vector field: changing vector magnitudes to make it conservative
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Consider a vector field $$vec{F}(x,y)=P(x,y)vec{i}+Q(x,y)vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives.
Under which conditions there exists a positive-valued function $mu (x,y)$ such that $$muvec{F}(x,y)=mu(x,y)P(x,y)vec{i}+mu(x,y)Q(x,y)vec{j}$$ is a conservative vector field?
Thanks for the help!
multivariable-calculus vector-analysis
asked Nov 15 at 0:16
FreddyM
162
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up vote
3
down vote
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Consider a vector field $$vec{F}(x,y)=P(x,y)vec{i}+Q(x,y)vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives.
Under which conditions there exists a positive-valued function $mu (x,y)$ such that $$muvec{F}(x,y)=mu(x,y)P(x,y)vec{i}+mu(x,y)Q(x,y)vec{j}$$ is a conservative vector field?
Thanks for the help!
multivariable-calculus vector-analysis
asked Nov 15 at 0:16
FreddyM
162
1
Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
2
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Consider a vector field $$vec{F}(x,y)=P(x,y)vec{i}+Q(x,y)vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives.
Under which conditions there exists a positive-valued function $mu (x,y)$ such that $$muvec{F}(x,y)=mu(x,y)P(x,y)vec{i}+mu(x,y)Q(x,y)vec{j}$$ is a conservative vector field?
Thanks for the help!
multivariable-calculus vector-analysis
asked Nov 15 at 0:16
FreddyM
162
Consider a vector field $$vec{F}(x,y)=P(x,y)vec{i}+Q(x,y)vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives.
Under which conditions there exists a positive-valued function $mu (x,y)$ such that $$muvec{F}(x,y)=mu(x,y)P(x,y)vec{i}+mu(x,y)Q(x,y)vec{j}$$ is a conservative vector field?
Thanks for the help!
multivariable-calculus vector-analysis
multivariable-calculus vector-analysis
asked Nov 15 at 0:16
FreddyM
162
asked Nov 15 at 0:16
FreddyM
162
edited Nov 15 at 0:35
asked Nov 15 at 0:16
FreddyM
162
asked Nov 15 at 0:16
FreddyM
162
asked Nov 15 at 0:16
FreddyM
162
162
1
Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
2
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40
add a comment |
1
Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
2
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40
1
1
Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
2
2
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40
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Well, you could start by applying any conditions that you might know of for a vector field to be conservative to $muvec F$.
– amd
Nov 15 at 0:21
What about $mu(x,y)=0$? (Yes, I know it is a "silly" objection, most likely that's not what you meant. But then you have to be clear about exactly what requirements you have on $mu$.)
– Arthur
Nov 15 at 0:22
First, thanks for the editing! Second, I would like $μ(x,y)>0$... The idea is that it should preserve the directions in the vector field. I apologize for my poor math knowledge.
– FreddyM
Nov 15 at 0:36
2
To amd. For the field $mu F$ to be conservative, I would need that $$ mu _y P + mu P_y = mu _x Q + mu Q_x $$ which remains difficult for me to investigate! Is the answer trivial?
– FreddyM
Nov 15 at 0:40