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What vector field property means “is the curl of another vector field?â€Â
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I know that a vector field $mathbfF$ is called irrotational if $nabla times mathbfF = mathbf0$ and conservative if there exists a function $g$ such that $nabla g = mathbfF$. Under suitable smoothness conditions on the component functions (so that Clairaut's theorem holds), conservative vector fields are irrotational, and under suitable topological conditions on the domain of $mathbfF$, irrotational vector fields are conservative.
Moving up one degree, $mathbfF$ is called incompressible if $nabla cdot mathbfF = 0$. If there exists a vector field $mathbfG$ such that $mathbfF = nabla times mathbfG$, then (again, under suitable smoothness conditions), $mathbfF$ is incompressible. And again, under suitable topological conditions (the second cohomology group of the domain must be trivial), if $mathbfF$ is incompressible, there exists a vector field $mathbfG$ such that $nabla timesmathbfG = mathbfF$.
It seems to me there ought to be a word to describe vector fields as shorthand for “is the curl of something†or “has a vector potential.†But a google search didn't turn anything up, and my colleagues couldn't think of a word either. Maybe I'm revealing the gap in my physics background. Does anybody know of such a word?
multivariable-calculus definition
edited Aug 6 at 22:46
amWhy
189k25219431
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
add a comment |Â
up vote
4
down vote
favorite
I know that a vector field $mathbfF$ is called irrotational if $nabla times mathbfF = mathbf0$ and conservative if there exists a function $g$ such that $nabla g = mathbfF$. Under suitable smoothness conditions on the component functions (so that Clairaut's theorem holds), conservative vector fields are irrotational, and under suitable topological conditions on the domain of $mathbfF$, irrotational vector fields are conservative.
Moving up one degree, $mathbfF$ is called incompressible if $nabla cdot mathbfF = 0$. If there exists a vector field $mathbfG$ such that $mathbfF = nabla times mathbfG$, then (again, under suitable smoothness conditions), $mathbfF$ is incompressible. And again, under suitable topological conditions (the second cohomology group of the domain must be trivial), if $mathbfF$ is incompressible, there exists a vector field $mathbfG$ such that $nabla timesmathbfG = mathbfF$.
It seems to me there ought to be a word to describe vector fields as shorthand for “is the curl of something†or “has a vector potential.†But a google search didn't turn anything up, and my colleagues couldn't think of a word either. Maybe I'm revealing the gap in my physics background. Does anybody know of such a word?
multivariable-calculus definition
edited Aug 6 at 22:46
amWhy
189k25219431
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
1
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I know that a vector field $mathbfF$ is called irrotational if $nabla times mathbfF = mathbf0$ and conservative if there exists a function $g$ such that $nabla g = mathbfF$. Under suitable smoothness conditions on the component functions (so that Clairaut's theorem holds), conservative vector fields are irrotational, and under suitable topological conditions on the domain of $mathbfF$, irrotational vector fields are conservative.
Moving up one degree, $mathbfF$ is called incompressible if $nabla cdot mathbfF = 0$. If there exists a vector field $mathbfG$ such that $mathbfF = nabla times mathbfG$, then (again, under suitable smoothness conditions), $mathbfF$ is incompressible. And again, under suitable topological conditions (the second cohomology group of the domain must be trivial), if $mathbfF$ is incompressible, there exists a vector field $mathbfG$ such that $nabla timesmathbfG = mathbfF$.
It seems to me there ought to be a word to describe vector fields as shorthand for “is the curl of something†or “has a vector potential.†But a google search didn't turn anything up, and my colleagues couldn't think of a word either. Maybe I'm revealing the gap in my physics background. Does anybody know of such a word?
multivariable-calculus definition
edited Aug 6 at 22:46
amWhy
189k25219431
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
I know that a vector field $mathbfF$ is called irrotational if $nabla times mathbfF = mathbf0$ and conservative if there exists a function $g$ such that $nabla g = mathbfF$. Under suitable smoothness conditions on the component functions (so that Clairaut's theorem holds), conservative vector fields are irrotational, and under suitable topological conditions on the domain of $mathbfF$, irrotational vector fields are conservative.
Moving up one degree, $mathbfF$ is called incompressible if $nabla cdot mathbfF = 0$. If there exists a vector field $mathbfG$ such that $mathbfF = nabla times mathbfG$, then (again, under suitable smoothness conditions), $mathbfF$ is incompressible. And again, under suitable topological conditions (the second cohomology group of the domain must be trivial), if $mathbfF$ is incompressible, there exists a vector field $mathbfG$ such that $nabla timesmathbfG = mathbfF$.
It seems to me there ought to be a word to describe vector fields as shorthand for “is the curl of something†or “has a vector potential.†But a google search didn't turn anything up, and my colleagues couldn't think of a word either. Maybe I'm revealing the gap in my physics background. Does anybody know of such a word?
multivariable-calculus definition
edited Aug 6 at 22:46
amWhy
189k25219431
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
edited Aug 6 at 22:46
amWhy
189k25219431
edited Aug 6 at 22:46
amWhy
189k25219431
edited Aug 6 at 22:46
amWhy
189k25219431
189k25219431
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
asked May 12 '15 at 13:29
Matthew Leingang
15k12143
15k12143
“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
1
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50
add a comment |Â
“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
1
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50
“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
1
1
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50
add a comment |Â
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“is a coboundary†?
– Circonflexe
May 12 '15 at 13:39
@Circonflexe: that would be correct; but my audience is multivariable calculus students and I was hoping there was a physics-inspired term outside of de Rham cohomology.
– Matthew Leingang
May 12 '15 at 13:41
1
As a fallback, you can always use “closed†vs. “exactâ€Â, hoping that 1. this will not cause confusion with the case of differential forms, and 2. any such confusion will not be too serious anyway. Btw, thanks for teaching me new terminology...
– Circonflexe
May 12 '15 at 13:42
Anyway, I lack the intuition about vector fields to help you here, but what is the idea behind the movement of a particle subject to a “coboundary†force? (The idea behind this is that, in degree 1, the mechanical energy will be conserved, hence “conservative†- here, maybe some angular momentum is conserved?)
– Circonflexe
May 12 '15 at 13:47
@Circonflexe: that's another question I have. I believe that the Fundamental Theorem of Line Integrals encapsulates conservation of energy, and the Divergence Theorem does the same for conservation of mass. That makes me think that Stokes's Theorem is conservation of momentum.
– Matthew Leingang
May 12 '15 at 13:50