Author: Simona Ranieri

*Mathematical Analysis II – starting Monday 29 September

Unfortunately, I won’t be able to be present this coming week, as such the classes of MA2 will start in the second week of the semester, Monday 29 September. I apogise for any inconvenience. Please share this information in any relevant channels. 

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

Electrical quantities and SI units. Electrical energy and electrical power. Passive and active sign convention. Passive and active elements. Ideal voltage and current sources. Basic ideal electric components: resistance, inductance, capacitance. Models of real components. Ohm-s law. Series and Parallel connection of components. Topological circuital laws: Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL). Mesh Current Method, Node Voltage Method. Sinusoidal functions: average and RMS (Root Mean Square) values.

Sinusoidal steady state circuit analysis. Phasors. Impedance and admittance. Analysis of circuits in AC steady state. Electrical power in the time domain and in sinusoidal steady state: active power, reactive power, complex power. Power factor correction. Maximum power transfer in AC. Application of superposition theorem in circuit analysis. Thevenin’s and Norton’s theorems.

Frequency response: first order electrical filters. Resonance: series and parallel resonant circuits.

Mutual inductance and ideal transformer. Three-Phase systems. Introduction to the power distribution and transportation grid. Time response and transient analysis. The unit step function, unit impulse function, exponential function, first-order circuits. Laplace transform method, Laplace transform of some typical functions, initial-value and final-value theorems, partial-fractions expansions, analysis of circuits in the s-domain. Network functions and circuit stability.

Electrical measurement bridges. Introductions to the electrical safety and electricity distribution system: description and prospects. Basics of designing a power plant. Effects of electricity on the human body and relative protection systems. Introduction to electrical machines: Tranformer and DC motor.

OBJECTIVES

LEARNING OUTCOMES:
Students will improve their knowledge under three main aspects: a) the comprehension and the related capability to articulate the arguments discussed throughout the course, using the proper scientific vocabulary and style; b) the methodological attitude toward complex problems, more specifically, to decompose a physical phenomenon into simpler elements to reach efficaciously a correct explanation; c) an elastic, fluid and reactive way of thinking, based on an inquisitive and hungry developed attitude toward new challenges.

KNOWLEDGE AND UNDERSTANDING:
Knowledge of the main arguments related to classical mechanics, gravitation, mechanical waves, elastic properties of solids and thermodynamics. Identification of the physical quantities and of the conservation principles needed to explain the main physical phenomena behind the above reported topics. Understanding scientific texts related to the course, and the acquisition of proper vocabulary and style.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Capability to achieve the correct solution of complex classical mechanical, gravitation, mechanical waves, elastic properties of solids and thermodynamics problems, through an effective and methodologically correct understanding and application of the principles governing the physical phenomena described. Capability to apply the acquired scientific way of thinking to new problems by comprehending, analysing and also modelling autonomously.

MAKING JUDGEMENTS:
Students are supported to improve their critical way of thinking and to develop an independent judgment able to pose, refine and elaborate scientific questions. The purpose is to pave the way to a free and active research spirit based on a questioning elastic and hungry mind.

COMMUNICATION SKILLS:
The students are expected to master, through familiarity with the arguments studied and through a developed practice refined in the framework of the course, a specific focus, style and proper scientific vocabulary.

LEARNING SKILLS:
Capability to understand, analyse, and model a physical phenomenon crossing different principles learnt during the course. Considering problems related to the topics of the course, students are expected to solve them and articulate the reasons behind the main assumptions followed. An inquisitive attitude, a proper scientific way of reasoning and communicative skills are therefore expected to be acquired during the course.

SYLLABUS

Mechanics (main arguments considered):
• Measurements and fundamental quantities; coordinate systems; elements of vector calculus;
• Relative motion and Galilean transformation; Inertial and non-inertial frames of reference; Fictitious forces;
• The point mass concept; kinematics; from rectilinear to general curvilinear motion on a plane; dynamics; the principle of conservation of momentum; Newton’s laws; the concept of force as interaction; angular momentum; torque; impulse; energy and work: conservative and dissipative forces; conservation principles; discussion of potential energy curves.
• From the point-mass to systems of particles: external and internal forces; Center of Mass (CM): definition, calculus and dynamics; angular momentum and torque for systems; the CM and laboratory frame of reference; König theorem part I and part II; energy of a system; reduced mass;
• Collisions: generality, impact parameter; elastic, partially inelastic and perfectly inelastic collision;
• From systems to the rigid body; moment of inertia: main properties and relationships; statics and dynamics; work on a rigid body; kinetic energy of a rigid body; pure rolling and sliding; rolling friction; systems with variable mass;
• Oscillatory motion: simple harmonic motion; energy of simple harmonic motion; superposition of simple harmonic motions; coupled oscillators; damped oscillations; forced oscillations;

Elastic properties of solids (main arguments considered):
• Young, Shear and Bulk moduli; elastic and plastic deformations; elastic hysteresis.

Gravitation (main arguments considered):
• Kepler’ s laws; Cavendish experiment; Newton’s law of universal gravitation; gravitational potential; a reduced mass approach; bounded and unbounded orbits; escape velocity; geostationary orbit;

Mechanical waves (main arguments considered):
• generality; transverse and longitudinal waves: velocity and equation of the waves; sound waves; intensity of waves; interference; Doppler effect; supersonic waves;

Fluids (main arguments considered):
• Introduction to fluids; mechanical actions on fluids; statics of fluids; Torricelli’s experiment; Archimede’ s principle; Pascal’s principle; fluid and conservative forces; fluids in non-inertial frames of reference; fluids dynamics: Lagrangian and Eulerian approaches; stationary motions; volumetric flow rate; Bernoulli’ s equation; Venturi’ s effect; considerations on real fluids: laminar and turbulent flows; viscosity; Reynolds’ number; considerations on drag forces in fluids;

Thermodynamics (main arguments considered):
• Main concepts of thermodynamics; zeroth principle of thermodynamics; the first principle of thermodynamics; internal energy, heat and work; Ideal gases and ideal gas law; P-V diagram; molar specific heats; heat capacity; main processes in thermodynamics; real gases; Wan der Waals equation; phase transitions; second principle of thermodynamics; Carnot cycle; Carnot theorem; Clausius theorem; entropy; thermodynamic Universe; Enthalpy, Helmholtz free energy, Gibbs free energy; Dalton’ s principle.

Texts

Students can use the book that best suits them. However, the references cited below are the ones considered during the course:
1. “Fundamental University Physics Volume 1: Mechanics”, by Alonso & Finn
2. “Fundamentals of Physics”, by Halliday & Resnick
3. “Physics for Scientists and Engineers”, by Serway & Jewett