Integral curve and first integral, what is it exactly?
$begingroup$
An integral curve of a vecteur field $V$ is a curve $gamma (t)$ s.t. $dot gamma (t)=V(gamma (t))$. Btw, why such a name for a curve ?
The a first integral is a curve s.t. $$left<V(delta (t)), dot delta (t)right>=0.$$
Could someone explain me what are these first integral ? I don't see it from the definition. Moreover, why is it called first integral ? Also, if $gamma (t)$ is an integral curve of $V$ and a first integral, then $|dotgamma (t)|=0$ and thus $gamma $ is constant. But in my course it's written that $V$ is constant along $gamma $, but I don't see the link...
real-analysis manifolds
asked Jan 12 at 11:23
user623855user623855
1507
$endgroup$
add a comment |
$begingroup$
An integral curve of a vecteur field $V$ is a curve $gamma (t)$ s.t. $dot gamma (t)=V(gamma (t))$. Btw, why such a name for a curve ?
The a first integral is a curve s.t. $$left<V(delta (t)), dot delta (t)right>=0.$$
Could someone explain me what are these first integral ? I don't see it from the definition. Moreover, why is it called first integral ? Also, if $gamma (t)$ is an integral curve of $V$ and a first integral, then $|dotgamma (t)|=0$ and thus $gamma $ is constant. But in my course it's written that $V$ is constant along $gamma $, but I don't see the link...
real-analysis manifolds
asked Jan 12 at 11:23
user623855user623855
1507
$endgroup$
1
$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51
add a comment |
$begingroup$
An integral curve of a vecteur field $V$ is a curve $gamma (t)$ s.t. $dot gamma (t)=V(gamma (t))$. Btw, why such a name for a curve ?
The a first integral is a curve s.t. $$left<V(delta (t)), dot delta (t)right>=0.$$
Could someone explain me what are these first integral ? I don't see it from the definition. Moreover, why is it called first integral ? Also, if $gamma (t)$ is an integral curve of $V$ and a first integral, then $|dotgamma (t)|=0$ and thus $gamma $ is constant. But in my course it's written that $V$ is constant along $gamma $, but I don't see the link...
real-analysis manifolds
asked Jan 12 at 11:23
user623855user623855
1507
$endgroup$
An integral curve of a vecteur field $V$ is a curve $gamma (t)$ s.t. $dot gamma (t)=V(gamma (t))$. Btw, why such a name for a curve ?
The a first integral is a curve s.t. $$left<V(delta (t)), dot delta (t)right>=0.$$
Could someone explain me what are these first integral ? I don't see it from the definition. Moreover, why is it called first integral ? Also, if $gamma (t)$ is an integral curve of $V$ and a first integral, then $|dotgamma (t)|=0$ and thus $gamma $ is constant. But in my course it's written that $V$ is constant along $gamma $, but I don't see the link...
real-analysis manifolds
real-analysis manifolds
asked Jan 12 at 11:23
user623855user623855
1507
asked Jan 12 at 11:23
user623855user623855
1507
asked Jan 12 at 11:23
user623855user623855
1507
asked Jan 12 at 11:23
user623855user623855
1507
asked Jan 12 at 11:23
user623855user623855
1507
1507
1
$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51
add a comment |
1
$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51
1
1
$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51
$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51
add a comment |
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$begingroup$
Your definition of first integral is totally confused. It's not a curve, it's a function $F(x)$ such that $F$ is constant along integral curves. When you write that $V$ is constant along $gamma$, that's not the same $V$ as you previously used for the vector field, it corresponds to what I called $F$.
$endgroup$
– Hans Lundmark
Jan 12 at 12:51