staniČna membrana i akcijski potencijal - sfzg.unizg.hr · variables – quantitive description of...
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Math functions
April,2018. S.Dolanski Babić
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Variables – quantitive description of chosen
system properties
Physical law – describes connection between
variables of one or several observed systems
Function – mathematical procedure of assignation
of variables from one set to the variables from the
other
1. Math functions
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Function
general form:
independent variables: tx,
1. Math functions
),( txfy
dependent variable:
function, f – describes how variable y
depends on variables x and t
y
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Function
1. Math functions
Modes function are:
1. graphical in coordinate frame
2. tabulation – two interconnected sequences of
numbers
3. analytical review – mathematical equation
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Linear function
mathematical representation is:
graphical representation is straight line
1. Math functions
baxy
a – slope
b – y-intercept
linear function may be:
- increasing for a > 0
- decreasing for a < 0
zero point has coordinates:
abx /
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Proportionality
axy
1. Math functions
To determine experimentally whether two physical quantities are directly
proportional, one performs several measurements and plots the resulting data
points in a Cartesian coordinate system. If the points lie on or close to a
straight line that passes through the origin (0, 0), then the two variables are
probably proportional, with the proportionality constant given by the line's
slope.
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Example: Motion with constant velocity
1. Math functions
s - dependent variable
v - slope const.v
t - independent variable
tstx v)(
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Inverse proportionality
mathematical form:
k - constant
there are no no zeros of the function; the line representing the function does not intercept the Y-axis
1. Math functions
function is defined for x<0, x>0
the first-quadrant is only
used in physics
function is not defined for x=0
x
ky
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Example: inverse proportionality
1. Math functions
pressure dependence of the volume for isothermal processes of ideal gas:
V
nRTp
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Exponential function
general form:
a – base, the most frequently is e (the base of natural logarithmic function)
b – rate changes
A=y(0) starting value
there are no zeros of the function
y>0 !
1. Math functions
bxAay
bxay
1b1b
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Example: exponential function
1. Math functions
the most frequent function in description of naturaI
process
increasing exponential function
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Examples: exponential function
1. Math functions
radioactive decay
decreasing exponential function
Ttmm /20
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Comparasion
Logarithmic function increases slower than linear function – helpful for presentation of variables which are changing in great intervals
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Periodic functions
breathing, heart beat, the changes of pressure in
the arteries
the function has the same value after period X:
time periodicity: ;
spatial periodicity: wavelength,
)()2( xfXxf
1. Math functions
)()( xfXxf
tx TX
f
vTv
http://en.wikipedia.org/wiki/Periodic_function
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Periodic functions
1. Math functions
- space distribution of pressure in
sound wave or electric and magnetic
fields in electromagnetic wave
tEE sin0
tBB sin0
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Harmonic function
the simplest periodic function: free oscillations
tAty sin)(
1. Math functions
)(ty – elongation
- amplitude A
f 2
Tf /1
http://www.animations.physics.unsw.edu.au/jw/oscillations.htm#Initial
frequency – the number of oscillation in a 1 second
general form )sin( tAy
F – the phase angle, determined by the starting conditions
sin)0( Ay
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Nonharmonic function
no analitical form
1. Math functions
periodic function
nonharmonic function
harmonic function http://en.wikipedia.org/wiki/Fourier_series
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Fourier theorem
1. Math functions
Fourier theorem – nonharmonic function can be represented by the sum of
harmonic functions
n
i
ii tAtx1
sin)(
1 ii
frequency of higher
harmonic frequency of ground harmonic
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Animations:
http://physics.mef.hr/Predavanja/java%20matem%20fje/index.html