Surface of the manifold defined by $xin(0,frac12)$, $yin(0,1)$, $z=x^2$
$begingroup$
I have to calculate the surface of a given manifold
M:= {(x,y,z) $in mathbb(R^3) | xin(0,frac1 2), yin(0,1), z=x^2$} $subset mathbb(R^3)$.
I already found a chart/an atlas (it is a one chart atlas) with
$phi: mathbb(R^3) to mathbb(R^2), phi: (x,y,z) to (x,y) $
respectively
$phi^{-1}: mathbb(R^2) to mathbb(R^3), phi^{-1}: (x,y) to (x,y,x^2)$
as well as the normal vectors in every (x,y,z)$in$M with
$nu(x,y,z)=(1,0,-frac 1 2 x)$.
With all these informations i have to calculate:
$int_U (phi^{-1})^{*}(det(nu,cdot,cdot))$
but sadly i have no idea how to do so.
Any kind of help would be very much appreciated.
integration analysis manifolds
edited Jan 20 at 15:07
Did
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
$endgroup$
add a comment |
$begingroup$
I have to calculate the surface of a given manifold
M:= {(x,y,z) $in mathbb(R^3) | xin(0,frac1 2), yin(0,1), z=x^2$} $subset mathbb(R^3)$.
I already found a chart/an atlas (it is a one chart atlas) with
$phi: mathbb(R^3) to mathbb(R^2), phi: (x,y,z) to (x,y) $
respectively
$phi^{-1}: mathbb(R^2) to mathbb(R^3), phi^{-1}: (x,y) to (x,y,x^2)$
as well as the normal vectors in every (x,y,z)$in$M with
$nu(x,y,z)=(1,0,-frac 1 2 x)$.
With all these informations i have to calculate:
$int_U (phi^{-1})^{*}(det(nu,cdot,cdot))$
but sadly i have no idea how to do so.
Any kind of help would be very much appreciated.
integration analysis manifolds
edited Jan 20 at 15:07
Did
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
$endgroup$
add a comment |
$begingroup$
I have to calculate the surface of a given manifold
M:= {(x,y,z) $in mathbb(R^3) | xin(0,frac1 2), yin(0,1), z=x^2$} $subset mathbb(R^3)$.
I already found a chart/an atlas (it is a one chart atlas) with
$phi: mathbb(R^3) to mathbb(R^2), phi: (x,y,z) to (x,y) $
respectively
$phi^{-1}: mathbb(R^2) to mathbb(R^3), phi^{-1}: (x,y) to (x,y,x^2)$
as well as the normal vectors in every (x,y,z)$in$M with
$nu(x,y,z)=(1,0,-frac 1 2 x)$.
With all these informations i have to calculate:
$int_U (phi^{-1})^{*}(det(nu,cdot,cdot))$
but sadly i have no idea how to do so.
Any kind of help would be very much appreciated.
integration analysis manifolds
edited Jan 20 at 15:07
Did
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
$endgroup$
I have to calculate the surface of a given manifold
M:= {(x,y,z) $in mathbb(R^3) | xin(0,frac1 2), yin(0,1), z=x^2$} $subset mathbb(R^3)$.
I already found a chart/an atlas (it is a one chart atlas) with
$phi: mathbb(R^3) to mathbb(R^2), phi: (x,y,z) to (x,y) $
respectively
$phi^{-1}: mathbb(R^2) to mathbb(R^3), phi^{-1}: (x,y) to (x,y,x^2)$
as well as the normal vectors in every (x,y,z)$in$M with
$nu(x,y,z)=(1,0,-frac 1 2 x)$.
With all these informations i have to calculate:
$int_U (phi^{-1})^{*}(det(nu,cdot,cdot))$
but sadly i have no idea how to do so.
Any kind of help would be very much appreciated.
integration analysis manifolds
integration analysis manifolds
edited Jan 20 at 15:07
Did
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
edited Jan 20 at 15:07
Did
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
edited Jan 20 at 15:07
Did
248k23224463
edited Jan 20 at 15:07
Did
248k23224463
edited Jan 20 at 15:07
Did
248k23224463
248k23224463
asked Jan 18 at 12:12
ZweisteinZweistein
64
asked Jan 18 at 12:12
ZweisteinZweistein
64
asked Jan 18 at 12:12
ZweisteinZweistein
64
64
add a comment |
add a comment |
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