Samenvatting - Mathematical Methods for Modern Physics (TN2986) - Minor Modern Physics
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Course
Mathematical Methods for Modern Physics (TN2986)
Institution
Technische Universiteit Delft (TU Delft)
Bij Mathematical Methods for Modern Physics (TN2986) wordt ingegaan op de benodigde wiskundige technieken voor het volgen van de minor Modern Physics. De behandelde onderwerpen zijn: differentiëren, coördinaatsystemen, complexe getallen, reeksen, integreren, differentiaalvergelijkingen, lineaire ...
Notebook 1 Differentiëren
Definitie
f ( x + ε ) − f ( x)
De definitie van de afgeleide luidt: f ' ( x ) = lim . Een functie die afgeleid kan
ε →0 ε
worden heet differentieerbaar.
Elementaire afgeleiden
f ( x) = xn f ' ( x ) = nx n−1
1
f ( x ) = log g ( x ) f '( x ) =
x ln ( g )
1
f ( x ) = ln ( x ) f '( x ) =
x
f ( x ) = ex f '( x ) = ex
f ( x ) = cos ( x ) f ' ( x ) = − sin ( x )
f ( x ) = sin ( x ) f ' ( x ) = cos ( x )
1
f ( x ) = tan ( x ) f ' ( x ) = 1+ tan 2 ( x ) =
cos 2 ( x )
1
f ( x ) = arctan ( x ) f '( x ) =
1+ x 2
d ( fg )
Productregel: = f ' g + fg' met f = f ( x ) en g = g ( x )
dx
df ( g ( x ))
Kettingregel: = f ' ( g ( x )) g' ( x )
dx
Inverse functies
Maak hier gebruik van de inverse functie waarvan de afgeleide bekend is.
VB
Bepaal de afgeleide van f ( x ) = arcsin ( x )
y = arcsin ( x )
x = sin ( y )
d d
dx
[ x ] = ⎡⎣sin ( y ) ⎤⎦
dx
dy
1 = cos ( y )
dx
dy
dx
= f '( x ) =
1
cos ( y )
=
1
2
( uit sin 2 ( y ) + cos 2 ( y ) = 1)
1− x
Taylorpolynoom
Je benadert een functie in een bepaald gebied met een polynoom.
∞
1
f ( x ) = f ( x0 ) + ∑ ( x − x0 ) f ( n ) ( x0 )
n
n=1 n!
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