The course provides the fundamental theoretical knowledge in developing mathematical modeling for mechanics and thermodynamics.
Curriculum
teacher profile teaching materials
1. Introduction
Physics as a theory of motion. Space, time, and mechanics in the modern view: quantum physics and the theory of relativity. Classical mechanics and its domain of validity. Point particle. Position. Degrees of freedom of a mechanical system. Reference frames and coordinates. Direction cosines. Physical quantities: operational definition, fundamental and derived quantities, units of measurement, dimensional analysis. Inertia and relativity.
2. Vectors
Vectors. Vector addition and multiplication by a scalar. Scalar product. Vector product. Derivative of a vector. Angular velocity.
3. Kinematics of a point particle
Kinematics. Equation of motion. Position, velocity, and acceleration. From acceleration to position as a function of time. Initial conditions. Uniform motions: definition. Arc length. Uniform rectilinear motion. Uniform circular motion. Acceleration in circular motion and in arbitrary planar motion. Examples. Motion with constant vector acceleration. Projectile motion: definition, equation of motion, time of flight, range, maximum height, trajectory. Example: the hunter and the monkey.
4. Principles of point-particle dynamics
Inertial reference frames. Principles of classical mechanics. Reference frames in uniform relative translational motion. Invariance of the second law of mechanics. Galilean transformations. Reference frames in arbitrary relative motion. Transport acceleration and Coriolis acceleration. Physical meaning and apparent forces. Linear momentum, angular momentum, and torque about a point. Fundamental equations of mechanics. The simple pendulum. Harmonic motion. Impulse and work: definitions.
5. Consequences of the second law of dynamics
The differential of a function. Impulse of a force and the impulse-momentum theorem. Work and the work-energy theorem. Examples. Conservative forces. Conservative forces and the potential function. Potential energy of a conservative field. Conservation of mechanical energy. Energy in the presence of non-conservative forces.
Functions of several variables: partial derivatives and total differential. Necessary and sufficient condition for a force field to be conservative. Line integrals and calculation of the potential. Conditions and types of equilibrium. Stationary points of the potential energy. Definition of power.
6. Force laws
Force laws: introduction. Gravitational force. Newton’s law for point masses and the gravitational field generated by a point mass. Spherical coordinates. Central forces. Conservativity. Gravitational potential energy. Gauss’s theorem. Gravitational field generated by a spherically symmetric mass distribution. Escape velocity. Elastic forces. Viscous drag forces and the falling motion of a body. Constraint forces; unilateral constraints. Bilateral constraints. Static and kinetic friction.
7. Laws of system dynamics
Dynamics of systems. Fundamental equations and the work-energy theorem for a system of point particles. Force couple. Newton’s third law and its consequence: action and reaction. Centre of mass. Fundamental equations of the dynamics of systems. Kinetic energy and König’s theorem. Angular momentum for systems of two particles rotating about an axis. The two-body problem. Reduced mass. Example: the Earth-Moon system. Equivalent systems of forces. Independence of the torque from the choice of reference point. Equivalence to a force and a couple. Systems of parallel forces. Centre of gravity.
8. Rigid-body systems
Rigid bodies: definition and degrees of freedom. Conditions for equilibrium. Importance of rotational motion. Angular momentum about an axis of symmetry. Moment of inertia about an axis. Calculation of moments of inertia about the centre of mass: homogeneous ring, disk, and rod. Angular momentum about a non-symmetry axis. Generalizations: central axes of inertia, principal axes of inertia, and the Huygens-Steiner theorem, with examples; general case. Projection of angular momentum along the rotation axis. Compound pendulum: equation of motion and resisting torques. Rigid bodies in contact with constraints. Systems composed of connected rigid bodies.
9. Collisions
Collisions: definition and conservation of linear momentum. Evaluation of average forces. Laboratory frame and centre-of-mass frame. Elastic collisions between spherical bodies: general case, equal masses, central elastic collision. Elastic collision with a wall. Inelastic collisions. Coefficient of restitution. Perfectly inelastic collisions. Collisions involving constrained and unconstrained material systems, with examples.
10. Fluid mechanics
Definition of a fluid. Mechanical actions on fluids. Pressure, shear stress, and body forces. Viscosity. Perfect fluids. Equations of fluid statics. Homogeneous fluids and Stevin’s law. Atmospheric pressure. Archimedes’ principle. Examples. Communicating vessels. Fluid statics in a conservative field. Equilibrium of immiscible fluids. Hydrodynamics. Perfect fluids. Lagrangian and Eulerian descriptions. Streamlines and flow lines. Conservation of mass. Flow rate. Bernoulli’s theorem. Examples and applications. Venturi effect. Lift.
THERMODYNAMICS
11. Thermodynamic systems and temperature
Thermodynamics: introduction and general principles. Microscopic foundations. Thermodynamic systems. Intensive and extensive parameters. Equilibrium states. Transformations between equilibrium states. Work. Work along a cyclic transformation. Ideal gases: equation of state and universal gas constant. Work during an isothermal expansion of an ideal gas. Temperature. Temperature scales.
12. Heat and the first law of thermodynamics
Heat. First law of thermodynamics. Local form of the first law along quasi-static transformations. Adiabatic transformations. Zero-work transformations. Specific heat of solid bodies. Equilibrium temperature of two isolated solid bodies. Ideal gases: equation of state. Clapeyron diagram and reversible transformations. Specific heats at constant pressure and constant volume. Reversible adiabatic transformations of ideal gases. Polytropic transformations. Specific heat along a polytropic transformation.
13. Second law of thermodynamics
Physics and probability. Second law of thermodynamics: Clausius and Kelvin-Planck statements. Elementary heat engines. Carnot cycle. Efficiency. Refrigeration cycle. Coefficient of performance. Carnot’s theorem. Absolute thermodynamic temperature. Clausius inequality. Entropy. Free expansion of a gas. Entropy and the second law of thermodynamics. Entropy of solids and ideal gases. Entropy as a state variable.
14. Thermodynamic potentials
Introduction and physical meaning. Legendre transform. Helmholtz free energy. Gibbs free energy. Enthalpy. Maxwell relations.
Programme
MECHANICS1. Introduction
Physics as a theory of motion. Space, time, and mechanics in the modern view: quantum physics and the theory of relativity. Classical mechanics and its domain of validity. Point particle. Position. Degrees of freedom of a mechanical system. Reference frames and coordinates. Direction cosines. Physical quantities: operational definition, fundamental and derived quantities, units of measurement, dimensional analysis. Inertia and relativity.
2. Vectors
Vectors. Vector addition and multiplication by a scalar. Scalar product. Vector product. Derivative of a vector. Angular velocity.
3. Kinematics of a point particle
Kinematics. Equation of motion. Position, velocity, and acceleration. From acceleration to position as a function of time. Initial conditions. Uniform motions: definition. Arc length. Uniform rectilinear motion. Uniform circular motion. Acceleration in circular motion and in arbitrary planar motion. Examples. Motion with constant vector acceleration. Projectile motion: definition, equation of motion, time of flight, range, maximum height, trajectory. Example: the hunter and the monkey.
4. Principles of point-particle dynamics
Inertial reference frames. Principles of classical mechanics. Reference frames in uniform relative translational motion. Invariance of the second law of mechanics. Galilean transformations. Reference frames in arbitrary relative motion. Transport acceleration and Coriolis acceleration. Physical meaning and apparent forces. Linear momentum, angular momentum, and torque about a point. Fundamental equations of mechanics. The simple pendulum. Harmonic motion. Impulse and work: definitions.
5. Consequences of the second law of dynamics
The differential of a function. Impulse of a force and the impulse-momentum theorem. Work and the work-energy theorem. Examples. Conservative forces. Conservative forces and the potential function. Potential energy of a conservative field. Conservation of mechanical energy. Energy in the presence of non-conservative forces.
Functions of several variables: partial derivatives and total differential. Necessary and sufficient condition for a force field to be conservative. Line integrals and calculation of the potential. Conditions and types of equilibrium. Stationary points of the potential energy. Definition of power.
6. Force laws
Force laws: introduction. Gravitational force. Newton’s law for point masses and the gravitational field generated by a point mass. Spherical coordinates. Central forces. Conservativity. Gravitational potential energy. Gauss’s theorem. Gravitational field generated by a spherically symmetric mass distribution. Escape velocity. Elastic forces. Viscous drag forces and the falling motion of a body. Constraint forces; unilateral constraints. Bilateral constraints. Static and kinetic friction.
7. Laws of system dynamics
Dynamics of systems. Fundamental equations and the work-energy theorem for a system of point particles. Force couple. Newton’s third law and its consequence: action and reaction. Centre of mass. Fundamental equations of the dynamics of systems. Kinetic energy and König’s theorem. Angular momentum for systems of two particles rotating about an axis. The two-body problem. Reduced mass. Example: the Earth-Moon system. Equivalent systems of forces. Independence of the torque from the choice of reference point. Equivalence to a force and a couple. Systems of parallel forces. Centre of gravity.
8. Rigid-body systems
Rigid bodies: definition and degrees of freedom. Conditions for equilibrium. Importance of rotational motion. Angular momentum about an axis of symmetry. Moment of inertia about an axis. Calculation of moments of inertia about the centre of mass: homogeneous ring, disk, and rod. Angular momentum about a non-symmetry axis. Generalizations: central axes of inertia, principal axes of inertia, and the Huygens-Steiner theorem, with examples; general case. Projection of angular momentum along the rotation axis. Compound pendulum: equation of motion and resisting torques. Rigid bodies in contact with constraints. Systems composed of connected rigid bodies.
9. Collisions
Collisions: definition and conservation of linear momentum. Evaluation of average forces. Laboratory frame and centre-of-mass frame. Elastic collisions between spherical bodies: general case, equal masses, central elastic collision. Elastic collision with a wall. Inelastic collisions. Coefficient of restitution. Perfectly inelastic collisions. Collisions involving constrained and unconstrained material systems, with examples.
10. Fluid mechanics
Definition of a fluid. Mechanical actions on fluids. Pressure, shear stress, and body forces. Viscosity. Perfect fluids. Equations of fluid statics. Homogeneous fluids and Stevin’s law. Atmospheric pressure. Archimedes’ principle. Examples. Communicating vessels. Fluid statics in a conservative field. Equilibrium of immiscible fluids. Hydrodynamics. Perfect fluids. Lagrangian and Eulerian descriptions. Streamlines and flow lines. Conservation of mass. Flow rate. Bernoulli’s theorem. Examples and applications. Venturi effect. Lift.
THERMODYNAMICS
11. Thermodynamic systems and temperature
Thermodynamics: introduction and general principles. Microscopic foundations. Thermodynamic systems. Intensive and extensive parameters. Equilibrium states. Transformations between equilibrium states. Work. Work along a cyclic transformation. Ideal gases: equation of state and universal gas constant. Work during an isothermal expansion of an ideal gas. Temperature. Temperature scales.
12. Heat and the first law of thermodynamics
Heat. First law of thermodynamics. Local form of the first law along quasi-static transformations. Adiabatic transformations. Zero-work transformations. Specific heat of solid bodies. Equilibrium temperature of two isolated solid bodies. Ideal gases: equation of state. Clapeyron diagram and reversible transformations. Specific heats at constant pressure and constant volume. Reversible adiabatic transformations of ideal gases. Polytropic transformations. Specific heat along a polytropic transformation.
13. Second law of thermodynamics
Physics and probability. Second law of thermodynamics: Clausius and Kelvin-Planck statements. Elementary heat engines. Carnot cycle. Efficiency. Refrigeration cycle. Coefficient of performance. Carnot’s theorem. Absolute thermodynamic temperature. Clausius inequality. Entropy. Free expansion of a gas. Entropy and the second law of thermodynamics. Entropy of solids and ideal gases. Entropy as a state variable.
14. Thermodynamic potentials
Introduction and physical meaning. Legendre transform. Helmholtz free energy. Gibbs free energy. Enthalpy. Maxwell relations.
Core Documentation
Non vengono adottati testi in lingua ingleseReference Bibliography
- The Feynman Lectures on Physics I -- Addison-Wesley - E. Fermi - Thermodynamics -- DoverAttendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
The assessment consists of a written examination followed by an oral examination. The written examination lasts two hours and requires students to solve three problems: one on the kinematics and dynamics of a point particle, one on the dynamics of systems (alternatively, on collisions or fluid mechanics), and one on thermodynamics. The written examination may be waived by successfully completing two midterm examinations of similar structure (two written midterm tests lasting two hours, each consisting of three problems), covering different portions of the course syllabus. In the written examinations, all three problems carry the same weight. The written examination papers, together with their solutions, are made available on the course Microsoft Teams channel. teacher profile teaching materials
1. Introduction
Physics as a theory of motion. Space, time, and mechanics in the modern view: quantum physics and the theory of relativity. Classical mechanics and its domain of validity. Point particle. Position. Degrees of freedom of a mechanical system. Reference frames and coordinates. Direction cosines. Physical quantities: operational definition, fundamental and derived quantities, units of measurement, dimensional analysis. Inertia and relativity.
2. Vectors
Vectors. Vector addition and multiplication by a scalar. Scalar product. Vector product. Derivative of a vector. Angular velocity.
3. Kinematics of a point particle
Kinematics. Equation of motion. Position, velocity, and acceleration. From acceleration to position as a function of time. Initial conditions. Uniform motions: definition. Arc length. Uniform rectilinear motion. Uniform circular motion. Acceleration in circular motion and in arbitrary planar motion. Examples. Motion with constant vector acceleration. Projectile motion: definition, equation of motion, time of flight, range, maximum height, trajectory. Example: the hunter and the monkey.
4. Principles of point-particle dynamics
Inertial reference frames. Principles of classical mechanics. Reference frames in uniform relative translational motion. Invariance of the second law of mechanics. Galilean transformations. Reference frames in arbitrary relative motion. Transport acceleration and Coriolis acceleration. Physical meaning and apparent forces. Linear momentum, angular momentum, and torque about a point. Fundamental equations of mechanics. The simple pendulum. Harmonic motion. Impulse and work: definitions.
5. Consequences of the second law of dynamics
The differential of a function. Impulse of a force and the impulse-momentum theorem. Work and the work-energy theorem. Examples. Conservative forces. Conservative forces and the potential function. Potential energy of a conservative field. Conservation of mechanical energy. Energy in the presence of non-conservative forces.
Functions of several variables: partial derivatives and total differential. Necessary and sufficient condition for a force field to be conservative. Line integrals and calculation of the potential. Conditions and types of equilibrium. Stationary points of the potential energy. Definition of power.
6. Force laws
Force laws: introduction. Gravitational force. Newton’s law for point masses and the gravitational field generated by a point mass. Spherical coordinates. Central forces. Conservativity. Gravitational potential energy. Gauss’s theorem. Gravitational field generated by a spherically symmetric mass distribution. Escape velocity. Elastic forces. Viscous drag forces and the falling motion of a body. Constraint forces; unilateral constraints. Bilateral constraints. Static and kinetic friction.
7. Laws of system dynamics
Dynamics of systems. Fundamental equations and the work-energy theorem for a system of point particles. Force couple. Newton’s third law and its consequence: action and reaction. Centre of mass. Fundamental equations of the dynamics of systems. Kinetic energy and König’s theorem. Angular momentum for systems of two particles rotating about an axis. The two-body problem. Reduced mass. Example: the Earth-Moon system. Equivalent systems of forces. Independence of the torque from the choice of reference point. Equivalence to a force and a couple. Systems of parallel forces. Centre of gravity.
8. Rigid-body systems
Rigid bodies: definition and degrees of freedom. Conditions for equilibrium. Importance of rotational motion. Angular momentum about an axis of symmetry. Moment of inertia about an axis. Calculation of moments of inertia about the centre of mass: homogeneous ring, disk, and rod. Angular momentum about a non-symmetry axis. Generalizations: central axes of inertia, principal axes of inertia, and the Huygens-Steiner theorem, with examples; general case. Projection of angular momentum along the rotation axis. Compound pendulum: equation of motion and resisting torques. Rigid bodies in contact with constraints. Systems composed of connected rigid bodies.
9. Collisions
Collisions: definition and conservation of linear momentum. Evaluation of average forces. Laboratory frame and centre-of-mass frame. Elastic collisions between spherical bodies: general case, equal masses, central elastic collision. Elastic collision with a wall. Inelastic collisions. Coefficient of restitution. Perfectly inelastic collisions. Collisions involving constrained and unconstrained material systems, with examples.
10. Fluid mechanics
Definition of a fluid. Mechanical actions on fluids. Pressure, shear stress, and body forces. Viscosity. Perfect fluids. Equations of fluid statics. Homogeneous fluids and Stevin’s law. Atmospheric pressure. Archimedes’ principle. Examples. Communicating vessels. Fluid statics in a conservative field. Equilibrium of immiscible fluids. Hydrodynamics. Perfect fluids. Lagrangian and Eulerian descriptions. Streamlines and flow lines. Conservation of mass. Flow rate. Bernoulli’s theorem. Examples and applications. Venturi effect. Lift.
THERMODYNAMICS
11. Thermodynamic systems and temperature
Thermodynamics: introduction and general principles. Microscopic foundations. Thermodynamic systems. Intensive and extensive parameters. Equilibrium states. Transformations between equilibrium states. Work. Work along a cyclic transformation. Ideal gases: equation of state and universal gas constant. Work during an isothermal expansion of an ideal gas. Temperature. Temperature scales.
12. Heat and the first law of thermodynamics
Heat. First law of thermodynamics. Local form of the first law along quasi-static transformations. Adiabatic transformations. Zero-work transformations. Specific heat of solid bodies. Equilibrium temperature of two isolated solid bodies. Ideal gases: equation of state. Clapeyron diagram and reversible transformations. Specific heats at constant pressure and constant volume. Reversible adiabatic transformations of ideal gases. Polytropic transformations. Specific heat along a polytropic transformation.
13. Second law of thermodynamics
Physics and probability. Second law of thermodynamics: Clausius and Kelvin-Planck statements. Elementary heat engines. Carnot cycle. Efficiency. Refrigeration cycle. Coefficient of performance. Carnot’s theorem. Absolute thermodynamic temperature. Clausius inequality. Entropy. Free expansion of a gas. Entropy and the second law of thermodynamics. Entropy of solids and ideal gases. Entropy as a state variable.
14. Thermodynamic potentials
Introduction and physical meaning. Legendre transform. Helmholtz free energy. Gibbs free energy. Enthalpy. Maxwell relations.
Programme
MECHANICS1. Introduction
Physics as a theory of motion. Space, time, and mechanics in the modern view: quantum physics and the theory of relativity. Classical mechanics and its domain of validity. Point particle. Position. Degrees of freedom of a mechanical system. Reference frames and coordinates. Direction cosines. Physical quantities: operational definition, fundamental and derived quantities, units of measurement, dimensional analysis. Inertia and relativity.
2. Vectors
Vectors. Vector addition and multiplication by a scalar. Scalar product. Vector product. Derivative of a vector. Angular velocity.
3. Kinematics of a point particle
Kinematics. Equation of motion. Position, velocity, and acceleration. From acceleration to position as a function of time. Initial conditions. Uniform motions: definition. Arc length. Uniform rectilinear motion. Uniform circular motion. Acceleration in circular motion and in arbitrary planar motion. Examples. Motion with constant vector acceleration. Projectile motion: definition, equation of motion, time of flight, range, maximum height, trajectory. Example: the hunter and the monkey.
4. Principles of point-particle dynamics
Inertial reference frames. Principles of classical mechanics. Reference frames in uniform relative translational motion. Invariance of the second law of mechanics. Galilean transformations. Reference frames in arbitrary relative motion. Transport acceleration and Coriolis acceleration. Physical meaning and apparent forces. Linear momentum, angular momentum, and torque about a point. Fundamental equations of mechanics. The simple pendulum. Harmonic motion. Impulse and work: definitions.
5. Consequences of the second law of dynamics
The differential of a function. Impulse of a force and the impulse-momentum theorem. Work and the work-energy theorem. Examples. Conservative forces. Conservative forces and the potential function. Potential energy of a conservative field. Conservation of mechanical energy. Energy in the presence of non-conservative forces.
Functions of several variables: partial derivatives and total differential. Necessary and sufficient condition for a force field to be conservative. Line integrals and calculation of the potential. Conditions and types of equilibrium. Stationary points of the potential energy. Definition of power.
6. Force laws
Force laws: introduction. Gravitational force. Newton’s law for point masses and the gravitational field generated by a point mass. Spherical coordinates. Central forces. Conservativity. Gravitational potential energy. Gauss’s theorem. Gravitational field generated by a spherically symmetric mass distribution. Escape velocity. Elastic forces. Viscous drag forces and the falling motion of a body. Constraint forces; unilateral constraints. Bilateral constraints. Static and kinetic friction.
7. Laws of system dynamics
Dynamics of systems. Fundamental equations and the work-energy theorem for a system of point particles. Force couple. Newton’s third law and its consequence: action and reaction. Centre of mass. Fundamental equations of the dynamics of systems. Kinetic energy and König’s theorem. Angular momentum for systems of two particles rotating about an axis. The two-body problem. Reduced mass. Example: the Earth-Moon system. Equivalent systems of forces. Independence of the torque from the choice of reference point. Equivalence to a force and a couple. Systems of parallel forces. Centre of gravity.
8. Rigid-body systems
Rigid bodies: definition and degrees of freedom. Conditions for equilibrium. Importance of rotational motion. Angular momentum about an axis of symmetry. Moment of inertia about an axis. Calculation of moments of inertia about the centre of mass: homogeneous ring, disk, and rod. Angular momentum about a non-symmetry axis. Generalizations: central axes of inertia, principal axes of inertia, and the Huygens-Steiner theorem, with examples; general case. Projection of angular momentum along the rotation axis. Compound pendulum: equation of motion and resisting torques. Rigid bodies in contact with constraints. Systems composed of connected rigid bodies.
9. Collisions
Collisions: definition and conservation of linear momentum. Evaluation of average forces. Laboratory frame and centre-of-mass frame. Elastic collisions between spherical bodies: general case, equal masses, central elastic collision. Elastic collision with a wall. Inelastic collisions. Coefficient of restitution. Perfectly inelastic collisions. Collisions involving constrained and unconstrained material systems, with examples.
10. Fluid mechanics
Definition of a fluid. Mechanical actions on fluids. Pressure, shear stress, and body forces. Viscosity. Perfect fluids. Equations of fluid statics. Homogeneous fluids and Stevin’s law. Atmospheric pressure. Archimedes’ principle. Examples. Communicating vessels. Fluid statics in a conservative field. Equilibrium of immiscible fluids. Hydrodynamics. Perfect fluids. Lagrangian and Eulerian descriptions. Streamlines and flow lines. Conservation of mass. Flow rate. Bernoulli’s theorem. Examples and applications. Venturi effect. Lift.
THERMODYNAMICS
11. Thermodynamic systems and temperature
Thermodynamics: introduction and general principles. Microscopic foundations. Thermodynamic systems. Intensive and extensive parameters. Equilibrium states. Transformations between equilibrium states. Work. Work along a cyclic transformation. Ideal gases: equation of state and universal gas constant. Work during an isothermal expansion of an ideal gas. Temperature. Temperature scales.
12. Heat and the first law of thermodynamics
Heat. First law of thermodynamics. Local form of the first law along quasi-static transformations. Adiabatic transformations. Zero-work transformations. Specific heat of solid bodies. Equilibrium temperature of two isolated solid bodies. Ideal gases: equation of state. Clapeyron diagram and reversible transformations. Specific heats at constant pressure and constant volume. Reversible adiabatic transformations of ideal gases. Polytropic transformations. Specific heat along a polytropic transformation.
13. Second law of thermodynamics
Physics and probability. Second law of thermodynamics: Clausius and Kelvin-Planck statements. Elementary heat engines. Carnot cycle. Efficiency. Refrigeration cycle. Coefficient of performance. Carnot’s theorem. Absolute thermodynamic temperature. Clausius inequality. Entropy. Free expansion of a gas. Entropy and the second law of thermodynamics. Entropy of solids and ideal gases. Entropy as a state variable.
14. Thermodynamic potentials
Introduction and physical meaning. Legendre transform. Helmholtz free energy. Gibbs free energy. Enthalpy. Maxwell relations.
Core Documentation
Non vengono adottati testi in lingua ingleseReference Bibliography
- The Feynman Lectures on Physics I -- Addison-Wesley - E. Fermi - Thermodynamics -- DoverAttendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
The assessment consists of a written examination followed by an oral examination. The written examination lasts two hours and requires students to solve three problems: one on the kinematics and dynamics of a point particle, one on the dynamics of systems (alternatively, on collisions or fluid mechanics), and one on thermodynamics. The written examination may be waived by successfully completing two midterm examinations of similar structure (two written midterm tests lasting two hours, each consisting of three problems), covering different portions of the course syllabus. In the written examinations, all three problems carry the same weight. The written examination papers, together with their solutions, are made available on the course Microsoft Teams channel.