Proving Surface Area Formula with Multiple Integrals

[Semo727]

Surface Area (help me to prove something:)

I was studying a bit about multiple integrals and found this theorem:
If we have function z=f(x,y) which is defined over the region R, surface S over the region is

$$S=\iint_R\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\,dA$$

I wanted to prove this, because I doesn't seem to me to be trivial and I had to use nasty gradients and nontrivial things. I wonder if there is some easier proof out there?

You know, if we want to count arc length of curve given by function y=f(x), integral looks similar

$$L=\int_R\sqrt{1+\left(\frac{df}{dx}\right)^2}\,dx$$

but in this case it is obvious...

 

[arildno]

In order to derive the surface element expression, think in terms of tangent vectors and the area of the parallellogram they span:
Given a surface z=f(x,y), we may set up the two tangent vectors:
$$\vec{t}_{x}=\vec{i}+\frac{\partial{f}}{\partial{x}}\vec{k}$$
$$\vec{t}_{y}=\vec{j}+\frac{\partial{f}}{\partial{y}}\vec{k}$$
Two vectors parallell to these, and with infinitesemal lengths are therefore:
$$d\vec{t}_{x}=(\vec{i}+\frac{\partial{f}}{\partial{x}}\vec{k})dx$$
$$d\vec{t}_{y}=(\vec{j}+\frac{\partial{f}}{\partial{y}}\vec{k})dy$$
These two infinitesemal vectors can be regarded to lie ON the surface in their entirety!

The area of the parallellogram they span is therefore given by the norm of their cross product:
$$dS=||d\vec{t}_{x}\times{d}\vec{t}_{y}||=||(-\frac{\partial{f}}{\partial{x}}\vec{i}-\frac{\partial{f}}{\partial{y}}\vec{j}+\vec{k}||dxdy=\sqrt{1+(\frac{\partial{f}}{\partial{x}})^{2}+(\frac{\partial{f}}{\partial{y}})^{2}}dxdy$$
which is the sought expression.

 

[DeadWolfe]

Well, the question becomes what you mean by surface area. For instance, if you try to define it in a way like the way arclength is defined (limit of polygonal arcs), then there are deep problems, as even the nicest surfaces have ill-defined surface area.

 

[Semo727]

Oh, thanks arildno!
that was something I wanted to see :) definitely much nicer than what I have done.

I was also thinking about tangent vectors
$$d\vec{t}_{x}=\left(\vec{i}+\frac{\partial{f}}{\partial{ x}}\vec{k}\right)dx$$
$$d\vec{t}_{y}=\left(\vec{j}+\frac{\partial{f}}{\partial{ y}}\vec{k}\right)dy$$
but just in a way that I realized that
$$dS\neq\Vert d\vec{t}_x\Vert\cdot\Vert d\vec{t}_y\Vert$$
I was close, I should just have thought about cross product :)

DeadWolfe: Well, I don't know any math definition of surface area (and guess that it is far too complicated for a poor high school guy like me - when I imagine all that topology stuff like manifolds...:) But nevertheless, I can imagine same surface and define their area based on intuition and see that mentioned formulas for surface area are right for nice surfaces.

 

[arildno]

Remember that what I presented is really how the surface element "ought" to be, if the surface is sufficiently "nice".

A minimum condition for this being true is that the tangent plane is defined at every point of the surface.
I don't remember whether this condition is also sufficient or merely necessary, perhaps somebody else might answer that?

 

[DeadWolfe]

Fair enough, I didn't know exactly what you were looking for.

BTW, you should be proud of being this far ahead in high school.

 

[Semo727]

oh, thanx, I don't know if I'm really proud, but I'm happy that I have learned some basics of calculus and abstract algebra so far and now, I can understand more interesting parts of physics, which a like most. For instance partial derivatives and variations are nothing difficult to understand at all and once you are familiar with it you can read and grasp such interesting things like lagrangian and hamiltonian mechanics. Or other examle: vector spaces for quantum mechanics or curl and electromagnetism .
In Slovakia (and Czechia), we have nice university students who treat us well with correspondence seminars they run for us - give us really difficult problems to solve and also write texts explaining some parts of physics and mathematics so that they can give us more difficult problems :)

 

[arildno]

Eastern Europe has an enviably good tradition on the maths&sciences education.
I only wish we had something like that here in Norway. Unfortunately, we don't.

 

[Semo727]

You should see some of my peers who stick their nose into things like groups or QFT. That's not normal anyway :)

 

[arildno]

Norwegian 16-year olds struggle with fractions and linear equations with a single variable.

That is the level of maths in Norway.

 

[Semo727]

I have been talking about slovakia and czechia which might have looked like I was talking in general, BUT not at all. Yes, we have rather good math oriented high school classes and some competitions here (but this occurs everywhere I think), but still it is only like a drop in the ocean. My math/physics teacher is absolutely terrible and people in my class hate math/physics and are not able to understand it. (and my math/physics teacher is not an exception at all)

But yes, we have oportunities fortunately.
Actually, I'm not very proud of my country (in most cases) and your words "enviably good tradition on the maths&sciences education" seemed very curious to hear for me :)

BTW, we don't have any Nobel Prize laureate :) (Norwey has 9:)
There are doubts about our education system and our universities...

 

[arildno]

A Norwegian high school student that knows about partial differentiation and finds it easy simply doesn't exist.

Norway is about 3-4 years behind other European countries in math competence, slightly above the Albanian level.

 

[disregardthat]

well that sucks, since I'm norwegian... Why is the level so low?

3-4 years? I seriously doubt that arildno. When I first learned that an x^2 in an equation has 2 answers, I doubt russia were doing integrals...

 

[arildno]

You'd better believe it. Norway is the very worst country in Europe in math competence, apart from Albania.

As for the reason, well..Øystein Djupedal is only a minor symptom of that disease.

For those not into Norwegian politics, Ø.D is the current minister of education. His latest good idea to get more girls interested in physics, was to make dolls part of the teaching aids. He meant it. Seriously.

 

[cliowa]

His latest good idea to get more girls interested in physics, was to make dolls part of the teaching aids. He meant it. Seriously.

Sounds like people at your schools are having a hell of a good time "learning"...
But you know, Norway has good social security, a fairly well stabilized social system for the future, compared to other european countries (never mind the states, where the term itself isn't even known). So maybe you people have plenty of time to work thru these things...

 

[disregardthat]

Hm, well.. We have an exchange student from America on our school, and he is one year ahead of us. So I doubt all countries are that far ahead.

Anyway, I do not find the tasks we get on school challenging enough. I understand why other countries lies ahead of us... But it is really demotivating! How are we supposed to survive in other countries if we want to study further?

If Ø.D said that, he is a joke, he know nothing of pedagogy in school...

But seriously arildno, how long ahead would you honestly estimate for example britain are ahead of norway? I am now thinking of high school (videregående)

 

[arildno]

That is true. The US is a couple of places above Norway in the latest international surveys. That amounts to about a year, as I've heard. I was talking about Europe.

What you really should do, is to enhance self-study.
The most important thing to do in that respect is to do LOTS of exercises, more than those few ordinarily present in a Norwegian math book on your level (the physics books are much better, in 2FY for example).
Since you can read English well, I advise you to visit, say, Akademika (or other university book-shop) and try to get hold of some pre-calculs/calculus books, depending on your level.

 

[disregardthat]

Well yeah, I am going to an english mathclass now, (although it's not harder than the norwegian one) And I am going to buy next years book and try some of the stuff standing there.

How about the other countries in europe? Would you estimate they are more than 1 year ahead? Are they confident with integrals in the first year of high school (videregående)?

 

[arildno]

I would think that first and foremost, the PRE-calculus competence is by far higher in other countries than in Norway.
Thus, other countries' pupils take the more advanced issues way more easily than Norwegian students, the majority of whom do not find fractions, algebra, linear equations with one variable, functions, graph drawing and coordinate systems to be TRIVIAL issues when they start at "videregående".

Since these issues ought to be felt trivial when starting with calculus, since they are essential base skills used, the Norwegian students will lag behind and bang their heads on these issues as well as on the new issues in calculus.

 

[disregardthat]

That's sad. Why is it like this, do you know of any reason? And do you think the level of learning flattens out in the late high school?

 

[arildno]

As long as you work assiduously, you should have no problems getting through high school. But you should prepare for an extremely steep learning curve if you are planning, say, to go to NTNU or UiO afterwards for further studies.

 

[disregardthat]

Well, those who want to study math further, they study more than that is planned for, they study deeper into the things they are going through, they study the advanced versions of it, they will be prepeared.

 

[arildno]

But seriously arildno, how long ahead would you honestly estimate for example britain are ahead of norway? I am now thinking of high school (videregående)

That depends on which part of the British educational system you are looking at. If you look at the "public" schools (expensive private institutions like Eton), then they are among the very best in Europe.
However, the rest of the school system (financed by the state/municipalities) is not very much better than the Norwegian schools, from what I've heard.

 

[disregardthat]

Well, is that only in britain? Do you still mean that the schools financed by state in other countries in europe are very much better than norway's?

 

[arildno]

Yes. The Finnish school is extremely good.
One of the things in the Finnish model is to require that at ALL age levels, you need to have majored in some subject (majored: hovedfag/master's degree).

In Norway, however, nannies without formal education are allowed to teach children in the age group 6-9. And those WITH a teacher's education have not majored in anything.

 

[disregardthat]

That's a major flaw in norways education.. Does this have a large effect ultimately do you think? By that I mean the number of graduates in universities in mathematics from norway.

 

[arildno]

Most likely, unfortunately.

 

[HallsofIvy]

When I was in Macedonia, our guide kept talking about how wonderful the entire Balkans region was "except Albania"! I pointed out that in the United States, we say "except West Virginia"!

(Ducking head while fire comes in from West Virginians!)

 

[disregardthat]

On my school there is one exchange student from Ukraina, and he told me that the level of maths is way over what we learn. I get good grades now because he remembers the stuff he learned 3 years ago(!)

That is wild, I suppose that most of the stuff is not that hard, it's just that we are not used to it. I guess the reason for that if we think the stuff we are learning now is hard, is because it is new. And not because it really is very difficult! The other reason is that all east-europeans is much more clever than scandinavians :tongue:

I think this is POOR for a resourceful country like Norway.