System of ODE's - Why does this vector function connect two constant vectors continuously?
Given the system of two ODE's
$frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$,
with
$lambda ({mathbf{w}}(xi )) = xi $
and
${mathbf{w}}(lambda ({{mathbf{u}}_L})) = {{mathbf{u}}_L}$,
${mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_R}$
where $xi=x/t$,
$x,;t in mathbb{R}$
${{mathbf{u}}_L},{{mathbf{u}}_R} in {mathbb{R}^2}$,
${mathbf{u}}:{mathbb{R}^2} to mathbb{R}^2$,
${mathbf{u}}(x,t) = {mathbf{w}}(x/t) = {mathbf{w}}(xi )$
why does that imply that the function ${mathbf{w(xi)}}$ continuously connects the constants ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ (both given) for a fixed t?
Looking at $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$ which gives the integral curves of the vector field ${mathbf{r}}$, I integrate
$$int_{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R}))}^{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L}))} {d{mathbf{w}}(xi ) = int_{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R})}^{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))} dxi } $$
which gives
$${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R} = int_{xi = lambda ({{mathbf{u}}_R})}^{xi = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))dxi }. $$
Since the integral on the right hand side is a continuous function of ${mathbf{w}}$ (${mathbf{r}}$ is continuous), we can write ${{mathbf{u}}_L} - {{mathbf{u}}_R} = {mathbf{f}}({mathbf{w}}(xi ))$, so that ${mathbf{w}}(xi )$ connects ${{mathbf{u}}_L}$ and ${{mathbf{u}}_R}$ continuously.
Is this explanation correct?
(This is from chapter 5.2, p.235 in the book 'Front Tracking for Hyperbolic Conservation Laws' by Risebro, Holden. Link to chapter 5: https://www.uio.no/studier/emner/matnat/math/MAT4380/v15/beskjeder/chapter5.pdf)
differential-equations systems-of-equations vector-analysis vector-fields
asked Nov 25 at 2:53
user619360
12
add a comment |
Given the system of two ODE's
$frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$,
with
$lambda ({mathbf{w}}(xi )) = xi $
and
${mathbf{w}}(lambda ({{mathbf{u}}_L})) = {{mathbf{u}}_L}$,
${mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_R}$
where $xi=x/t$,
$x,;t in mathbb{R}$
${{mathbf{u}}_L},{{mathbf{u}}_R} in {mathbb{R}^2}$,
${mathbf{u}}:{mathbb{R}^2} to mathbb{R}^2$,
${mathbf{u}}(x,t) = {mathbf{w}}(x/t) = {mathbf{w}}(xi )$
why does that imply that the function ${mathbf{w(xi)}}$ continuously connects the constants ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ (both given) for a fixed t?
Looking at $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$ which gives the integral curves of the vector field ${mathbf{r}}$, I integrate
$$int_{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R}))}^{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L}))} {d{mathbf{w}}(xi ) = int_{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R})}^{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))} dxi } $$
which gives
$${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R} = int_{xi = lambda ({{mathbf{u}}_R})}^{xi = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))dxi }. $$
Since the integral on the right hand side is a continuous function of ${mathbf{w}}$ (${mathbf{r}}$ is continuous), we can write ${{mathbf{u}}_L} - {{mathbf{u}}_R} = {mathbf{f}}({mathbf{w}}(xi ))$, so that ${mathbf{w}}(xi )$ connects ${{mathbf{u}}_L}$ and ${{mathbf{u}}_R}$ continuously.
Is this explanation correct?
(This is from chapter 5.2, p.235 in the book 'Front Tracking for Hyperbolic Conservation Laws' by Risebro, Holden. Link to chapter 5: https://www.uio.no/studier/emner/matnat/math/MAT4380/v15/beskjeder/chapter5.pdf)
differential-equations systems-of-equations vector-analysis vector-fields
asked Nov 25 at 2:53
user619360
12
I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30
add a comment |
Given the system of two ODE's
$frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$,
with
$lambda ({mathbf{w}}(xi )) = xi $
and
${mathbf{w}}(lambda ({{mathbf{u}}_L})) = {{mathbf{u}}_L}$,
${mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_R}$
where $xi=x/t$,
$x,;t in mathbb{R}$
${{mathbf{u}}_L},{{mathbf{u}}_R} in {mathbb{R}^2}$,
${mathbf{u}}:{mathbb{R}^2} to mathbb{R}^2$,
${mathbf{u}}(x,t) = {mathbf{w}}(x/t) = {mathbf{w}}(xi )$
why does that imply that the function ${mathbf{w(xi)}}$ continuously connects the constants ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ (both given) for a fixed t?
Looking at $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$ which gives the integral curves of the vector field ${mathbf{r}}$, I integrate
$$int_{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R}))}^{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L}))} {d{mathbf{w}}(xi ) = int_{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R})}^{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))} dxi } $$
which gives
$${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R} = int_{xi = lambda ({{mathbf{u}}_R})}^{xi = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))dxi }. $$
Since the integral on the right hand side is a continuous function of ${mathbf{w}}$ (${mathbf{r}}$ is continuous), we can write ${{mathbf{u}}_L} - {{mathbf{u}}_R} = {mathbf{f}}({mathbf{w}}(xi ))$, so that ${mathbf{w}}(xi )$ connects ${{mathbf{u}}_L}$ and ${{mathbf{u}}_R}$ continuously.
Is this explanation correct?
(This is from chapter 5.2, p.235 in the book 'Front Tracking for Hyperbolic Conservation Laws' by Risebro, Holden. Link to chapter 5: https://www.uio.no/studier/emner/matnat/math/MAT4380/v15/beskjeder/chapter5.pdf)
differential-equations systems-of-equations vector-analysis vector-fields
asked Nov 25 at 2:53
user619360
12
Given the system of two ODE's
$frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$,
with
$lambda ({mathbf{w}}(xi )) = xi $
and
${mathbf{w}}(lambda ({{mathbf{u}}_L})) = {{mathbf{u}}_L}$,
${mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_R}$
where $xi=x/t$,
$x,;t in mathbb{R}$
${{mathbf{u}}_L},{{mathbf{u}}_R} in {mathbb{R}^2}$,
${mathbf{u}}:{mathbb{R}^2} to mathbb{R}^2$,
${mathbf{u}}(x,t) = {mathbf{w}}(x/t) = {mathbf{w}}(xi )$
why does that imply that the function ${mathbf{w(xi)}}$ continuously connects the constants ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ (both given) for a fixed t?
Looking at $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$ which gives the integral curves of the vector field ${mathbf{r}}$, I integrate
$$int_{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R}))}^{{mathbf{w}}(xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L}))} {d{mathbf{w}}(xi ) = int_{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_R}) = lambda ({{mathbf{u}}_R})}^{xi = lambda ({mathbf{w}}(xi ) = {{mathbf{u}}_L}) = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))} dxi } $$
which gives
$${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R} = int_{xi = lambda ({{mathbf{u}}_R})}^{xi = lambda ({{mathbf{u}}_L})} {{mathbf{r}}({mathbf{w}}(xi ))dxi }. $$
Since the integral on the right hand side is a continuous function of ${mathbf{w}}$ (${mathbf{r}}$ is continuous), we can write ${{mathbf{u}}_L} - {{mathbf{u}}_R} = {mathbf{f}}({mathbf{w}}(xi ))$, so that ${mathbf{w}}(xi )$ connects ${{mathbf{u}}_L}$ and ${{mathbf{u}}_R}$ continuously.
Is this explanation correct?
(This is from chapter 5.2, p.235 in the book 'Front Tracking for Hyperbolic Conservation Laws' by Risebro, Holden. Link to chapter 5: https://www.uio.no/studier/emner/matnat/math/MAT4380/v15/beskjeder/chapter5.pdf)
differential-equations systems-of-equations vector-analysis vector-fields
differential-equations systems-of-equations vector-analysis vector-fields
asked Nov 25 at 2:53
user619360
12
asked Nov 25 at 2:53
user619360
12
edited Nov 25 at 3:01
asked Nov 25 at 2:53
user619360
12
asked Nov 25 at 2:53
user619360
12
asked Nov 25 at 2:53
user619360
12
12
I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30
add a comment |
I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30
I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30
add a comment |
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I'm having trouble working out the meaning of "connects $mathbf u_L$ and $mathbf u_R$ continuously." You start by mentioning "a system of two ODEs" by which I assume is meant $frac{{d{mathbf{w}}}}{{dxi }} = {mathbf{r}}({mathbf{w}}(xi ))$. But thereafter we are only told $mathbf r$ is a (continuous) "vector field", so it's hard to know what reasoning we can apply. I then make out that $mathbf u_L$ and $mathbf u_R$ serve (as constant vectors) to supply initial(?) conditions of a kind. Toward the end of the Question there is a line that subtracts those two conditions, to get...
– hardmath
Nov 25 at 3:11
... ${mathbf{w}}(lambda ({{mathbf{u}}_L})) - {mathbf{w}}(lambda ({{mathbf{u}}_R})) = {{mathbf{u}}_L} - {{mathbf{u}}_R}$. That much is certainly not surprising. But what does "connects... continuously" mean here?
– hardmath
Nov 25 at 3:11
I think it means that we can draw a continuous curve between the points ${{mathbf{u}}_R}$ and ${{mathbf{u}}_L}$ in ${mathbf{u}} = ({u_1},{u_2})$ - space, but I'm not sure. See figure 5.2 p.239 in the link i posted above where the curves ${R_1}$ and ${R_2}$ connect the point ${mathbf{L}} = {{mathbf{u}}_L}$ to any point ${{mathbf{u}}_R}$ on ${R_1}$ or ${R_2}$ continuously.
– user619360
Nov 25 at 3:25
Perhaps what you are asking about is related to the system of ODEs being autonomous?
– hardmath
Nov 25 at 3:30