Surface integral (Flux) with cylinder and plane intersections

CAF123

Quote by Dick

Assuming we are still on polar coordinates ## \vec{r}=(r cos \theta, r sin \theta, 0)## , ##|\vec{r}_r\times \vec{r}_\theta|=r##, ##dA= dr d\theta##. So ##|\vec{r}_r\times \vec{r}_\theta| dr d\theta## IS ##r dr d\theta##.

I see that this implies that dS = r dr dθ. But in the example from the website they also have dA = r dr dθ. How does this follow if dA = dr dθ? How can it equal two different things?

Dick

Quote by CAF123

I see that this implies that dS = r dr dθ. But in the example from the website they also have dA = r dr dθ. How does this follow if dA = dr dθ? How can it equal two different things?

They are two different uses of the symbol 'dA'. They are two different things.

CAF123

Quote by Dick

They are two different uses of the symbol 'dA'. They are two different things.

Ok, thanks. I think this also answers one of my other questions: The form dA=dr dθ is not really an area element because the dimensions don't check out. What is the difference between the two forms?

Dick

Quote by CAF123

Ok, thanks. I think this also answers one of my other questions: The form dA=dr dθ is not really an area element because the dimensions don't check out. What is the difference between the two forms?

We've been over that several times already. dr dθ is the area element without the jacobian part 'r'. You use that in the cross product form of the surface integral because the cross product contains the jacobian part, 'r'. If you are just integrating a function in polar coordinates you want the full area element, i.e. with jacobian r dr dθ.

CAF123

Quote by Dick

We've been over that several times already. dr dθ is the area element without the jacobian part 'r'. You use that in the cross product form of the surface integral because the cross product contains the jacobian part, 'r'. If you are just integrating a function in polar coordinates you want the full area element, i.e. with jacobian r dr dθ.

Ok, everything makes sense. Thanks so much!

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