthermore, they have very large molecular structures. These materials typically have low densities and may be extremely flexible.

                                

COMPOSITES

A number of composite materials have been engineered that consist of more than one material type. Fiberglass is a familiar example, in which glass fibers are embedded within a polymeric material. A composite is designed to display a combination of the best characteristics of each of the component materials. Fiberglass acquires strength from the glass and flexibility from the polymer. Many of the recent material developments have involved composite materials.

 

 SEMICONDUCTORS

Semiconductors have electrical properties that are intermediate between the electrical conductors and insulators. Furthermore, the electrical characteristics of these materials are extremely sensitive to the presence of minute concentrations of impurity atoms, which concentrations may be controlled over very small spatial regions. The semiconductors have made possible the advent of integrated circuitry that has totally revolutionized the electronics and computer industries (not to mention our lives) over the past two decades.

 

BIOMATERIALS

Biomaterials are employed in components implanted into the human body forreplacement of diseased or damaged body parts. These materials must not produce toxic substances and must be compatible with body tissues (i.e., must not cause adverse biological reactions).

 

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D.C. Circuits: Ohm's Law, junction, node, circuit elements classification: Linear & nonlinear, active & passive, lumped & distributed, unilateral & bilateral with examples. KVL, KCL, Loop and node-voltage analysis of resistive circuit. Star-Delta transformation for resistors.

Network Theorems: Superposition, Thevenin's, Norton's and Maximum power transfer theorems in a resistive network. AC Fundamentals: Mathematical representation of various wave functions. Sinusoidal periodic signal, instantaneous and peak values, polar & rectangular form of representation of impedances and phasor quantities. Addition & subtraction of two or more phasor sinusoidal quantities using component resolution method.RMS and average values of various waveforms.

A.C. Circuits: Behavior of various components fed by A.C. source (steady state response of pureR, pure L, pure C, RL, RC, RLC series with waveforms of instantaneous voltage, current & power on simultaneous time axis scale and corresponding phasor diagrams), power factor, active, reactive & apparent power. Frequency response of Series & Parallel RLC ckts including resonance, Q factor, cut-off frequency & bandwidth. Generation of alternating emf.

Balanced Three Phase Systems:  Generation of alternating 3- phaseemf). 3-phase balanced circuits, voltage and current relations in star and delta connections. Measurement of 3-phase power by two wattmeter method for various types of star & delta connected balanced loads.

Single Phase Transformer (qualitative analysis only): Concept of magnetic circuits Relation between MMF & Reluctance. Hysteresis& Eddy current phenomenon. Principle, construction & emf equation Phasor diagram at ideal, no load and on load conditions. Losses & Efficiency, regulation OC & SC test, equivalent circuit, concept of auto transformer.

Electrical Machines (qualitative analysis only): Construction and working of dc machine with commutator action, speed control of dc shunt motor. Generation of rotating magnetic fields, Construction and working of a three-phase induction motor, Significance of torque-slip characteristic. Basics of Single-phase induction motor, capacitor start capacitor run Single-phase induction motor working. Basic construction and working of synchronous generator and motor.

Electrical Installations (LT Switchgear): Switch Fuse Unit (SFU), MCB, ELCB, MCCB, Types of Wires and Cables, Earthing.

 

 

 

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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 Parameters, Applications To Engineering Problems (Without Derivation).

Series Solution And Special Functions: Series Solution Of Second Order Ordinary Differential Equations With Variable Coefficient (Frobenius Method), Bessel And Legendre Equations And Their Series Solutions, Properties Of Bessel Function And Legendre Polynomials.

Laplace Transform: Laplace Transform, Existence Theorem, Laplace Transforms Of Derivatives And Integrals, Initial And Final Value Theorems, Unit Step Function, Dirac- Delta Function, Laplace Transform Of Periodic Function, Inverse Laplace Transform, Convolution Theorem, Application To Solve Simple Linear And Simultaneous

Differential Equations: Fourier Series And Partial Differential Equations Periodic Functions, Fourier Series Of Period 2, Euler's Formulae, Functions Having Arbitrary Periods, Change Of Interval, Even And Odd Functions, Half Range Sine And Cosine Series, Harmonic Analysis. Solution Of First Order Partial Differential Equations By Lagrange's Method, Solution Of Second Order Linear Partial Differential Equations With Constant Coefficients.

Applications Of Partial Differential Equations: Classification Of Second Order Partial Differential Equations, Method Of Separation Of Variables For Solving Partial Differential Equations, Solution Of One And Two Dimensional Wave And Heat Conduction Equations, Laplace Equation In Two Dimension, Equation Of Transmission Lines.

 

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What are the assumptions of "quantum free electron theory" ?

 In 1929, Somerfield stated to apply quantum mechanics to explain conductivity phenomenon in metal. He has improved the Drude - Lorentz theory by quantizing the free electron energy and retained the classical concept of free motion of electron at a random.

 ASSUMPTIONS:-

·         The electrons are free to move within the metal like gaseous molecules. They are confined to the metal due to surface potential.

·         The velocity distribution of the free electrons is described by Fermi-Dirac Statistics because electrons are spin half particles.

·         The free electrons would go into the different energy levels by following Pauli’s exclusion Principle which states that no two electrons have same set of Quantum numbers.

·         The motion of electrons is associated with a complex wave called matter wave, according to De-Broglie hypothesis.

·         The electrons cannot have all energies but will have discrete energies according to the equation, E = n² h² / 8ma².

 

Drawbacks:

Conductivity: According to Quantum free electron theory, the conductivity of a metal is

σ = μne, here ‘μ’ is the mobility of electrons, ‘n’ is the free electron concentration and ‘e’ is the electron charge.

According to the above equation, polyvalent metals like Aluminum (Al) should be more conductive than monovalent metals like copper (Cu). But experimentally it is not so.

 Hall coefficient: According to the free electron theory, the hall coefficients for all metals is negative where as there are certain metals like Be, Cd, Zn for which the Hall coefficient is + ve. Free electron theory could not explain why certain substances behave as insulators and some other substances as semiconductors; in spite of they have free electrons in them.

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