- #1
mahler1
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Homework Statement .
Let ##C## be the curve in the plane ##xz## given in polar coordinates by:
##r(\phi)=\frac{4√3}{9}(2-cos(2\phi)), \frac{π}{6}≤\phi≤\frac{5π}{6}## (##\phi## being the angle between the radius vector and the positive z-semiaxis). Let ##S## the surface obtained by the revolution of this curve around the ##z## axis. Calculate the flux through the surface in the "external" orientation of the field ##F(x,y,z)=(x,y,-2z)##.
The attempt at a solution.
First, I am not so sure if I parametrized the surface in the correct way, I hope so:
##T:[\frac{π}{6},\frac{5π}{6}]\times[0,2π] \to \mathbb R^3##, with
##T(\phi,θ)=(r(\phi)cos(θ),r(\phi)sin(θ),\phi)##
I suppose I must relate the flux of the vector field through this surface with the volume integral of the divergence of the region bounded by the surface. The problem is that one of the hypothesis of the divergence theorem is not satisfied: if you see the image I've attached, this surface is clearly not closed.
My idea was to apply the divergence theorem on the surface ##S'=S \cup D_1 \cup D_2## where ##S## is the original surface and ##D_1## and ##D_2## are the top and bottom disks, so the flux through ##S## would be:
##\iint_S (F.n)dS=\iint_{S'} (F.n)dS-2\iint_D (F.n)dS=\iiint_{W'} (divF)dV-2\iint_D (F.n)dS##, where ##D## is one of the two disks, it doesn't matter which since the integral of any of the two gives the same result.
If all of the things I've said are correct, then my doubts are:
how do I know for instance the radius of the disks? I need to get that information in order to parametrize them as surfaces and calculate the integrals over the disks, but I don't know how to get that information.
The other doubt that I have is: in the problem it says that the surface has the "external" orientation (the normal vector pointing outwards), I've parametrized the surface with ##T##, I know how to check if ##T## preserves or not the orientation but when I want to calculate ##\iiint_{W'} (divF)dV##, how do I describe the region bounded by ##S'## and how do I know it preserves the orientation?
Let ##C## be the curve in the plane ##xz## given in polar coordinates by:
##r(\phi)=\frac{4√3}{9}(2-cos(2\phi)), \frac{π}{6}≤\phi≤\frac{5π}{6}## (##\phi## being the angle between the radius vector and the positive z-semiaxis). Let ##S## the surface obtained by the revolution of this curve around the ##z## axis. Calculate the flux through the surface in the "external" orientation of the field ##F(x,y,z)=(x,y,-2z)##.
The attempt at a solution.
First, I am not so sure if I parametrized the surface in the correct way, I hope so:
##T:[\frac{π}{6},\frac{5π}{6}]\times[0,2π] \to \mathbb R^3##, with
##T(\phi,θ)=(r(\phi)cos(θ),r(\phi)sin(θ),\phi)##
I suppose I must relate the flux of the vector field through this surface with the volume integral of the divergence of the region bounded by the surface. The problem is that one of the hypothesis of the divergence theorem is not satisfied: if you see the image I've attached, this surface is clearly not closed.
My idea was to apply the divergence theorem on the surface ##S'=S \cup D_1 \cup D_2## where ##S## is the original surface and ##D_1## and ##D_2## are the top and bottom disks, so the flux through ##S## would be:
##\iint_S (F.n)dS=\iint_{S'} (F.n)dS-2\iint_D (F.n)dS=\iiint_{W'} (divF)dV-2\iint_D (F.n)dS##, where ##D## is one of the two disks, it doesn't matter which since the integral of any of the two gives the same result.
If all of the things I've said are correct, then my doubts are:
how do I know for instance the radius of the disks? I need to get that information in order to parametrize them as surfaces and calculate the integrals over the disks, but I don't know how to get that information.
The other doubt that I have is: in the problem it says that the surface has the "external" orientation (the normal vector pointing outwards), I've parametrized the surface with ##T##, I know how to check if ##T## preserves or not the orientation but when I want to calculate ##\iiint_{W'} (divF)dV##, how do I describe the region bounded by ##S'## and how do I know it preserves the orientation?
