# Thread: Vector field and flow

1. ## Vector field and flow

Vector field $\displaystyle F(x,y,z) = xi + 0j + zk$ and the cylinder obtained by rotation the straight $\displaystyle x=1, y=t, z=0$ with $\displaystyle -1 \leq t \leq 1$ around the OY axis. Determine the flow of the field F in the direction normal outside of the cylinder. (the normal vector points to out)

2. If you know the divergence theorem it is simply 2*V = 4pi

Even though this is not a closed surface the flux through the right and left faces is 0 (j component of F is 0) you can close up the cylinder and use the divergence theorem

If Not

The equation of the cylinder is then x^2 +z^2 = 1 -1< y <1

You'll have to break this up into the upper half and lower half over the rectangle -1 < x < 1 and -1 < y <1

For the upper half N = x/z i + k For the lower half N = -x/z i - k

You'll get 2pi in both cases

3. Originally Posted by Calculus26
If you know the divergence theorem it is simply 2*V = 4pi

Even though this is not a closed surface the flux through the right and left faces is 0 (j component of F is 0) you can close up the cylinder and use the divergence theorem

If Not

The equation of the cylinder is then x^2 +z^2 = 1 -1< y <1

You'll have to break this up into the upper half and lower half over the rectangle -1 < x < 1 and -1 < y <1

For the upper half N = x/z i + k For the lower half N = x/z i - k

You'll get 2pi in both cases
divergence theorem is:
$\displaystyle \frac{d(x)}{dx} + \frac{d(0)}{dy} + \frac{d(z)}{dz} = 2$
$\displaystyle 2 \int_0^{2 \pi} ds = 2 \pi$ it is ?

Sory
I do not understand
" For the upper half N = x/z i + k For the lower half N = x/z i - k "

I do not know well integral surface.
What are the steps to calculate?

4. Divergence Theorem : Flux is triple integral of divergence

Here div = 2 volume is pi (1)^2 *2 = 4pi

Not sure where the integral comes from but even with this integral is 2*2pi is still 4pi

If you're not going to use the divergence theorem

then you're just going to have to look up calculating N for a surface

where z = f(x,y)---do the work

5. If you're using the formula for a flux integral over a parameterized surface

I see where comes from but this does not involve the divergence---

6. Originally Posted by Calculus26
If you're using the formula for a flux integral over a parameterized surface

I see where comes from but this does not involve the divergence---
How can I calculate the normal ?
Is there a trick to parameterization ?

7. I'll tell you what --we've seen 3 different ways of showing the flux is 4pi.

Submit your work and expanation of your work and then I'll comment on what you are doing right or not right