- #1
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- 10,463
Hi, everyone. I am new here, so I hope I am follow the protocols. Please
let me know otherwise. Also, I apologize for not knowing Latex yet, tho
I hope to learn it soon.
am trying to show that the vector field:
X^2(del/delx)+del/dely
Is not a complete vector field. I think this is
from John Lee's book, but I am not sure (it was in my
class notes.)
From what I understand, we need to find the
integral curves for the vector field first, i.e
we need to solve the system:
dx/dt=[x(t)]^2
and
dy/dt=1
I found the solutions to be given by (1/(x+c),y+c')
c,c' real constants.
In my notes ( 2-yrs old, unfortunately) , there is a solution:
Phi(x,t)=(1/(1-tx), y+t)
somehow in function of (x,t)
In addition, there is a statement that Phi(x,t)
satisfies:
Phi(x,t+s)=Phi(x,t)oPhi(x,t) (o = composition)
and that Phi satisfies certain initial conditions
(which were not given explicitly for the problem).
I suspect this has to see with one-parameter groups,
but I am not sure of it, and I don't understand them
that well, nor the relation with complete V.Fields.
I would appreciate any explanation or help.
let me know otherwise. Also, I apologize for not knowing Latex yet, tho
I hope to learn it soon.
am trying to show that the vector field:
X^2(del/delx)+del/dely
Is not a complete vector field. I think this is
from John Lee's book, but I am not sure (it was in my
class notes.)
From what I understand, we need to find the
integral curves for the vector field first, i.e
we need to solve the system:
dx/dt=[x(t)]^2
and
dy/dt=1
I found the solutions to be given by (1/(x+c),y+c')
c,c' real constants.
In my notes ( 2-yrs old, unfortunately) , there is a solution:
Phi(x,t)=(1/(1-tx), y+t)
somehow in function of (x,t)
In addition, there is a statement that Phi(x,t)
satisfies:
Phi(x,t+s)=Phi(x,t)oPhi(x,t) (o = composition)
and that Phi satisfies certain initial conditions
(which were not given explicitly for the problem).
I suspect this has to see with one-parameter groups,
but I am not sure of it, and I don't understand them
that well, nor the relation with complete V.Fields.
I would appreciate any explanation or help.