Flux Calculation for Vertical Cylinder with Top Cut Off

[DryRun]

Homework Statement
http://s1.ipicture.ru/uploads/20120119/gW4x6Mxi.jpg

The attempt at a solution
I drew the graph. It's a vertical cylinder, with its axis along the z-axis, with the top cut off by the plane z=3 but the cylinder goes down to infinity. The radius of the cylinder is 1.

\phi(x,y,z)=x^2+y^2-1
∇\vec{\phi}=2x\vec{i}+2y\vec{j}
Since the direction of unit normal vector is downward, -∇\vec{\phi}=-2x\vec{i}-2y\vec{j}
\hat{n}=\frac{-∇\vec{\phi}}{|-∇\vec{\phi}|}
\hat{n}=\frac{-x-y}{\sqrt{x^2+y^2}}
Flux=\int\int\frac{-x^2-y^2}{\sqrt{x^2+y^2}}\,.d \sigma=-\sqrt{x+y}\,.d \sigma
Projecting the surface S onto the yz-plane (i cannot project onto the xy-plane as the cylinder starts from z=0 and goes down to infinity):
x=\sqrt{1-y^2}
x_y=-\frac{y}{\sqrt{1-y^2}}
x_z=0
d\sigma=\frac{1}{\sqrt{1-y^2}}\,.dzdy
S.A.=2\int\int \frac{1}{\sqrt{1-y^2}}\,.dzdy
The limits are -∞≤z≤3 and -1≤y≤1 but i think i messed up somewhere above, as these limits look suspicious.

When i visualize this thin slice of the surface at the plane z=3, it seems to me that if i project the thin slice onto the yz-plane, the slice would become invisible, as its thickness is zero. Am i right? But since we're dealing with flux here, then this slice does not have a thickness of zero. Correct?

 

[lanedance]

can you explain a bit?

isn't the surface a disk radius one centred on the z axis and in the z=3 plane, giving \hat{n}=(0,0,-1)^T

 

[DryRun]

can you explain a bit?

isn't the surface a disk radius one centred on the z axis and in the z=1 plane, giving \hat{n}=(0,0,-1)^T

No, it's in the plane z=3 and i don't understand how you got \hat{n} directly?
From my notes, i must always follow this method:
\hat{n}=\frac{∇\vec{\phi}}{|∇\vec{\phi}|}

 

[lanedance]

yeah was editing as you replied

 

[lanedance]

i'm not sure what you mean about a cylinder, isn't it just the disk as described?

 

[lanedance]

then only the z component of the field will contribute to the integral

 

[DryRun]

OK, i think i understand. For flat surfaces, the outward unit normal is +1 is the direction is along the positive side of the axis, otherwise it's negative.
So, for example, for plane x=2, \hat{n}=(1,0,0)^T
And, for plane y=-3, \hat{n}=(0,-1,0)^T
Are these correct?

So, for flat surfaces, there is no need to use this formula:
\hat{n}=\frac{∇\vec{\phi}}{|∇\vec{\phi}|}
Correct?

But if i do use the formula, then i don't get -1. Instead i get this:
\hat{n}=\frac{-x-y}{\sqrt{x^2+y^2}}
Using x^2+y^2=1, i can simply:
\hat{n}=\frac{-x-y}{\sqrt{x^2+y^2}}=-x-y
But i don't know how it becomes -1.

 

[lanedance]

well the normal can be in the positive or negative direction depending on the surface orientation you choose, and that gets important when you start looking at stokes theorem, but yeah they are valid normals

the problem is the function phi you defined, when phi=0, it gives the unit cylinder, NOT the plane z=3.

and though the method is a good one, it's not really required as the surface normal is constant and easily to calculate

if you really want to use it the method, you should start with
phi=z-3

 

[lanedance]

just to clarify, you might want to re-read the question, the surface is z=3, the boundary is given by x^2+y^2=1 (the unit circle)

 

[DryRun]

Yes, i did a mistake with \phi.
\phi(x,y,z)=z-3∇\vec{\phi}=\vec{k}-∇\vec{\phi}=-\vec{k}\hat{n}=-\vec{k}Flux=\int\int \vec{F}.\hat{n}\,.d \sigma=\int\int-z\,.d \sigma
Now, I'm not sure, but i think the only plane where i can project the surface z=3 is the xy-plane. Am i right?
So, projecting on the xy-plane:
z_x=0z_y=0\sigma=\sqrt{0+0+1}\,.dxdyS.A.=\int\int(-z)\,.dxdyS.A.=\int\int(-3)\,.dxdy
Transforming to polar coordinates:S.A.=\int\int(-3)\,.dxdy=\int^{2\pi}_0\int^1_0(-3)\,.rdrd\theta=-3\pi
The answer is wrong.

 

[lanedance]

why is it wrong?

 

[DryRun]

Thanks for answering, lanedance. I've been waiting for confirmation the whole day.

Well, the answer in my notes is: \frac{3\pi}{2}
Furthermore, the answer in my notes is positive, whereas the answer i got is negative.

 

[lanedance]

i agree with you rworking

see the following
http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx

the 2nd part (S2) of the first example has a similar problem

 

[DryRun]

I checked out the example in your link and there is this last part that i can't understand from the website:

At this point we can acknowledge that D is a disk of radius 1 and this double integral is nothing more than the double integral that will give the area of the region D so there is no reason to compute the integral. Here is the value of the surface integral.

My question is: How can he directly know that the answer is \pi without evaluating the integral?
At this point, I'm having a hard time to fully understand, as I'm still learning the fundamentals in this chapter, and my notes are apparently wrong.

 

[lanedance]

So the full surface vector integral is
<br /> \int\int \vec{F}\cdot\vec{dS}<br />

the surface differential is equivalent to an infintesimal area on the surface in the direction of the normal
<br /> \vec{dS}<br /> =\hat{n}dA<br />

Thus the integral becomes
<br /> \int\int \vec{F}\cdot\hat{n}dA<br />

In our case the dot product is equal to -z, and z=3 on the surface
<br /> \vec{F}\cdot\hat{n}=-z=-3<br />

So the integral becomes
<br /> \int\int \vec{F}\cdot\hat{n}dA<br /> =-3\int\int dA<br />

Now what is the area of our surface, a circle of radius 1? you could integrate, as you did with the change to polar coords, or just use the area \pi r^2=p\i
<br /> \int\int \vec{F}\cdot\hat{n}dA<br /> =\int\int dA<br />

 

[DryRun]

I see it now.
\int\int \vec{F}\cdot\hat{n}dA=-3 \int\int dA
But since \int\int dA={\pi}{r}^2 where r=1
\int\int \vec{F}\cdot\hat{n}dA=-3 \int\int dA=-3\pi
Thanks for the help, lanedance.