Warwick PX420 Solar MHD 2011-2012: Useful Mathematics
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Useful Mathematics
1. Linear first-order ODE
The linear first-order ODE
dy
+ a(x)y = 0
dx
where y(x) is a given function. The solution is
Z
y(x) = A exp
a(x) dx ,
where A is an arbitrary constant to be determined from initial conditions.
In particular, when the initial condition is
y(x = 0) = A0 ,
the particular solution is
y(x) = A0 exp
Z
0
x
a(x′ ) dx′ .
2. Harmonic Oscillator
The harmonic oscillator ODE
d2 y
+ k2 y = 0
dx2
where y(x) is an unknown function and k is constant, has the solution
y(x) = B cos kx + C sin kx,
where B and C are arbitrary constants. These constants can be determined from initial or boundary
conditions.
An alternative form of the solution is
y(x) = A cos(kx + φ),
where A and φ are arbitrary constants. In this form, A can be considered as an amplitude of the harmonic
oscillation and φ is an initial phase.
Also, the solution can be written with the use of the complex numbers,
y(x) = a exp(ikx) + b exp(−ikx),
where a and b are arbitrary constants and i is the imaginary unit.
When the equation is of the form
d2 y
− k 2 y = 0,
dx2
it has the solution
y(x) = B exp(kx) + C exp(−kx),
where B and C are arbitrary constants.
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Warwick PX420 Solar MHD 2011-2012: Useful Mathematics
3. The ∇-operator
3.1 In Cartesian Coordinates (x, y, z)
Nabla-operator
∂
∂
∂
ex +
ey +
ez
∂x
∂y
∂z
∇=
Gradient of a scalar function
∇f = grad f =
∂f
∂f
∂f
ex +
ey +
ez
∂x
∂y
∂z
Divergence of a vector function
∇ · B = div B =
Curl of a vector function
∂By
∂Bz
∂Bx
+
+
∂x
∂y
∂z
ex
∂
∇ × B = curl B = ∂x
Bx
ey
∂
∂y
By
ez ∂ ∂z Bz This is the compact form using the determinant, it allows to determine the components of the vector.
The components actually are
∂By
∂Bz
−
ex +
∇ × B = curl B =
∂y
∂z
∂By
∂Bx
∂Bx ∂Bz
ey +
ez
−
−
∂z
∂x
∂x
∂y
Laplasian
∇2 f =
∂2f
∂2f
∂2f
+ 2 + 2
2
∂x
∂y
∂z
3.2 In Cylindrical Coordinates (r, φ, z)
∇A =
∇·B=
∇×B=
1 ∂A
∂A
∂A
er +
eφ +
ez .
∂r
r ∂φ
∂z
1 ∂
1 ∂Bφ
∂Bz
(rBr ) +
+
r ∂r
r ∂φ
∂z
∂Br
∂Bφ
∂Bz
1 ∂Bz
er +
eφ
−
−
r ∂φ
∂z
∂z
∂r
1 ∂
1 ∂Br
+
ez
(rBφ ) −
r ∂r
r ∂φ
1 ∂
∇ A=
r ∂r
2
∂A
1 ∂2A ∂2A
r
+ 2
+
∂r
r ∂φ2
∂z 2
Warwick PX420 Solar MHD 2011-2012: Useful Mathematics
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Bφ2
∂Br
∂Br
Bφ ∂Br
(B · ∇)B = Br
er +
+
−
+ Bz
∂r
r ∂φ
r
∂z
Br ∂
∂Bφ
Bφ ∂Bφ
+
eφ +
(rBφ ) +
+ Bz
r ∂r
r ∂φ
∂z
∂Bz
Bφ ∂Bz
∂Bz
ez
+ Br
+
+ Bz
∂z
∂r
r ∂φ
3.3 Some properties
For arbitrary vectors V and U, and a scalar φ:
div(φV) = φdivV + grad φ · V,
div(U × V) = V(curl U) − U(curl V),
div grad φ = ∇2 φ,
curl(φV) = φcurl V + (grad φ) × V,
curl curl V = grad div V − ∇2 V,
∇2 V = (∇2 V1 , ∇2 V2 , ∇2 V3 ),
where 1,2 and 3 are, e.g., x,y and z,
curl grad φ = ∇ × ∇φ = 0,
div curl V = ∇ · ∇ × V = 0.
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