Energy Systems

Energy Systems
3 YEAR1 semester6 CREDITS
Prof. Michele Manno2019-20
MANNO MICHELE 2020-21
Code: 8037964
SSD: ING-IND/09

OBJECTIVES

LEARNING OUTCOMES:
After completing the course, the students should acquire a good knowledge of the fundamental operating principles of energy conversion systems, and they should be able to analyze the layout and evaluate the performance and efficiency of thermal and hydroelectric power plants.

KNOWLEDGE AND UNDERSTANDING:
Students are expected to understand the fundamental principles underlying the operation of energy conversion systems.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Students are expected to be able to assess the performance of energy conversion systems.

MAKING JUDGEMENTS:
Students are expected to be able to choose the most suitable energy conversion system and its operating parameters, given a particular application.

COMMUNICATION SKILLS:
Students are expected to be able to describe and illustrate the operating principles of energy conversion systems.

LEARNING SKILLS:
Students are expected to be able to read and fully understand technical literature related to energy conversion systems.

COURSE SYLLABUS

Students will be introduced to the main principles of energy conversion systems, with particular reference to steam and gas turbine power plants, combined cycle power plants,
hydroelectric power generation.

More specifically, the following topics will be addressed:

Introduction

  • Review of fluid properties and equations of state.
  • Analysis of combustion processes.
  • Analysis of energy conversion systems based on 1st and 2nd Laws of Thermodynamics.
  • Thermodynamic cycles: definition of network output and thermal efficiency; external and internal irreversibilities; efficiency factors.

Steam power plants

  • Analysis of ideal and real thermodynamic cycles.
  • Choice of operating parameters.
  • Techniques to improve plant efficiency: steam reheating, regenerative feed heating.
  • Plant layouts, applications.

Gas turbine power plants

  • Analysis of ideal and real thermodynamic cycles.
  • Choice of operating parameters and techniques to improve performance: regenerative heat exchanger, reheaters, intercoolers.
  • Layout of heavy-duty and aeroderivative turbines, applications.

Combined cycle power plants

  • Analysis of “topping” (gas turbine) and “bottoming” sections, definition of recovery efficiency.
  • Thermodynamic optimization of bottoming sections with variable temperature heat input.
  • Plant layout, applications.

Hydroelectric power generation

  • Hydraulic turbines: classification, operating parameters, performance characteristics.
  • Hydroelectric plant classification and layouts, applications.

Fluid Machinery

Fluid Machinery
3 YEAR1 semester6 CREDITS
Prof. Vincenzo Mulone e
Roberto Verzicco
2019-20
VERZICCO ROBERTO
MULONE VINCENZO
2020-21
Code: 8037967
SSD: ING-IND/08

LEARNING OUTCOMES: This course aims at providing the fundamentals of fluid dynamics applied to fluid machines. More in detail, it deals with the fluid dynamics equations applied to energy-consuming and energy-producing machines, characterized by both axial and radial flows. It also deals with the understanding of systems connected to fluid machines.

KNOWLEDGE AND UNDERSTANDING: The student will be able to develop simple but useful calculations of fluid machines in terms of flow, work and power, along with solving practical problems of interest. The student will also learn the basics of the control of fluid machines with respect to the flow rate, work exchanged and power output or input The knowledge developed will help the student for both the design of fluid machines and of the systems connected to the machines.

APPLYING KNOWLEDGE AND UNDERSTANDING: The student will apply the knowledge and understanding developed to the analysis of practical problems. This would imply critical knowledge in terms of size and power output/input; the same thing will be done for the systems connected to the machine.

MAKING JUDGEMENTS: The student will have to prove his critical awareness with respect to the simplifying assumptions useful to describe and calculate fluid machines, as well as his critical awareness of the correct order of magnitude of performance parameters while dealing or designing fluid machines.

COMMUNICATION SKILLS: The student will prove, mostly during the oral test, his capacity of describing the operation and functioning of fluid machines, convening of the knowledge developed.

LEARNING SKILLS: The student will get familiar with the schematization of practical problems, mostly during the development of his skills for the written test. This mainly concerns fluid machines (e.g. wind turbines, steam turbines, hydraulic turbines, hydraulic pumps, gas compressors, etc) and the systems connected to the machines (e.g. hydraulic power plants, pumping systems, air distribution systems, etc).

DETAILED SYLLABUS  

Introduction 

Classification of machines. Turbines, compressors, volumetric, rotary machines and their applications to industrial practical cases. Analysis of performance: power, specific work, efficiency.

Basics of fluid mechanics 

Material and spatial description of the flow field. Translation, deformation and rotation. Reynolds’ transport theorem. Principles of conservation and balance (mass, momentum, energy, entropy) in differential form. Mass, momentum, thermal and mechanical energy in stationary and rotating frames of reference. 

Basics of fluid mechanics applied to turbomachinery 

Integral balances in turbomachines (mass, momentum, moment of momentum, energy) and basic applications. 
Gas dynamics equations, speed of sound, Mach number. Applications to nozzles in supersonic conditions, normal shock waves. 

Velocity diagrams coupled to stator and rotor blades for energy producing and consuming machines. Moment of Momentum balance. Energy transfer and different expressions of the Euler work. Trothalpy, degree of reaction, utilization for a turbine. 

Applications 

Scaling and similitude: dimensionless parameters, specific speed and diameter, Cordier curve. Scaling and similitude for compressible flow machines. 

Axial turbines: stage analysis, flow and loading coefficients, reaction ratio, special cases of 0 and 0.5 reaction ratio designs. Off-design operation and performance maps. 

Axial compressors: stage analysis, flow and loading coefficients, reaction ratio. De Haller design criterion and its effect on blade design. Off-design operation and performance maps. 

Centrifugal compressors: analysis of velocity diagrams, effect of blade shape on performance maps, stability and efficiency. Slip factor. Vaneless and vaned diffuser. Flow control (variable speed, IGV and throttling). 

Centrifugal pumps operation into systems: definition of head and volumetric flow rate. Head-flow rate performance map and effects on velocity diagrams, blade design and efficiency. System head curves for simple and multi-branched open-ended and closed-circuit systems. Friction factor and expression of dimensional friction losses. Flow control by variable speed and throttling.  

Cavitation: physical description; effects of system design on cavitation, Net Positive Suction Head, suction specific speed.

TEXTBOOKS AND MATERIAL 

S. Korpela. Principles of Turbomachinery, Wiley 2019. 

Karassik et al., Pump handbook, McGraw Hill. 

Powerpoint slides and videos are available on the MS-team website. 

Kinematics and Dynamics of Mechanisms

Kinematics and Dynamics of Mechanisms
3 YEAR1 semester9 CREDITS
Prof. Marco Ceccarelli2019-20
CECCARELLI MARCO 2020-21
Code: 8037957
SSD: ING-IND/13

OBJECTIVES

LEARNING OUTCOMES: The course aims to teach students the knowledge and tools that are needed to address the issues that are related to the identification, modeling, analysis, design of multi-body planar systems, and in particular some transmission organs in English language and terminology

KNOWLEDGE AND UNDERSTANDING: modeling and procedures to recognize the structure and characteristics of mechanisms and machines

APPLYING KNOWLEDGE AND UNDERSTANDING: acquisition of analysis procedures for the understanding of kinematic and dynamic characteristics of mechanisms and machines

MAKING JUDGEMENTS: possibility of judging the functionality of mechanisms and machines with their own qualitative and quantitative assessments

COMMUNICATION SKILLS: learning of technical terminology and procedures for presenting the performance of mechanisms

LEARNING SKILLS: learning of technical terminology and procedures for the presentation of the performance of mechanisms

COURSE SYLLABUS

  • Structure and classification of planar mechanical systems, kinematic modeling, mobility analysis, graphical approaches of kinematics analysis, kinematic analysis with computer-oriented algorithms, fundamentals of mechanism synthesis, trajectory generation; dynamics and statics modeling, graphical approaches of dynamics analysis, dynamic analysis with computer-oriented algorithms, performance evaluation; elements of mechanical transmissions with gears, belts, brakes, and flywheels.

Digital Electronics

Digital Electronics
3 YEAR1 semester9 CREDITS
Prof. Marco Re2019-20
RE MARCO 2020-21
Code: 8037956
SSD: ING-INF/01

OBJECTIVES

LEARNING OUTCOMES

This course aims at providing the fundamentals of DIGITAL ELECTRONICS. More in detail, it deals with the characterization and design of combinational circuits starting from gates. The target technology is CMOS. Starting from the study of the CMOS circuits and the implementation of memory cells the course will face the design and characterization of sequential circuits.

KNOWLEDGE AND UNDERSTANDING

The student will be able to analyze and design combinational and sequential circuits.
Starting from these blocks the student will be able to write a high-level description of a complex digital system based on a computational unit and a control unit.

APPLYING KNOWLEDGE AND UNDERSTANDING

The student will apply the knowledge and understanding developed to the analysis of practical problems. This would imply critical knowledge in terms of silicon real estate and speed for both combinational and sequential systems.
MAKING JUDGEMENTS: The student will have to prove his critical awareness with respect to the simplifying assumptions useful to describe and analyze combinational and sequential systems as well as his critical awareness of the correct order of magnitude of performance parameters while dealing or designing digital circuits.

COMMUNICATION SKILLS

The student will prove, mostly during the oral test, his capacity of describing the operation and functioning of digital systems.

LEARNING SKILLS

The student will get familiar with the schematization of practical problems, mostly during the development of his skills for the written test. This mainly concerns combinational systems and sequential systems

COURSE SYLLABUS

  • This course constitutes an introduction to the engineering of digital systems.
  • Starting with data representation in digital form, it goes on to provide students with the ability to design a circuit for a given algorithmic information processing task. For this purpose, Boolean functions and combinational design are covered, followed by sequential logic design through Finite State Machines. Moreover standard MSI blocks (sequential and combinational are illustrated) up the description of algorithmic state machines.
  • The student should be able to understand the structure of a complex digital system and able to design the architecture and the internal blocks of the system. In the course, a brief introduction to the electrical measurements for digital systems is given (oscilloscope, Logic State Analyzer, Pattern Generator).

Thermodynamics and Heat Transfer

Thermodynamics and Heat Transfer
2 YEAR2 semester9 CREDITS
Prof. Paolo Coppa2019-20
BOVESECCHI GIANLUIGI 2020-21
Michela Gefulsa2021-22
Code: 8039146
SSD: ING-IND/10

OBJECTIVES

LEARNING OUTCOMES: The course aims to provide students with the basic principles, physical laws, and applications of thermodynamics, thermo-fluid dynamics and heat transfer, with the dual purpose of preparing them to afford more applicative courses, and use the acquired knowledge for design and sizing simple components and thermal systems.

KNOWLEDGE AND UNDERSTANDING

Students will have to understand the physical laws of applied thermodynamics and heat transfer, and understand the structure and operation of the simplest components and systems. They will also demonstrate that they have acquired the basic methodologies for verifying and designing the studied devices.

APPLYING KNOWLEDGE AND UNDERSTANDING

Students must be able to afford courses for which this course is preparatory (for example Thermotechnique, or Machines) and to size or verify simple components and systems, topics covered by the course, such as air conditioning systems, engine systems, fins and heat exchangers.

MAKING JUDGEMENTS

Students must assume the autonomous capacity to face the subsequent studies for which this course is preparatory and to draw up simple projects of thermal systems that use the studied components. They will also need to be able to evaluate projects written by other parties, checking that the project requirements are satisfied.

COMMUNICATION SKILLS

Students must be able to illustrate in a complete and exhaustive way the acquired information, the results of their study and of their project activity, also through the normally used means of communication (discussion of the results obtained, report on the performed activity, PowerPoint presentations, etc.).

LEARNING SKILLS

Students must be able to apply the physical laws underlying the studied phenomena, and to face further studies that use the acquired knowledge. They will have to be able to expand the already owned information through the analysis of technical-scientific literature and to modify their curricula choosing future knowledge to be acquired on the base of their knowledge and tendency.

COURSE SYLLABUS

  • Fundamental laws of thermodynamics: zeroth law, first law for open and closed systems, second law, entropy definition and Clausius integral, Maxwell equations, Claperyron equation
  • Thermodynamic diagrams: P-v, T-s, H-s, P-h
  • Thermodynamic cycles for close and open systems: engine cycles: Otto Cycle, Diesel cycle, Joule Brighton cycle, Rankine and Hirn cycle, refrigeration cycle
  • Air and steam mixtures, design of conditioning air systems
  • Basic laws of fluid dynamics: Bernoulli equation, the motion of fluids in ducts, major and minor pressure drops, dimensional analysis for turbulent flow friction factor.
  • Heat transfer mechanisms. Conduction: Fourier law, basic conduction heat transfer equation, solution for simple geometries with and without heat generation, lumped parameter problems, cooling fins. Convection: dimension analysis for forced and free convection. Radiation heat transfer: basic laws of radiation, radiation exchanges between black bodies and grey bodies, configuration factors, electric analogy. Heat exchangers: types, size problem and rate problem.

Mechanics of Materials and Structures

Mechanics of Materials and Structures
2 YEAR2 semester9 CREDITS
Prof. Micheletti Andrea
Prof. Artioli Edoardo
A.Y. 2019-20
MICHELETTI ANDREA
ARTIOLI EDOARDO
2020-21
Code: 8037955
SSD: ICAR/08

OBJECTIVES

LEARNING OUTCOMES: The goal of this course is to provide the student with basic knowledge of the mechanics of linearly elastic structures and of the strength of materials. By completing this class successfully, the student will be able to compute simple structural elements and reasonably complex structures.

KNOWLEDGE AND UNDERSTANDING: At the end of this course, the student will be able to:

  • compute constraint reactions and internal actions in rigid-body systems and beams subjected to point/distributed forces and couples
  • compute centroid position and central principal second-order moments of area distributions
  • understand the formal structure of the theory of linear elasticity for both discrete and continuous systems (beams and 3D bodies)
  • analyze strain and stress states in 3D bodies
  • compute the stress state in beams subjected to uniaxial bending, biaxial bending, eccentric axial force
  • understand the behavior of beams subjected to shear with bending, and torsion
  • understand how to compute displacements/rotations in isostatic beam systems, how to solve statically underdetermined systems, how to apply yield criteria, and how to design beams against buckling

APPLYING KNOWLEDGE AND UNDERSTANDING: The student will apply the knowledge and understanding skills developed during the course to the analysis of practical problems. This includes the analysis of linearly elastic structures and structural members in terms of strength and stiffness.

MAKING JUDGEMENTS: The student will have to demonstrate his awareness of the modeling assumptions useful to describe and calculate structural elements, as well as his critical judgment on the static response of elastic structures under loads, in terms of stresses, strains, and displacements.

COMMUNICATION SKILLS: The student will demonstrate, mostly during the oral test, his capacity of analyzing and computing the static response of linearly elastic structures, as well as his knowledge of the underlying theoretical models.

LEARNING SKILLS: The student will get familiar with the modeling of structures and structural elements in practical problems, mostly during the development of his skills for the written test. This mainly concerns discrete systems, beams, and three-dimensional bodies.

COURSE SYLLABUS

  • Review of basic notions of vector and tensor algebra and calculus.
  • Kinematics and statics of rigid-body systems.
  • Geometry of area distributions.
  • Discrete linearly elastic systems, static-kinematic duality, solution methods.
  • Strain and stress in 3D continuous bodies and beam-like bodies.
  • Virtual power and virtual work equation for discrete systems, beams, and 3D bodies.
  • One-dimensional beam models: Bernoulli-Navier model, Timoshenko model, constitutive equations, governing differential equations.
  • Constitutive equation for linearly elastic and isotropic bodies, material moduli.
  • Hypothesis in linear elasticity, equilibrium problem for linearly elastic discrete systems, beams, and 3D bodies.
  • Three-dimensional beam model: the Saint-Venant problem, uniaxial and biaxial bending, eccentric axial force, shear and bending, torsion.
  • Elastic energy of beams and 3D bodies, work-energy theorem, Betti’s reciprocal theorem, Castigliano’s theorem.
  • Yield criteria (maximum normal stress, maximum tangential stress, maximum elastic energy, maximum distortion energy).
  • Buckling instability, bifurcation diagrams, load and geometry imperfections, Euler buckling load, design against buckling.
  • Basic notions on the finite element method and structural analysis software.

Feedback Control Systems

Feedback Control Systems
2 YEAR2 semester9 CREDITS
Prof. Cristiano Maria Verrelli2019-20
VERRELLI CRISTIANO MARIA 2020-21
Code: 8037953
SSD: ING-INF/04

OBJECTIVES

LEARNING OUTCOMES:
The theory of differential equations is successfully used to gain profound insight into the fundamental mathematical control design techniques for linear and nonlinear dynamical systems.

KNOWLEDGE AND UNDERSTANDING:
Students should be able to deeply understand (and be able to use) the theory of differential equations and of systems theory, along with related mathematical control techniques.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Students should be able to design feedback controllers for linear (and even nonlinear) dynamical systems.

MAKING JUDGEMENTS:
Students should be able to identify the specific design scenario and to apply the most suitable techniques. Students should be able to compare the effectiveness of different controls, while analyzing theoretical/experimental advantages and drawbacks.

COMMUNICATION SKILLS: Students are expected to be able to read and capture the main results of a technical paper concerning the topics of the course, as well as to effectively communicate in a precise and clear way the content of the course. Tutor-guided individual projects (including Maple and Matlab-Simulink computer simulations as well as visits to labs) invite an intensive participation and ideas exchange.

LEARNING SKILLS:
Being enough skilled in the specific field to undertake following studies characterized by a high degree of autonomy.

COURSE SYLLABUS

Linear systems
The matrix exponential; the variation of constants formula. Computation of the matrix exponential via eigenvalues and eigenvectors and via residual matrices. Necessary and sufficient conditions for exponential stability: Routh-Hurwitz criterion. Invariant subspaces. Impulse responses, step responses and steady state responses to sinusoidal inputs. Transient behaviors. Modal analysis: mode excitation by initial conditions and by impulsive inputs; modal observability from output measurements; modes which are both excitable and observable. Popov conditions for modal excitability and observability. Autoregressive moving average (ARMA) models and transfer functions.
Kalman reachability conditions, gramian reachability matrices and the computation of input signals to drive the system between two given states. Kalman observability conditions, gramian observability matrices and the computation of initial conditions given input and output signals. Equivalence between Kalman and Popov conditions. Kalman decomposition for non-reachable and non-observable systems.
Eigenvalues assignment by state feedback for reachable systems. Design of asymptotic observers and Kalman filters for state estimation of observable systems. Design of dynamic compensators to stabilize any reachable and observable system. Design of regulators to reject disturbances generated by linear exosystems.
Introduction to adaptive control. Introduction to tracking control. Minimum phase systems and Proportional Integral Derivative (PID) control.
Bode plots. Static gain, system gain and high-frequency gain. Zero-pole cancellation. Nyquist plot and Nyquist criterion. Root locus analysis. Stability margins. Frequency domain design. Realization theory.

Introduction to nonlinear systems Nonlinear models and nonlinear phenomena. Fundamental properties. Lyapunov stability. Linear systems and linearization. Center manifold theorem. Stabilization by linearization.

Analogue Electronics

Analogue Electronics
2 YEAR2 semester9 CREDITS
Prof. Rocco Gioffré2019-20
GIOFRE’ ROCCO 2020-21
Code: 8037954
SSD: ING-INF/01

OBJECTIVES

COURSE SYLLABUS

  • Classification of electrical systems and requirements.
  • Analysis of transitory and frequency behavior.
  • Distortion in electronic systems and Bode diagrams.
  • Diode semiconductor devices and circuit applications: clipper, clamper, peak detector, etc. Bipolar Junction and Field-Effect Transistors.
  • Biasing techniques for Transistors. Amplifiers classification, analysis and circuit design.
  • Frequency response of single and cascaded amplifiers.
  • Differential amplifiers and Cascode.
  • Current mirrors.
  • Feedback amplifiers and stability issues. Power amplifiers.
  • Operational amplifiers and related applications.
  • Oscillator circuits. Integrated circuits and voltage waveform generators.

Physics II

Physics II
2 YEAR1 semester9 CREDITS
Prof. Vittorio Foglietti2019-20
FOGLIETTI VITTORIO 2020-21
Code: 8037952
SSD: FIS/01

OBJECTIVES

LEARNING OUTCOMES: Learning the basic elements of Electromagnetism and fundamental physical principles of quantum mechanics.

KNOWLEDGE AND UNDERSTANDING:
Knowledge of the basic principles of electromagnetism and quantum mechanics useful for the own field of study. Understanding of advanced books on the arguments treated during the course.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Capacity to develop autonomously basic conceptual ideas using arguments treated during the course.

MAKING JUDGEMENTS:
Capacity to evaluate autonomously ideas or project using the knowledge acquired in the course.

COMMUNICATION SKILLS:
Capacity to share informations and ideas on the basis of knwoledge acquired in the course.
Comprehension of specific problems and relative solutions proposed.

LEARNING SKILLS:
The knowledge acquired in the course must be of help for the student in future courses, improving the capacity of autonomous learning.

COURSE SYLLABUS

1) Electric Charge and Electric Field : Conductors, Insulators, and Induced Charges.
Coulomb’s Law. Electric Field and Electric Forces. Electric Field Lines. Charge and Electric Flux, Gauss’’ s Law. Charges on Conductors.
2) Electric Potential: Electric Potential Energy, Electric Potential, Equipotential Surfaces, Potential Gradient. Definition of electric dipole. Approximated formula for the electric potential of a dipole at large distances.
3) Capacitors and Capacitance. Capacitors in series and parallel configuration. Electrostatic Potential Energy of a Capacitor. Polarization in Dielectrics. Induced Dipoles. Alignment of Polar Molecules. Electric Field inside a dielectric material. Relative dielectric constant. Capacitors with dielectric materials.
4) Electric current, Vector current density J, Resistivity (ρ) and conductivity ( σ) of materials, Ohm’s law in vector and scalar form, Resistors and resistance, Microscopic theory of electric transport in metals (Drude model). Differences between thermal velocity and drift velocity of charge carriers. Thermal coefficient of resistivity for metal and semiconductors. Resistors in parallel. Kirchhoff current law and the conservation of charge. Resistors in series. Kirchhoff voltage law ( KVL) and the conservative nature of electric field. Resistor and capacitor in series. Charging a capacitor. Solving the equation for current and voltage in RC circuits, time constant.
5) Introduction to magnetism, historical notes. Magnetic Force on a moving charged particle in a Magnetic Field. Definition of the vector ( cross ) product. Vector product expressed by the formal determinant and calculated by Sarrus Rule. Thomson’s q/m experiment and the discovery of the electron. Magnetic force on a current carrying conductor. Local equation for the magnetic force, the second formula of Laplace. Introduction to current loops, the torque. Force and Torque on a current loop in presence of a constant magnetic field. The magnetic dipole moment. Torque in vector form. Stable and unstable equilibrium states. Equivalence between a magnetic dipole of a current loop and the dipole of a magnet. Potential energy of a dipole moment in a magnetic field. Force exerted on a magnetic dipole in a non-uniform magnetic field. Working principle of a dc motor. Generalization of a magnetic dipole to current loops with irregular area. Magnetic dipole of a coil consisting of n loops in series. The Hall effect.
6) Historical introduction to the Biot Savart Laplace equation. Electric current as sources of magnetic field, the current element. The Biot Savat Laplace (BSL) equation. BSL equation for an infinitely long wire with an electric current flow. The flux of the magnetic field B. The Gauss Law for the magnetic field. Forces acting on wires with electric current flow. Magnetic field on the axis of a current loop and a coil. Ampere Circuital Law. Definition of a Solenoid. Magnetic field from a long cylindrical conductor. Magnetic field from a toroidal coil. The Bohr magneton. Magnetic materials. Paramagnetism, Diamagnetism, Ferromagnetism.
7) Magnetic induction experiments. Faraday Law. Lenz Law. Flux swept by a coil and Motional Electromotive Force. Induced Electric Field. Displacement current. The four Maxwell equations in integral form. Symmetry of the Maxwell equation. Self induction. Inductors. Inductor as circuit element.Self inductance of a coil. Magnetic Field Energy. The R-L circuit. The LC circuit. The RLC series circuit.
8) The electromagnetic waves. Derivation of EM waves from Maxwell Equation. The electromagnetic spectrum. Electromagnetic energy flow and the Poynting vector. Energy in a sinusoidal wave. Electromagnetic momentum flow. Standing Electromagnetic waves.
9) Light waves behaving as particles. The photocurrent experiment. Threshold frequency and Stopping Potential. Einstein’s explanation of Light absorbed as “Photons”. Light Emitted as Photons: X-Ray Production. Light Scattered as Photons: Compton Scattering.
10) Interference and diffraction of waves. The Wave Particle Duality. De Broglie wavelength. The x-ray diffraction from a crystal lattice, the Bragg’s Law. The electron diffraction experiment of Davisson and Germer. The double slit experiment with electrons. Waves in one dimension: Particle Waves, the one-dimensional Schrödinger equation. Physical interpretation of the Wave Function. Wave Packets. Uncertainty principle. Particle in a box. Energy-levels and wave functions for a particle in a box. The tunneling effect.

Mathematical Analysis II

Mathematical Analysis II
2 YEAR1 semester9 CREDITS
Prof. TANIMOTO YOH –
Prof. BUTTERLEY OLIVER JAMES
2019-20
BUTTERLEY OLIVER JAMES 2020-21
Code: 8037950
SSD: MAT/05

OBJECTIVES

LEARNING OUTCOMES:
One learns power series, differential calculus of several variables, line integral, multiple integral and surface and volume integral. One obtains the ability to calculate partial derivatives of elementary and composed functions, calculate various integrals and apply theorems of Green, Gauss and Stokes to facilitate the computations.

KNOWLEDGE AND UNDERSTANDING:
To know the definitions of basic conepts (convergence of series, partial derivatives, extremal points, multiple integral, line integal, surface integral and volume integral) and apply various theorems to execute concrete computations.

APPLYING KNOWLEDGE AND UNDERSTANDING:
To identify the theorems and techniques to apply to the given problems and execute computations correctly.

MAKING JUDGEMENTS:
To understand mathematical concepts for the given problems and to divide them into smaller problems that can be solved with the knowledge obtained during the course.

COMMUNICATION SKILLS:
To frame the problems in the obtained concepts, express the logic and general facts that are used during the computations.LEARNING SKILLS:

To know precisely basic mathematical concepts and apply them to some simple examples in physics.

COURSE SYLLABUS

  • Sequences and series of functions, Taylor series
  • Differential calculus of scalar and vector fields
  • Applications of differential calculus, extremal points
  • Basic differential equations
  • Line integrals
  • Multiple integrals
  • Surface integrals, Gauss and Stokes theorems