Surface Integrals Explained for Beginners

[brendan_foo]

At the risk of sounding imbecilic, I'm going to pose this question anyway.

If I integral a vector function over a surface {a defined region R on a surface S} then what in fact am I doing? I know it sounds bizarre but I can see the logic of the process to find surface areas..but what does this actually represent.. I know its the integral of the vector function and the unit normal vector dotted together, but what is this actually doing? Is this saying how much area this function will trace out in this defined region or what?

I am reading Div, Grad, Curl by H M Schey and I get the idea in the main, but what stumps me is when the author says:

"We evaluate the function F(x,y,z) and this point and form its dot product with $\mathbf{\hat{n}}$. The resulting quantity is then multiplied by the area $\Delta S$"

In this case he's talking about dividing up the surface into N faces, then taking the limit of the sum to form the integral etc..etc..

But I don't understand the essence...I can do the algebra and the calculus; that's not an issue..but the underlying essence of it I cannot grasp. If I integrate : $\iint_s \mathbf{F}(x,y,z)\cdot\mathbf{\hat{n}} dS$ then just what the heck is going on, what does the resulting quantity represent?

Sorry if I sound like a fool, but there's probably something obvious I've yet to have spotted.

Thanks guys!

 

[dextercioby]

The quantity
$$\Phi =:\int\int_{S} \vec{F}(\vec{r})\cdot \vec{n} dS$$
is defined as THE FLUX OF THE VECTOR FIELD "F" THROUGH AN ORIENTABLE OPEN SURFACE "S" WHICH HAS IN EACH POINT A UNIT VECTOR "n".
For a closed surface:call it Sigma
$$\Phi=:\oint_{\Sigma} \vec{F}(\vec{r})\cdot \vec{n} dS$$

So that's what u're doing.Computing a flux of a vector field.

Daniel.

PS.Think about the magnetic flux:u can visualize magnetic field lines and tangent vectors B.Then the magnetic flux can be thought of being the number of magnetic field lines crossing through an orientable closed/open surface.

 

[brendan_foo]

Yeah I read on a bit and it was talking about the "flux", but surely they should've let you know that at an earlier stage, or do they just expect you to keep plodding along and taking it at face value.

Well now I know how to evalute the integral (projection methods) I suppose I can begin to tackle the divergence stuff.

Anymore comments to add please feel free to do so..

Cheers guys
HAPPY NEW YEAR :)

 

[QuantumTheory]

Yeah I read on a bit and it was talking about the "flux", but surely they should've let you know that at an earlier stage, or do they just expect you to keep plodding along and taking it at face value.

Well now I know how to evalute the integral (projection methods) I suppose I can begin to tackle the divergence stuff.

Anymore comments to add please feel free to do so..

Cheers guys
HAPPY NEW YEAR :)

Well I apologize as this comment is not very helpful to you as I am not that far yet in my journey with calculus :)

Happy new years to you as well! I was wondering about what 'level' is this caluclus? Calculus III? Calculus IV? V?

Colleage Calculus?

Or the advanced high school stuff? <--- (Lord help me if this is true, as I am going to take calculus in school, and hope to god they don't put me in vectors quite yet ;) )

 

[FulhamFan3]

It's Calculus III and Vector Analysis. In my high school the most advanced it ever got was integration by parts.

 

[brendan_foo]

In my pre-university study, the hardest it got with integral calculus was integration by parts and substitution and that was about it. For differential calculus we touched second-order differential equations as part of the mechanics module, with springs + a damping force. The outcomes were pretty much just learned and certainly not derived.

Interesting though; I'm doing electronic engineering now and although for my first year its not really a major part of the course, I feel the urge to learn vast amounts, so I started studying vector analysis and higher level calculus, and its been very rewarding so far at least.

Hope everyone had a good new year by the way

All the best!