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- Thread starter andytoh
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No way. Even though non-mathematicians may not learn the proofs, they still need to learn from a quality source, and more importantly,a pedagogical source. You don't see that in Wikipedia. It is a bunch of articles, written by different people, that are being edited almost on a daily basis. Some articles may be rigorous, while others may be much less so. Wikipedia in most cases (or all?) does not even provide you any insights. You can't say the same thing about a book like Boas'. Even if it so happened that you had no option but to choose a web source, I would recommend a site like Mathworld, Planet Math, or Hyper-Math over Wikipedia.

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That pretty much sounds like a description of the text of Boas or Arfken and Weber...can a physicist learn his math tools from an informal source that gives definitions, qualitative explanations, simple examples, and states the most important results. No exercises, proofs, or rigorous treatment. Just learn the hardcore facts.

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What's the use then? No proofs, no exercises, no use*. Its sounds more like one of those books with integral tables and such.Ok, what I meant was: can a physicist learn his math tools from an informal source that gives definitions, qualitative explanations, simple examples, and states the most important results.No exercises, proofs, or rigorous treatment. Just learn the hardcore facts. Wikipedia was just the first example I thought of.

*To someone who wishes to become a physicist.

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JasonRox

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Just extra jargon a crackpot can use in his proofs.What's the use then? No proofs, no exercises, no use*. Its sounds more like one of those books with integral tables and such.

*To someone who wishes to become a physicist.

From the mathematical model so-and-so we see that the big bang was contructed through the continuous dynamical so-and-so.

Note: And the funny part is when it comes to a crackpot, is that because he learned on the mathematics on Wikipedia and did no exercises or proofs. He has no idea how to apply. So it's no surprise that when a crackpot puts "mathematics" into his "model" it makes no sense.

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Compare Wikipedia with (1) Mary Boas's "Mathematical Methods in the Physical Science" and (ii) Arfken's "Mathematical Physics". One would be out of one's mind to dump those two in favor of a very

This fascination and obsession with Wikipedia needs to go out of fashion very quickly.You also have a very strange understanding about "mathematics" and the

Zz.

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I personally find http://mathworld.wolfram.com/" [Broken] to be a very valuable resource, and much more reliable than Wikipedia.

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I consider learning to use a book like Abramowitz & Stegun or Gradshteyn & Ryzhik to be one of the milestones in a physicist's professional development.What's the use then? No proofs, no exercises, no use*. Its sounds more like one of those books with integral tables and such.

*To someone who wishes to become a physicist.

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I didn't say it wasn't. :) We are discussing a milestone that comes before one starts using such resources. In a sense, it's like learning to play the violin and learning to play a tough musical piece on the violin.I consider learning to use a book like Abramowitz & Stegun or Gradshteyn & Ryzhik to be one of the milestones in a physicist's professional development.

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radou

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You don't need to hate to say it, since it absolutely true.I hate to say it, but nothing beats a textbook.

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So now I ask if physicists can use an informal source to learn the math tools, a source that avoids the rigour and gives the basic facts, results, and a nice general overview. And now you guys are saying that it's not enough, you must study from a textbook, contradicting the first answer.

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Since when is a rigorous study of the TECHNIQUE implies one way or the other?

So now I ask if physicists can use an informal source to learn the math tools, a source that avoids the rigour and gives the basic facts, results, and a nice general overview. And now you guys are saying that it's not enough, you must study from a textbook, contradicting the first answer.

Think of a tradesman building a house. Do you think it is fine to grab someone off the street, tell this person what each tool does, and then let him go build a house? If you think this is fine, I'd like to build your next house, and I'm CHEAP!

You have to LEARN how to use mathematics properly. Mathematics is not a subject in which you can simply sit back and read! It requires SKILL. You need to have a FEEL for the mathematics to know what to do and what not to do. These techniques should almost be automatic by the time you have to use it in solving a problem. You do not get such skills simply by reading!

For some odd reason, you simply refuses to look into one of the mathematical physics texts that I've mentioned here and previously elsewhere. Why don't you? You seem to have such difficulty grasping what really is required as far as mathematical skills goes for physicists. Why won't you open these texts and see for yourself to what extent is the mathematics skills and knowledge that is needed here? You could have saved a lot of time and effort by simply satisfying this curiosity yourself.

There IS a LARGE SPACE in between "Learn proofs from ground zero" to "learn mathematics simply by reading Wikipedia". I can't believe that you only think that there are only these two extremes that exist.

Oy!

Zz.

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AlephZero

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And Wikipedia is an incomplete "dictionary" with lots of typos...

Wiki is OK for quick reference, provided you know enough to spot the mistakes - at least the truth of maths propositions are not a matter of opinion, unlike some other parts of Wiki. But it's not a teaching tool, and not intended to be one.

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nrqed

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Doing physics is a bit like playing violin. The physics is analogue to th emusic produced and the maths are analogue to the instrument being used, the violin. It is NOT enough to read about what a violin is and the basic idea of how it is used to play music if you want to become a violonist! You have to PRACTICE. It's as if you are talking about becoming a violonist just by reading about the principles of how a violin is played on Wikipedia. If you are then handed a violin, you will not get anything worthwhile out of it this way! You would have to practice for thousand of hours before being good at playing. The same thing for maths in physics. You have to do thousands of exercises and calculations before you start mastering enough the maths to be able to actually use them to do physics.

So now I ask if physicists can use an informal source to learn the math tools, a source that avoids the rigour and gives the basic facts, results, and a nice general overview. And now you guys are saying that it's not enough, you must study from a textbook, contradicting the first answer.

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This is simply not true, a few well thought out exercises are often sufficient to be able to work out any future cases which one encounters.You have to do thousands of exercises and calculations before you start mastering enough the maths to be able to actually use them to do physics.

We are talking about Math, why should we be concerned with figures of authority? If an equation from wikipedia contains errors, big deal, you can see them and you can fix them when things don't make sense, this is an important skill for physicists. Anyone who blindly obeys Arfken without mentally testing the formulae for sensibility is in danger of making an error, just as is the case with Wikipedia.very dubious and unverified source that Wikipedia is.

If we are talking about students, then who cares if what they learn is incorrect in some minor detail? Will their skills evaporate when the detail is corrected? Unlikely, in fact any time spent concentrating is good for the student, even if the object of their concentration contains elements of nonsense/incorrectness.

This phenomenon of established authority figures bashing wikipedia has got to stop. Physicist are no longer human computers, and absolute correctness need not be such a high priority as it was in the last generation, those details can easily be checked in any case that "actually matters".

To the OP, I would answer that yes, one can gain the knowledge of a working physicist with an undergraduate education from wikipedia alone. This is because (in America) the undergraduate experience is merely backround information to prepare one for on the job training.

In the same vein, I know professors of Mathematics who have far less general knowledge then would a student of wikipedia, and whose research is a sham designed to maintain a career built off of what they got out of their graduate advisor. In this sense, Wikipedia is better than good enough for some mathematicians.

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After making a side by side comparison of wikipedia and mathworld, I find wikipedia to be much more thorough in content, examples, and links. I don't know why people here are bashing out at wikipedia so much. Is it because wikipedia is for the general public, while mathworld is not?

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Are you hijacking your own thread?After making a side by side comparison of wikipedia and mathworld, I find wikipedia to be much more thorough in content, examples, and links. I don't know why people here are bashing out at wikipedia so much. Is it because wikipedia is for the general public, while mathworld is not?

If you wish to talk about the "validity" of wikipedia, please join the existing threads in General Discussion.

Zz.

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Are you saying what I think you are saying? Are you putting texts such as Arken on the SAME level on the same topic that you find in Wikipedia?We are talking about Math, why should we be concerned with figures of authority? If an equation from wikipedia contains errors, big deal, you can see them and you can fix them when things don't make sense, this is an important skill for physicists. Anyone who blindly obeys Arfken without mentally testing the formulae for sensibility is in danger of making an error, just as is the case with Wikipedia.

I'm sorry, but this is nonsense. Any instructor can tell you the pain and suffering one has to go through in correcting wrong information and impression that a student has gained.If we are talking about students, then who cares if what they learn is incorrect in some minor detail? Will their skills evaporate when the detail is corrected? Unlikely, in fact any time spent concentrating is good for the student, even if the object of their concentration contains elements of nonsense/incorrectness.

This is highly dubious. It is the DETAILS that differentiate between a superficial knowledge of something and understanding something well. And what exactly would one use to "check" these details, since you have dismissed "authority figures".This phenomenon of established authority figures bashing wikipedia has got to stop. Physicist are no longer human computers, and absolute correctness need not be such a high priority as it was in the last generation, those details can easily be checked in any case that "actually matters".

Right.. and we really don't care if your background information is correct or not.To the OP, I would answer that yes, one can gain the knowledge of a working physicist with an undergraduate education from wikipedia alone. This is because (in America) the undergraduate experience is merely backround information to prepare one for on the job training.

This is absurd. You are using anecdotal instances and exceptions, and turning them into a rule.In the same vein, I know professors of Mathematics who have far less general knowledge then would a student of wikipedia, and whose research is a sham designed to maintain a career built off of what they got out of their graduate advisor.

Zz.

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Once you get into specific areas of Physics; each his it's own separate language to go with it essentially. Such as Calculus is really the "language" of classical/general physics. But later down the road you may use tensors, you may use different things. You really can't get this from Wikipedia.

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I disagree. I first studied tensors years ago. Then couple of years later I studied tensors again and was confused at how different it was. I was never given an explanation of the reason for the difference, but accepted it nevertheless. Only last year did it all make sense to me, thanks to wikipedia:

Once you get into specific areas of Physics; each his it's own separate language to go with it essentially. Such as Calculus is really the "language" of classical/general physics. But later down the road you may use tensors, you may use different things. You really can't get this from Wikipedia.

"There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.

The classical approach

The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the values in the array. This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.

However, to count as a tensor, the arrays need to transform correctly when the reference co-ordinate system is changed. This transformation is a generalisation of the relationship which holds for vector components, and is similarly an expression of the independence of the underlying entity from the reference frame in which it is expressed.

The modern approach

The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. This treatment has attempted to replace the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector. You could say that the slogan is 'tensors are elements of some tensor space'. Nevertheless, a component-free approach has not become fully popular, owing to the difficulties involved with giving a geometrical interpretation to higher-rank tensors.

The intermediate treatment of tensors article attempts to bridge the two extremes, and to show their relationships.

In the end the same computational content is expressed, both ways. See glossary of tensor theory for a listing of technical terms."

- #23

mathwonk

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i thought the question was " can a physicist learn?" and wanted to hear the opinions.

JUST KIDDING GUYS! I KNOW PHYSICISTS ARE SMARTER THAN US PURE MATH TYPES.

no need to tap dance on my head, but it is allowed.

JUST KIDDING GUYS! I KNOW PHYSICISTS ARE SMARTER THAN US PURE MATH TYPES.

no need to tap dance on my head, but it is allowed.

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"There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition as directions of curves

Suppose M is a Ck manifold (k ≥ 1) and x is a point in M. Pick a chart φ : U → Rn where U is an open subset of M containing x. Suppose two curves γ1 : (-1,1) → M and γ2 : (-1,1) → M with γ1(0) = γ2(0) = x are given such that φ o γ1 and φ o γ2 are both differentiable at 0. Then γ1 and γ2 are called tangent at 0 if the ordinary derivatives of φ o γ1 and φ o γ2 at 0 coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at x. The equivalence class of the curve γ is written as γ'(0). The tangent space of M at x, denoted by TxM, is defined as the set of all tangent vectors; it does not depend on the choice of chart φ.

To define the vector space operations on TxM, we use a chart φ : U → Rn and define the map (dφ)x : TxM → Rn by (dφ)x(γ'(0)) = (φ o γ)'(0). It turns out that this map is bijective and can thus be used to transfer the vector space operations from Rn over to TxM, turning the latter into an n-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not.

Definition via derivations

Suppose M is a C∞ manifold. A real-valued function f : M → R belongs to C∞(M) if f o φ-1 is infinitely often differentiable for every chart φ : U → Rn. C∞(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

Pick a point x in M. A derivation at x is a linear map D : C∞(M) → R which has the property that for all f, g in C∞(M):

D(fg) = D(f)·g(x) + f(x)·D(g)

modeled on the product rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space TxM.

The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is D(f) = (f o γ)'(0) (where the derivative is taken in the ordinary sense, since f o γ is a function from (-1,1) to R).

Definition via the cotangent space

Again we start with a C∞ manifold M and a point x in M. Consider the ideal I in C∞(M) consisting of all functions f such that f(x) = 0. Then I and I 2 are real vector spaces, and TxM may be defined as the dual space of the quotient space I / I 2. This latter quotient space is also known as the cotangent space of M at x.

While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry.

If D is a derivation, then D(f) = 0 for every f in I2, and this means that D gives rise to a linear map I / I2 → R. Conversely, if r : I / I2 → R is a linear map, then D(f) = r((f - f(x)) + I 2) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space."

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I don't know what books you looked at but from the ones I have, Kuhnel's Differential Geometry, Lee's Introduction to smooth manifolds, Abraham,Marsden and Ratiu's Manifold, Tensor Analysis and Applications all have discussions about some of the several definitions, 3 or 4 different ones each. Abraham's book refers to Spivak's for more information. I have also seen this in several of my Riemannian geometry books (though only as a comment, but that's still better than not pointing anything out.) You probably need to get better books.Here's another example. I studied the tangent space of a manifold from 3 different textbooks, each presenting it in different ways, which confused the hell out of me. Of course, no explanation was given for the different presentations...until Wikipedia explained it to me:

Also the professor for my Manifold theory course went over this as well. He even warned us on the syllabus that the concept of a tangent vector is often the most difficult in the course. Then again this was one of the better geometry professors at my school, so I'm sure that made a difference.

You may find that some of the articles on wikipedia are OK, maybe some are even good. But you can probably always get a better explanation from a good textbook and almost always a better explanation from a good professor (who studies this field).

EDIT:Let me add that if a physicist or mathematician wanted to learn about several complex variables, say in order to later study complex or kahler geometry, they could not do so by going to wikipedia.

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