Physics - Lecture Notes
Notes:
Lecture 3 RELATIVITY OF MOTION
Antiderivative ? Definite integral
Lecture 3 - 1
G( x )
d
( F ( x )) = G ( x ) F ( x ) → G ( x )
dx
Learn more:
http://ocw.mit.edu/ans7870/resources/Strang/str
angtext.htm
F ( x)
derivative
x
Calculus, Gilbert Strang: e-book
diferentation
x
x
G ( x ) → F ( x ) ??
antiderivative
∫ G ( x )dx = F ( x ) + C
integrand function
Bogdan Żółtowski 2012
d
dx
integration
indefinite integral
[∫ F ( x )dx ] = F ( x )
primary function
Integration (properties):
∫ K ⋅ f ( x )dx = K ⋅ ∫ f ( x )dx
∫ [ f ( x ) + g( x )]dx = ∫ f ( x )dx + ∫ g( x )dx
∫x
n
dx =
x n +1
+C
n+1
( n ≠ −1)
1
∫ x dx = ln x + C
∫e
kx
dx =
1 x
e +C
k
∫ sin dx = − cos x + C ∫ cos xdx = sin x + C
1
∫ sin 2 x dx = tan x + C
Key terms:
antiderivative, integrand function, primary function, constant of integration
- 34 -
Physics - Lecture Notes
Bogdan Żółtowski 2012
Notes:
Differential equations
Lecture 3 - 2
Example
Unique solution of E calls for the additional
information concerning values of the function
and its derivatives at given variable value –
“Initial conditions”
Volume of that information depends on the order
of DE.
d2y
dy
−2 + y=0
2
dx
dx
y ( x ) = xe x
y( x ) ?
Learn more:
dny
⇒ y( x )
= f ( x , y , y ' , y ' ' , y n −1 )
n
dx
y ( x o ) = y o ; y ' ( x o ) = y1 ; ... y n −1 ( x o ) = y n −1
solving DE
initial value problem (IVP)
Ordinary differential equations ODE: function of one variable
Partial differential equations PDE: function of more than one variable
∂2E y
∂x 2
= µ 0ε 0
∂2E y
∂t 2
Key terms:
initial conditions
- 35 -
http://ocw.mit.edu/ans7870/resources/Strang/str
angtext.htm
Calculus, Gilbert Strang: e-book