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old wiki Curvilinear Coordinates
Aaron Meurer edited this page Dec 2, 2011
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This page shows the curvilinear_coordinates.py example, which demonstrates how to use sympy to calculate metric tensors (and other things like a laplace operator) in curvilinear coordinates.
$ python examples/advanced/curvilinear_coordinates.py
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Transformation: polar
ρ = ρ⋅cos(φ)
φ = ρ⋅sin(φ)
Jacobian:
⎡cos(φ) -ρ⋅sin(φ)⎤
⎢ ⎥
⎣sin(φ) ρ⋅cos(φ) ⎦
metric tensor g_{ij}:
⎡1 0 ⎤
⎢ ⎥
⎢ 2⎥
⎣0 ρ ⎦
inverse metric tensor g^{ij}:
⎡1 0 ⎤
⎢ ⎥
⎢ 1 ⎥
⎢0 ──⎥
⎢ 2⎥
⎣ ρ ⎦
det g_{ij}:
2
ρ
Laplace:
2
d d
──(f(ρ, φ)) ─────(f(ρ, φ)) 2
dρ dφ dφ d
─────────── + ────────────── + ─────(f(ρ, φ))
ρ 2 dρ dρ
ρ
________________________________________________________________________________
Transformation: cylindrical
ρ = ρ⋅cos(φ)
φ = ρ⋅sin(φ)
z = z
Jacobian:
⎡cos(φ) -ρ⋅sin(φ) 0⎤
⎢ ⎥
⎢sin(φ) ρ⋅cos(φ) 0⎥
⎢ ⎥
⎣ 0 0 1⎦
metric tensor g_{ij}:
⎡1 0 0⎤
⎢ ⎥
⎢ 2 ⎥
⎢0 ρ 0⎥
⎢ ⎥
⎣0 0 1⎦
inverse metric tensor g^{ij}:
⎡1 0 0⎤
⎢ ⎥
⎢ 1 ⎥
⎢0 ── 0⎥
⎢ 2 ⎥
⎢ ρ ⎥
⎢ ⎥
⎣0 0 1⎦
det g_{ij}:
2
ρ
Laplace:
2
d d
──(f(ρ, φ, z)) ─────(f(ρ, φ, z)) 2 2
dρ dφ dφ d d
────────────── + ───────────────── + ─────(f(ρ, φ, z)) + ─────(f(ρ, φ, z))
ρ 2 dρ dρ dz dz
ρ
________________________________________________________________________________
Transformation: spherical
ρ = ρ⋅cos(φ)⋅sin(θ)
θ = ρ⋅sin(φ)⋅sin(θ)
φ = ρ⋅cos(θ)
Jacobian:
⎡cos(φ)⋅sin(θ) ρ⋅cos(φ)⋅cos(θ) -ρ⋅sin(φ)⋅sin(θ)⎤
⎢ ⎥
⎢sin(φ)⋅sin(θ) ρ⋅cos(θ)⋅sin(φ) ρ⋅cos(φ)⋅sin(θ) ⎥
⎢ ⎥
⎣ cos(θ) -ρ⋅sin(θ) 0 ⎦
metric tensor g_{ij}:
⎡1 0 0 ⎤
⎢ ⎥
⎢ 2 ⎥
⎢0 ρ 0 ⎥
⎢ ⎥
⎢ 2 2 ⎥
⎣0 0 ρ ⋅sin (θ)⎦
inverse metric tensor g^{ij}:
⎡1 0 0 ⎤
⎢ ⎥
⎢ 1 ⎥
⎢0 ── 0 ⎥
⎢ 2 ⎥
⎢ ρ ⎥
⎢ ⎥
⎢ 1 ⎥
⎢0 0 ──────────⎥
⎢ 2 2 ⎥
⎣ ρ ⋅sin (θ)⎦
det g_{ij}:
4 2
ρ ⋅sin (θ)
Laplace:
2 2
d d d d
─────(f(ρ, θ, φ)) 2⋅──(f(ρ, θ, φ)) ─────(f(ρ, θ, φ)) ──(f(ρ, θ, φ))⋅cos(θ) 2
dθ dθ dρ dφ dφ dθ d
───────────────── + ──────────────── + ───────────────── + ───────────────────── + ─────(f(ρ, θ, φ))
2 ρ 2 2 2 dρ dρ
ρ ρ ⋅sin (θ) ρ ⋅sin(θ)
________________________________________________________________________________
Transformation: rotating disk
t = t
x = x⋅cos(t⋅w) - y⋅sin(t⋅w)
y = x⋅sin(t⋅w) + y⋅cos(t⋅w)
z = z
Jacobian:
⎡ 1 0 0 0⎤
⎢ ⎥
⎢-w⋅x⋅sin(t⋅w) - w⋅y⋅cos(t⋅w) cos(t⋅w) -sin(t⋅w) 0⎥
⎢ ⎥
⎢w⋅x⋅cos(t⋅w) - w⋅y⋅sin(t⋅w) sin(t⋅w) cos(t⋅w) 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
metric tensor g_{ij}:
⎡ 2 2 2 2 ⎤
⎢1 + w ⋅x + w ⋅y -w⋅y w⋅x 0⎥
⎢ ⎥
⎢ -w⋅y 1 0 0⎥
⎢ ⎥
⎢ w⋅x 0 1 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
inverse metric tensor g^{ij}:
⎡ 1 w⋅y -w⋅x 0⎤
⎢ ⎥
⎢ 2 2 2 ⎥
⎢w⋅y 1 + w ⋅y -x⋅y⋅w 0⎥
⎢ ⎥
⎢ 2 2 2 ⎥
⎢-w⋅x -x⋅y⋅w 1 + w ⋅x 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
det g_{ij}:
1
Laplace:
2 2 2 2 2
⎛ 2 2⎞ d ⎛ 2 2⎞ d d d d
⎝1 + w ⋅x ⎠⋅─────(f(t, x, y, z)) + ⎝1 + w ⋅y ⎠⋅─────(f(t, x, y, z)) + w⋅y⋅─────(f(t, x, y, z)) + w⋅y⋅─────(f(t, x, y, z)) - w⋅x⋅─────(f(t, x, y, z)) - w⋅x⋅
dy dy dx dx dx dt dt dx dy dt
2 2 2 2 2
d 2 d 2 d d d
─────(f(t, x, y, z)) - x⋅y⋅w ⋅─────(f(t, x, y, z)) - x⋅y⋅w ⋅─────(f(t, x, y, z)) + ─────(f(t, x, y, z)) + ─────(f(t, x, y, z))
dt dy dy dx dx dy dt dt dz dz
________________________________________________________________________________
Transformation: parabolic
σ = σ⋅τ
2 2
τ σ
τ = ── - ──
2 2
Jacobian:
⎡τ σ⎤
⎢ ⎥
⎣-σ τ⎦
metric tensor g_{ij}:
⎡ 2 2 ⎤
⎢σ + τ 0 ⎥
⎢ ⎥
⎢ 2 2⎥
⎣ 0 σ + τ ⎦
inverse metric tensor g^{ij}:
⎡ 1 ⎤
⎢─────── 0 ⎥
⎢ 2 2 ⎥
⎢σ + τ ⎥
⎢ ⎥
⎢ 1 ⎥
⎢ 0 ───────⎥
⎢ 2 2⎥
⎣ σ + τ ⎦
det g_{ij}:
2 2 4 4
2⋅σ ⋅τ + σ + τ
Laplace:
2 2
d d ⎛ 2 3⎞ d ⎛ 2 3⎞ d
─────(f(σ, τ)) ─────(f(σ, τ)) ⎝4⋅σ⋅τ + 4⋅σ ⎠⋅──(f(σ, τ)) ⎝4⋅τ⋅σ + 4⋅τ ⎠⋅──(f(σ, τ))
dσ dσ dτ dτ dσ dτ
────────────── + ────────────── + ───────────────────────────────── + ─────────────────────────────────
2 2 2 2 ⎛ 2 2⎞ ⎛ 2 2 4 4⎞ ⎛ 2 2⎞ ⎛ 2 2 4 4⎞
σ + τ σ + τ ⎝σ + τ ⎠⋅⎝4⋅σ ⋅τ + 2⋅σ + 2⋅τ ⎠ ⎝σ + τ ⎠⋅⎝4⋅σ ⋅τ + 2⋅σ + 2⋅τ ⎠
________________________________________________________________________________
Transformation: elliptic
μ = a⋅cos(ν)⋅cosh(μ)
ν = a⋅sin(ν)⋅sinh(μ)
Jacobian:
⎡a⋅cos(ν)⋅sinh(μ) -a⋅cosh(μ)⋅sin(ν)⎤
⎢ ⎥
⎣a⋅cosh(μ)⋅sin(ν) a⋅cos(ν)⋅sinh(μ) ⎦
metric tensor g_{ij}:
⎡ 2 2 2 2 2 2 ⎤
⎢a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν) 0 ⎥
⎢ ⎥
⎢ 2 2 2 2 2 2 ⎥
⎣ 0 a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎦
inverse metric tensor g^{ij}:
⎡ 2 2 2 2 2 2
⎢ a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)
⎢────────────────────────────────────────────────────────────────────────────────── 0
⎢ 4 2 2 2 2 4 4 4 4 4 4
⎢2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)
⎢
⎢ 2 2 2 2 2 2
⎢ a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)
⎢ 0 ──────────────────────────────────────────────────────────────────────
⎢ 4 2 2 2 2 4 4 4 4
⎣ 2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
────────────⎥
4 4 ⎥
(μ)⋅sin (ν)⎦
det g_{ij}:
4 2 2 2 2 4 4 4 4 4 4
2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)
Laplace:
2 2
⎛ 2 2 2 2 2 2 ⎞ d ⎛ 2 2 2 2 2 2 ⎞ d
⎝a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎠⋅─────(f(μ, ν)) ⎝a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎠⋅─────(f(μ, ν))
dμ dμ dν dν
────────────────────────────────────────────────────────────────────────────────── + ──────────────────────────────────────────────────────────────────────
4 2 2 2 2 4 4 4 4 4 4 4 2 2 2 2 4 4 4 4
2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν) 2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh
⎛ 2 2 2 2 2 2 ⎞ ⎛ 4 3 4 4 4 3 4 2 3 2
⎝a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎠⋅⎝- 4⋅a ⋅cos (ν)⋅sinh (μ)⋅sin(ν) + 4⋅a ⋅cosh (μ)⋅sin (ν)⋅cos(ν) - 4⋅a ⋅cosh (μ)⋅sin (ν)⋅sinh (μ)⋅
──────────── + ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
4 4 ⎛ 4 2 2 2 2 4 4 4 4 4 4 ⎞ ⎛ 4 2 2 2 2
(μ)⋅sin (ν) ⎝2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎠⋅⎝4⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) +
4 3 2 2 ⎞ d ⎛ 2 2 2 2 2 2 ⎞ ⎛ 4 4 3 4 3
cos(ν) + 4⋅a ⋅cos (ν)⋅cosh (μ)⋅sinh (μ)⋅sin(ν)⎠⋅──(f(μ, ν)) ⎝a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin (ν)⎠⋅⎝4⋅a ⋅cos (ν)⋅sinh (μ)⋅cosh(μ) + 4⋅a ⋅cosh (μ)⋅si
dν
─────────────────────────────────────────────────────────── + ─────────────────────────────────────────────────────────────────────────────────────────────
4 4 4 4 4 4 ⎞ ⎛ 4 2 2 2 2 4 4 4 4 4 4
2⋅a ⋅cos (ν)⋅sinh (μ) + 2⋅a ⋅cosh (μ)⋅sin (ν)⎠ ⎝2⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + a ⋅cos (ν)⋅sinh (μ) + a ⋅cosh (μ)⋅sin
4 4 2 3 2 4 2 2 3 ⎞ d
n (ν)⋅sinh(μ) + 4⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh(μ) + 4⋅a ⋅cos (ν)⋅sin (ν)⋅sinh (μ)⋅cosh(μ)⎠⋅──(f(μ, ν))
dμ
──────────────────────────────────────────────────────────────────────────────────────────────────────────
⎞ ⎛ 4 2 2 2 2 4 4 4 4 4 4 ⎞
(ν)⎠⋅⎝4⋅a ⋅cos (ν)⋅cosh (μ)⋅sin (ν)⋅sinh (μ) + 2⋅a ⋅cos (ν)⋅sinh (μ) + 2⋅a ⋅cosh (μ)⋅sin (ν)⎠