1. Lecture 1 Introduction, vector calculus, functions of more variables, Ing. Jaroslav Jíra, CSc. Physics for informatics

2. Introduction Source of information: http://aldebaran.feld.cvut.cz/, section Physics for OIhttp://aldebaran.feld.cvut.cz/ Lecturers: prof. Ing. Stanislav Pekárek, CSc., pekarek@fel.cvut.cz, room 49Apekarek@fel.cvut.cz Ing. Jaroslav Jíra, CSc., jira@fel.cvut.cz, room 42jira@fel.cvut.cz Textbooks: Physics I, Pekárek S., Murla M. Physics I - seminars, Pekárek S., Murla M. Internet tests: https://fyzika.feld.cvut.cz/auth/oitesthttps://fyzika.feld.cvut.cz/auth/oitest

3. Conditions for assessment: - to gain at least 40 points, - to measure six laboratory experiments and submit reports from them 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. Points can be gained by: - written tests, max. 50 points. Two tests by 25 points max. (8 th and 13 th week) - laboratory reports, max. 30 points. The last five lab reports are marked by up to 6 points each - tests on the internet, max. 20 points. There are 10 electronical tests available on the internet, each consisting of 8 questions. Correct answering of ALL eight questions results in 2 point gain for the student.

4. Number of problems to solvePoints from the semester 190 and more 275 – 89 365 – 74 455 – 64 5less than 55 Examination – first part: Every student must solve certain number of problems according to his/her points from the semester.

5. 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 this way. Then the oral part of the exam follows and each student defends a mark according to the table below. The column resulting in better mark is taken into account. written examsemester + written exam Aexcellent125120 Bvery good1-23110 Cgood220100 Dsatisfactory2-1890 Esufficient31580

6. 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.

7. Adding and subtracting vectors Multiplying a vector by a scalar Example of multiplying of a vector by a scalar in a plane

8. Multiplication of a vector by a scalar in the Mathematica

9. Example of addition of three vectors in a plane The vectors are given: Numerical addition gives us Graphical solution:

10. Addition of three vectors in the Mathematica

11. Example of subtraction of two vectors a plane The vectors are given: Numerical subtraction gives us Graphical solution:

12. Subtraction of two vectors in the Mathematica

13. Time derivation and time integration of a vector function

14. Determine for any time t: a) b) the tangential and the radial accelerations Example of the time derivation of a vector The motion of a particle is described by the vector equation

15. Time derivation of a vector in the Mathematica

16. 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

17. 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 v 0 =(10,20) m/s and g=(0,-9.81) m/s 2.

18. Time integration of a vector in the Mathematica Projectile motion - trajectory: Study of balistic projectile motion, when components of initial velocity are given

19. Scalar product (dot product) – is defined as Where Θ is a smaller angle between vectors a and b and S is a resulting scalar. Scalar product For three component vectors we can write Geometric interpretation – scalar product is equal to the area of rectangle having a and b.cosΘ as sides. Blue and red arrows represent original vectors a and b. Basic properties of the scalar product

20. Vector product Basic properties of the vector product Vector product (cross product) – is defined as Where Θ 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

21. Scalar product and vector product in the Mathematica

22. Direction of the resulting vector of the vector product can be determined either by the right hand rule or by the screw rule Vector triple product Scalar triple product Geometric interpretation of the scalar triple product is a volume of a paralellepiped V

23. 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) = −(cos 2 x + cos 2 y) 2 depicted as a projected vector field on the bottom plane.

24. 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

25. Extremes of the scalar field in the Mathematica

26. Vector operators Gradient (Nabla operator) Laplacian Divergence Curl

27. Basic mechanical quantities and relations and their analogies in linear and rotational motion Linear motionRotational motion s, rpath, position vector[ m ]φangle[ rad ] vvelocity[ m*s -1 ]ωanglular velocity[ rad*s -1 ] aacceleration[ m*s -2 ]εangular acceleration[ rad*s -2 ] Fforce[ N ]Mtorque[ N*m] mmass[ kg ]Jmoment of inertia[ kg*m 2 ] plinear momentum[ kg*m*s -1 ]bangular momentum[kg*m 2 *s -1 ] Work W= F sWork W= M φ Kinetic energy E k = ½ m v 2 Kinetic energy E k = ½ J ω 2 Equation of motion F = m aEquation of motion M = J ε