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Exercise about differential form
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(1) Determine $g:(0, +infty)to mathbbR$ such that $g(1)=1$ and the differential form $omega=2xg(x^2+y^2)dx + frac2y(x^2+y^2)^2dy$ is closed.
(2) Before determining the potentials, prove that for such a g the form $omega$ is exact.
(3) Determine the potential U such that $U(0,1)=2$.
(4) Using U determine the solution of the Cauchy problem: $y'(x)=-x/y$, $y(0)=1$.
My attempt:
(1) Let F be the field associated to $omega$ we must have $partial_xF_2= partial_yF_1$, that is $g'(t)=-2/t^3$, therefore $g(t)=1/t^2$.
(2) The form is defined on $mathbbR^2-0$, which is not a star domain or a simply connected set... I don't know other criteria to obtain the thesis without finding a potential. I can only say that $omega$ is locally exact. Can you give me a hint?
(3) $U(x,y)=U(0,1) + int_1^yF_2(0,t)dt + int_0^xF_1(t,y)dt = 3-frac1x^2+y^2$ which is a potential in $mathbbR^2-0$.
(4) Here I would also like a hint.
differential-geometry differential-forms
edited Jul 31 at 16:43
John Ma
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asked Jul 31 at 14:14
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up vote
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(1) Determine $g:(0, +infty)to mathbbR$ such that $g(1)=1$ and the differential form $omega=2xg(x^2+y^2)dx + frac2y(x^2+y^2)^2dy$ is closed.
(2) Before determining the potentials, prove that for such a g the form $omega$ is exact.
(3) Determine the potential U such that $U(0,1)=2$.
(4) Using U determine the solution of the Cauchy problem: $y'(x)=-x/y$, $y(0)=1$.
My attempt:
(1) Let F be the field associated to $omega$ we must have $partial_xF_2= partial_yF_1$, that is $g'(t)=-2/t^3$, therefore $g(t)=1/t^2$.
(2) The form is defined on $mathbbR^2-0$, which is not a star domain or a simply connected set... I don't know other criteria to obtain the thesis without finding a potential. I can only say that $omega$ is locally exact. Can you give me a hint?
(3) $U(x,y)=U(0,1) + int_1^yF_2(0,t)dt + int_0^xF_1(t,y)dt = 3-frac1x^2+y^2$ which is a potential in $mathbbR^2-0$.
(4) Here I would also like a hint.
differential-geometry differential-forms
edited Jul 31 at 16:43
John Ma
37.5k93669
asked Jul 31 at 14:14
user
1319
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
(1) Determine $g:(0, +infty)to mathbbR$ such that $g(1)=1$ and the differential form $omega=2xg(x^2+y^2)dx + frac2y(x^2+y^2)^2dy$ is closed.
(2) Before determining the potentials, prove that for such a g the form $omega$ is exact.
(3) Determine the potential U such that $U(0,1)=2$.
(4) Using U determine the solution of the Cauchy problem: $y'(x)=-x/y$, $y(0)=1$.
My attempt:
(1) Let F be the field associated to $omega$ we must have $partial_xF_2= partial_yF_1$, that is $g'(t)=-2/t^3$, therefore $g(t)=1/t^2$.
(2) The form is defined on $mathbbR^2-0$, which is not a star domain or a simply connected set... I don't know other criteria to obtain the thesis without finding a potential. I can only say that $omega$ is locally exact. Can you give me a hint?
(3) $U(x,y)=U(0,1) + int_1^yF_2(0,t)dt + int_0^xF_1(t,y)dt = 3-frac1x^2+y^2$ which is a potential in $mathbbR^2-0$.
(4) Here I would also like a hint.
differential-geometry differential-forms
edited Jul 31 at 16:43
John Ma
37.5k93669
asked Jul 31 at 14:14
user
1319
(1) Determine $g:(0, +infty)to mathbbR$ such that $g(1)=1$ and the differential form $omega=2xg(x^2+y^2)dx + frac2y(x^2+y^2)^2dy$ is closed.
(2) Before determining the potentials, prove that for such a g the form $omega$ is exact.
(3) Determine the potential U such that $U(0,1)=2$.
(4) Using U determine the solution of the Cauchy problem: $y'(x)=-x/y$, $y(0)=1$.
My attempt:
(1) Let F be the field associated to $omega$ we must have $partial_xF_2= partial_yF_1$, that is $g'(t)=-2/t^3$, therefore $g(t)=1/t^2$.
(2) The form is defined on $mathbbR^2-0$, which is not a star domain or a simply connected set... I don't know other criteria to obtain the thesis without finding a potential. I can only say that $omega$ is locally exact. Can you give me a hint?
(3) $U(x,y)=U(0,1) + int_1^yF_2(0,t)dt + int_0^xF_1(t,y)dt = 3-frac1x^2+y^2$ which is a potential in $mathbbR^2-0$.
(4) Here I would also like a hint.
differential-geometry differential-forms
edited Jul 31 at 16:43
John Ma
37.5k93669
asked Jul 31 at 14:14
user
1319
edited Jul 31 at 16:43
John Ma
37.5k93669
edited Jul 31 at 16:43
John Ma
37.5k93669
edited Jul 31 at 16:43
John Ma
37.5k93669
37.5k93669
asked Jul 31 at 14:14
user
1319
asked Jul 31 at 14:14
user
1319
asked Jul 31 at 14:14
user
1319
1319
add a comment |Â
add a comment |Â
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