Lecture 1 Introduction, vector calculus, functions of more variables,

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Lecture 1Introduction, vector calculus,
functions of more variables,
Physics for informatics
Ing. Jaroslav Jíra, CSc.
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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
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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.
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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.
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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
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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
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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.
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Adding and subtracting vectors
Multiplying a vector by a scalar
Example of multiplying of a vector by a scalar in
a plane
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Multiplication of a vector by a scalar in the
Mathematica
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Example of addition of three vectors in a plane
The vectors are given
Numerical addition gives us
Graphical solution
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Addition of three vectors in the Mathematica
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Example of subtraction of two vectors a plane
The vectors are given
Numerical subtraction gives us
Graphical solution
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Subtraction of two vectors in the Mathematica
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Time derivation and time integration of a vector
function
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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
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Time derivation of a vector in the Mathematica
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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
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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.
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Time integration of a vector in the Mathematica
Study of balistic projectile motion, when
components of initial velocity are given
Projectile motion - trajectory
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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
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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
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Scalar product and vector product in the
Mathematica
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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
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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.
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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
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Extremes of the scalar field in the Mathematica
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Vector operators
Gradient (Nabla operator)
Divergence
Curl
Laplacian
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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