Mixing
Looking for explanation to solution - divergence theorem, differential equations
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What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system
$begin{cases}dot x = cos(x+y+z) \ dot y = -sin(y+z) \ dot z = -cos(x+y+z)+sin(y+z)+2zend{cases}$
Solution
Let $Omega(t)$ be the domain at time $t$. At $t=0$ this is the cube we were given, and the volume is $V(0) = int_{Omega(0)}dxdydz = 2^3 = 8$
Notice that the divergence of this system (the right side of the system) is $2$, and so:
$dot V = int_{Omega(t)}text{div}(F)dxdydz = 2int_{Omega(t)}dxdydz = 2V$
Hence $V(t) = V(0)e^{2t} = 8e^{2t}$ so our answer is $V(20) = 8e^{40}$
My issue is
I'm not sure why $dot V = int_{Omega(t)}text{div}(F)dxdydz$. I understood everything else.
calculus ordinary-differential-equations proof-explanation volume divergence
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
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add a comment |
$begingroup$
What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system
$begin{cases}dot x = cos(x+y+z) \ dot y = -sin(y+z) \ dot z = -cos(x+y+z)+sin(y+z)+2zend{cases}$
Solution
Let $Omega(t)$ be the domain at time $t$. At $t=0$ this is the cube we were given, and the volume is $V(0) = int_{Omega(0)}dxdydz = 2^3 = 8$
Notice that the divergence of this system (the right side of the system) is $2$, and so:
$dot V = int_{Omega(t)}text{div}(F)dxdydz = 2int_{Omega(t)}dxdydz = 2V$
Hence $V(t) = V(0)e^{2t} = 8e^{2t}$ so our answer is $V(20) = 8e^{40}$
My issue is
I'm not sure why $dot V = int_{Omega(t)}text{div}(F)dxdydz$. I understood everything else.
calculus ordinary-differential-equations proof-explanation volume divergence
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
$endgroup$
$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26
add a comment |
$begingroup$
What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system
$begin{cases}dot x = cos(x+y+z) \ dot y = -sin(y+z) \ dot z = -cos(x+y+z)+sin(y+z)+2zend{cases}$
Solution
Let $Omega(t)$ be the domain at time $t$. At $t=0$ this is the cube we were given, and the volume is $V(0) = int_{Omega(0)}dxdydz = 2^3 = 8$
Notice that the divergence of this system (the right side of the system) is $2$, and so:
$dot V = int_{Omega(t)}text{div}(F)dxdydz = 2int_{Omega(t)}dxdydz = 2V$
Hence $V(t) = V(0)e^{2t} = 8e^{2t}$ so our answer is $V(20) = 8e^{40}$
My issue is
I'm not sure why $dot V = int_{Omega(t)}text{div}(F)dxdydz$. I understood everything else.
calculus ordinary-differential-equations proof-explanation volume divergence
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
$endgroup$
What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system
$begin{cases}dot x = cos(x+y+z) \ dot y = -sin(y+z) \ dot z = -cos(x+y+z)+sin(y+z)+2zend{cases}$
Solution
Let $Omega(t)$ be the domain at time $t$. At $t=0$ this is the cube we were given, and the volume is $V(0) = int_{Omega(0)}dxdydz = 2^3 = 8$
Notice that the divergence of this system (the right side of the system) is $2$, and so:
$dot V = int_{Omega(t)}text{div}(F)dxdydz = 2int_{Omega(t)}dxdydz = 2V$
Hence $V(t) = V(0)e^{2t} = 8e^{2t}$ so our answer is $V(20) = 8e^{40}$
My issue is
I'm not sure why $dot V = int_{Omega(t)}text{div}(F)dxdydz$. I understood everything else.
calculus ordinary-differential-equations proof-explanation volume divergence
calculus ordinary-differential-equations proof-explanation volume divergence
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
edited Jan 31 at 20:41
Oria Gruber
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
asked Jan 31 at 20:34
Oria GruberOria Gruber
6,53732463
6,53732463
$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26
add a comment |
$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26
$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26
$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26
add a comment |
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$begingroup$
If you want a rigorous proof for this, take a look at Meiss' Differential dynamical systems (9.2, Volume preserving flows) or Strogatz's Nonlinear Dynamics and Chaos ... (9.2, Simple properties of Lorenz Equations, Volume Contraction).
$endgroup$
– Evgeny
Feb 1 at 9:26