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Lecture 1Introduction, vector calculus,

functions of more variables,

Physics for informatics

Ing. Jaroslav Jíra, CSc.

Introduction

Lecturers prof. Ing. Stanislav Pekárek, CSc.,

pekarek_at_fel.cvut.cz , room 49A Ing. Jaroslav

Jíra, CSc., jira_at_fel.cvut.cz , room 42

Source of information http//aldebaran.feld.cvut.

cz/ , section Physics for OI

Textbooks Physics I, Pekárek S., Murla

M. Physics I - seminars, Pekárek S., Murla M.

Scoring system of the Physics for OI The maximum

reachable amount of points from semester is 100.

Points from semester go with each student to the

exam, where they create a part of the final grade

according to the exam rules.

Conditions for assessment - to gain at least 40

points, - to measure specified number of

laboratory works, - to submit semester work

OI Seminars Labs OI Seminars Labs

Week Program of the lab measurement or program of the seminar

1 Introduction, safety instructions, laboratory rules, list of experiments,

2 - 3 Introduction to electric circuit analysis (osciloscope), measuring devices. Exercise at the computer - kinematics and dynamics of a particle, analytical and numerical derivation.

4 - 5 Exercise at the computer - work and energy. Measurement of simple electronic circuit response.

6 - 7 Exercise at the computer. Measurement for the semester work.

8 Test 1

9 - 10 Exercise at the computer. Analysis of a spiral type stability and instability.

11 - 12 Exercise at the computer. Strange attractor analysis.

13 Test 2

14 Final grading. Assessment.

OI Lectures OI Lectures

Week Topic

1 Mathematical apparatus of physics, vector calculus, physical fields.

2 Differential equations, particle kinematics.

3 - 9 Newton's laws, equations of motion, work, power, kinetic and potential energy, Mechanical oscillating systems. Simple and damped harmonic motion. Forced oscillations. Resonance of displacement and velocity. Waves and their mathematical description, dispersion, interference. System of n-particles, conservation of momentum and energy. Rigid bodies, equations of motion, rotation of the rigid body with respect to the fixed axis. Moment of inertia, parallel axis theorem

10 - 13 Classification of dynamical systems. Phase portraits, phase trajectory, fixed points, dynamical flow. Stability of linear systems. Mathematical description of linear dynamical systems. Nonlinear dynamical systems. Bifurcation, logistic equation, deterministic chaos.

14 Description of complex system (physics of plasma, biological systems, nonlinear acoustics), discussion.

Points can be gained by - written tests, max.

50 points. Two tests by 25 points max. (8th and

13th week) - semester work for max. 30 points

(lab report program) - activity on exercises,

problem solving, voluntary homeworks, max. 20

points

Examination first part Every student must

solve certain number of problems according to

his/her points from the semester.

Number of problems to solve Points from the semester

1 90 and more

2 75 89

3 65 74

4 55 64

5 less than 55

Examination - second part Student answers

questions in written form during the written

exam. The answers are marked and the total of 30

points can be gained in this way. Then follows

oral part of the exam and each student defends a

mark according to the table below. The column

resulting in better mark is taken into account.

written exam semester written exam

A excellent 1 25 120

B very good 1- 23 110

C good 2 20 100

D satisfactory 2- 18 90

E sufficient 3 15 80

Vector calculus - basics

A vector standard notation for three dimensions

Unit vectors i,j,k are vectors of magnitude 1 in

directions of the x,y,z axes.

Magnitude of a vector

Position vector is a vector r from the origin to

the current position

where x,y,z, are projections of r to the

coordinate axes.

Adding and subtracting vectors

Multiplying a vector by a scalar

Example of multiplying of a vector by a scalar in

a plane

Multiplication of a vector by a scalar in the

Mathematica

Example of addition of three vectors in a plane

The vectors are given

Numerical addition gives us

Graphical solution

Addition of three vectors in the Mathematica

Example of subtraction of two vectors a plane

The vectors are given

Numerical subtraction gives us

Graphical solution

Subtraction of two vectors in the Mathematica

Time derivation and time integration of a vector

function

Example of the time derivation of a vector

The motion of a particle is described by the

vector equation

Determine for any time t a) b)

the tangential and the radial accelerations

Time derivation of a vector in the Mathematica

Time derivation of a vector in the Mathematica

-continued

What would happen without Assuming and Refine

What would happen without Simplify

Graphical output of the

Example of the time integration of a vector

Evaluate the time dependence of the velocity and

the position vector for the projectile motion.

Initial velocity v0(10,20) m/s and g(0,-9.81)

m/s2.

Time integration of a vector in the Mathematica

Study of balistic projectile motion, when

components of initial velocity are given

Projectile motion - trajectory

Scalar product

- Scalar product (dot product) is defined as
- Where T is a smaller angle between vectors
- a and b and S is a resulting scalar.

For three component vectors we can write

Geometric interpretation scalar product is

equal to the area of rectangle having a and

b.cosT as sides. Blue and red arrows represent

original vectors a and b.

Basic properties of the scalar product

Vector product

Vector product (cross product) is defined

as Where T is the smaller angle between vectors

a and b and n is unit vector perpendicular to

the plane containing a and b.

Geometric interpretation - the magnitude of the

cross product can be interpreted as the positive

area A of the parallelogram having a and b as

sides

Component notation

Basic properties of the vector product

Scalar product and vector product in the

Mathematica

Direction of the resulting vector of the vector

productcan be determined either by the right

hand rule or by the screw rule

Vector triple product

Geometric interpretation of the scalar triple

product is a volume of a paralellepiped V

Scalar triple product

Scalar field and gradient

Scalar field associates a scalar quantity to

every point in a space. This association can be

described by a scalar function f and can be also

time dependent. (for instance temperature,

density or pressure distribution).

The gradient of a scalar field is a vector field

that points in the direction of the greatest rate

of increase of the scalar field, and whose

magnitude is that rate of increase.

Example the gradient of the function f(x,y)

-(cos2x cos2y)2 depicted as a projected vector

field on the bottom plane.

Example 2 finding extremes of the scalar field

Find extremes of the function

Extremes can be found by assuming

In this case

Answer there are two extremes

Extremes of the scalar field in the Mathematica

Vector operators

Gradient (Nabla operator)

Divergence

Curl

Laplacian

Basic mechanical quantities and relations and their analogies in linear and rotational motion Basic mechanical quantities and relations and their analogies in linear and rotational motion Basic mechanical quantities and relations and their analogies in linear and rotational motion Basic mechanical quantities and relations and their analogies in linear and rotational motion Basic mechanical quantities and relations and their analogies in linear and rotational motion Basic mechanical quantities and relations and their analogies in linear and rotational motion

Linear motion Linear motion Linear motion Rotational motion Rotational motion Rotational motion

s, r path, position vector m f angle rad

v velocity ms-1 ? anglular velocity rads-1

a acceleration ms-2 e angular acceleration rads-2

F force N M torque Nm

m mass kg J moment of inertia kgm2

p linear momentum kgms-1 b angular momentum kgm2s-1

Work W F s Work W F s Work W F s Work W M f Work W M f Work W M f

Kinetic energy Ek ½ m v2 Kinetic energy Ek ½ m v2 Kinetic energy Ek ½ m v2 Kinetic energy Ek ½ J ?2 Kinetic energy Ek ½ J ?2 Kinetic energy Ek ½ J ?2

Equation of motion F m a Equation of motion F m a Equation of motion F m a Equation of motion M J e Equation of motion M J e Equation of motion M J e