Is Calculus Just Math or Voodoo? Exploring the Controversy

[loom91]

Hi,

I'm sure I'm wrong, but calculus appears voodoo to me. I can usually get the right answers, but it all looks like a castle of clouds to me. There's no internallogic, things are forced into place to make them work. In particular, dy/dx is reffered to and defined as an operator d/dx acting on y, but then it is frequently treatted as an actual quotent of two actual algebraic quantities, particularly in integral calculus and solution of differential equations. What sort of black magic is this? Thanks.

Molu

 

[chroot]

The reason it's confusing is because you're being taught by teachers who are either (a) lazy or (b) incompetent.

It is unfortunate that calculus can be taught almost like it's black magic -- you put this symbol here, put that symbol there, and you get the right answer. It is, however, a rigorous subject that does not actually include any "black magic."

Once you reach the level of differential forms and real analysis, all of the "black magic" features of calculus will be revealed as very strict, sensible constructions. "dx," for example, is actually a one-form, though no Calc I teacher will ever explain that to you.

- Warren

 

[CRGreathouse]

It seems that way because you are given tools to work with without explanations. Real Analysis covers the "why"s in muc more detail, but that's usually a senior-level course in college.

 

[Daverz]

Try to find the little book Gravity by George Gamow. It will give you a more physical feel for calculus using the original application. Also, there are some visual calculus tutorials on the net, just google "calculus".

As for the infinitisemals, I don't know what to tell you. They are never put on a solid foundation in the typical calculus courses, but you'll still see them used all the time in Physics, and Physicists are expected to pick it up by osmosis. So you still need to play with the simple dy/dx picture of little infinitesimal triangles, or at least Physicists do.

The infinitesimals were given a rigourous foundation in the last century. There's an entire calculus book using infinitesimals, with all the usual applications, listed at the bottom of that page, and Dover has a couple short books on the subject. But I don't know if that would help or just be a distraction for you at this point.

 

[matt grime]

You do not have to treat dx as an infinitesimal: it is a 1-form.

 

[chroot]

In fact, it's much more concrete to approach calculus as an application of differential forms, never once referring to dx and dy as infinitesimals...

- Warren

 

[Daverz]

I can see how differential forms make sense out of

dy = f' dx

but it's not clear to me how they make sense out of

f' = dy/dx

as an actual ratio rather than just notation that looks like a ratio but isn't. That's what non-standard analysis does.

Also, there's a huge literature out there that makes naive use of infinitesimals. Pure math students may be lucky enough not to encounter it in modern textbooks, but students in the sciences are not so lucky.

 

[loom91]

So dy/dx is better treated as a ratio of two "1-forms" rather than as an operator acting on y? That means that the definition we use for first-principle calculations of derivative would have to be abandoned. Our high-school course in calculus is particularly lacking in rigour. We learn the applications of single variable real calculus up to second-order differential equations, but most of it is application of formulas given without proof (for example the limit theorems and standard limits).

In physics, we rely on intuition to get a right answer treating dx as an infinitesimal change, but because of the lack of a properly understood foundation it's all like groping in the dark and it can get very confusing in problems requiring complicated use of calculus.

I've heard about 1-forms before. What actually are these and how do they relate to the definition of derivative we are taught? Thanks.

Molu

 

[Daverz]

You can get a free book on forms here:

http://pzacad.pitzer.edu/~dbachman/

The author is a forum member.

 

[Daverz]

So dy/dx is better treated as a ratio of two "1-forms" rather than as an operator acting on y?

I don't believe it's meaningful to divide 1-forms that way.

That means that the definition we use for first-principle calculations of derivative would have to be abandoned.

That's not the case if you're talking about the usual

f'(x) = lim (f(x+h) - f(x))/h
        h->0

definition of the derivitive.

Our high-school course in calculus is particularly lacking in rigour. We learn the applications of single variable real calculus up to second-order differential equations, but most of it is application of formulas given without proof (for example the limit theorems and standard limits).

I don't think that's a bad approach. It can take some mathematical maturity to really "get" the idea of limits.

In physics, we rely on intuition to get a right answer treating dx as an infinitesimal change, but because of the lack of a properly understood foundation it's all like groping in the dark and it can get very confusing in problems requiring complicated use of calculus.

And wait until you get to how variational problems are traditionally handled in Physics (e.g. in Goldstein). That can really be confusing.

 

[Office_Shredder]

Actually, using dx and dy as just really small distances is a great intuitive way of dealing with physics

 

[coalquay404]

I think that you're all trying to over-analyse the OP's problem. Differential calculus of functions is *not* easily discussed in terms of differential forms. In fact, a moment's thought should be enough to convince you that any expression of the form

f'(x) = (d/dx)f(x)

cannot be reliably interpreted in terms of differential forms. Differential calculus of this type is the study of the limiting behaviour of the rate of change of functions -- this is analysis, not differential geometry.

I do agree though that anybody who has trouble understanding calculus is most probably being taught by a lazy teacher.

 

[Mickey]

I don't think that's a bad approach. It can take some mathematical maturity to really "get" the idea of limits.

Well, it's obviously a bad approach for the mathematically mature, then.

 

[loom91]

I will be trying out that linked book now.

By the way, I think I 'get' the epsilon-delta definition of limits and the infinitesimal treatment in Physics appears rather forced and illogical to me.

In any case, what is dy/dx really if not the ratio of two 1-forms?

 

[matt grime]

If y is a function of x then dy/dx is the limit of y(x+e)/y(x) as e tends to zero. That is what it 'really' is, and I imagine what you already know. And if you're still wondering how it can be 'treated as a fraction' when doing integrals, then just think of the following example (latex is still broken I think)

if g(y)dy/dx = f(x), then integrating both sides gives wrt x

int g(y)(dy/dx)dx = int f(x)dx

but the LHS is just the same as int g(y)dy by the chain rule.

 

[mathwonk]

defnitely voodoo. famous religious figures even tried to expose Newton as a heathen mathematician in the old days. fortunately he was not burned at the stake.

 

[loom91]

You mean that the fraction thingy is just a notation? No meaning is assigned to the intermediate steps in the solution of a differential equation where we manipulate the dys and dxs separately?

 

[matt grime]

No meaning needs to be assigned to that treatment. The fact is that all you're doing is a omitting some steps, in an attempt to make it intuitively easy to solve the DE. There is no need to manipulate the dys and dxs separately at all. However, since the net effect is the same as treating them as entitities that are manipualable, if that's a word, that is what is taught when it is the ends and not the means that matter. Not something I condone, by the way.All this talk of infinitesimals, and 1-forms are rigorous ways of justifying this treatment, but at the level we're talking about here there is no need to use it.

 

[nocturnal]

And if you're still wondering how it can be 'treated as a fraction' when doing integrals, then just think of the following example (latex is still broken I think)

if g(y)dy/dx = f(x), then integrating both sides gives wrt x

int g(y)(dy/dx)dx = int f(x)dx

but the LHS is just the same as int g(y)dy by the chain rule.

No meaning needs to be assigned to that treatment. The fact is that all you're doing is a omitting some steps, in an attempt to make it intuitively easy to solve the DE. There is no need to manipulate the dys and dxs separately at all. However, since the net effect is the same as treating them as entitities that are manipualable, if that's a word, that is what is taught when it is the ends and not the means that matter. Not something I condone, by the way.


All this talk of infinitesimals, and 1-forms are rigorous ways of justifying this treatment, but at the level we're talking about here there is no need to use it.

Nicely put. I wish my teacher had mentioned this when I was taking intro to calc. This simple fact was the source of a lot of confusion and it took me a better part of a year to figure this out for myself. However, I don't think my math teacher knew too much about math. When asked if I would ever encounter a class where calculus would be put on a rigorous foundation and everything would be proved (beyond the brief hand-wavy proofs that were given in the text) I was told no, that this intro class was all there was. Imagine my amazement (and relief) when I learned about a class called real analysis where we would start with the axioms for the real numbers and build everything up from there.

I am also surprised that many textbooks introduce the methods "separation of variables" and "u-substitution" without any mention of the chain rule, but rather show that we can manipulate dx's and dy's after we're explicity told that dx and dy have no meaning on their own and that dy/dx is not to be treated as a fraction!

Anyways, I empathize with the OP and suggest that if your current text lacks the rigour you desire, you might try picking up a better one such as the ones written by Spivak, or Apostol, or even an intro analysis text.

 

[loom91]

The explanation is logical, but it leaves a distinct feeling of unfulfillment. Anyway, thanks.

 

[mathwonk]

i have explained this elsewhere here and it is also treated in some excellent clasic ode books such as tenenbaum and pollard.briefly: consider the tangent line at (p,f(p)) as the graph of the best linear approximation to f, near p. i.e. for x near p, the linear function f'(p)(x-p) is an excellent approximation to the function f(x)-f(p).

i.e. .the linear function is called df(p), and is a good approximation to the difference function called "deltaf"(p).

thus the function f(x) = x also has a linear approximation at each point, namely dx(p) is the lienar function with value 1(x-p) on (x-p). it is a superb approximation to the diference function (x-p).thus the diferential df, is a function whose value at p is the linear function df(p), and the diferential dx is a function whose value at p is the linear function dx(p).

when you divide these 2 linear functions you get the expression
df(p)/dx(p) whose value at every x (except x=p), is the constant quotient f'(p)(x-p)/1(x-p) = f'(p).so it is true that df/dx = f', and also that df = f'dx.but now you know both why matt said it is unnecessary to know this, and why your teacher did not explain it.the explanation is more trouble than its worth, except to salve your conscience that an explanation is possible.tenenbaum and pollard probably make it look more palatable however.

 

[loom91]

I have heard good things about the following books. Which one should I use at 11-12th Grade level (keeping in mind I aspire to seat for the IIT-JEE, renowned as the world's most difficult examination)? Thanks.

1)Michael Spivak
2)Thomas and Finney
3)James Stewert

I also imagine there may be good soviet authors who are not well-known in USA.

 

[mathwonk]

you have listed them roughly in the order of difficulty and sophistication.

spivak is miles above the other two.you mihjt also consider the three volume set by goursat, recommended by the russian prifessor v.i. arnold.

 

[matt grime]

thomas and finney, and stewart are rubbish. To clarify mathowonk's post, you have listed them in decreasing order of sophistication and difficulty.

 

[loom91]

Does Spivak have difficult to crack problems that are within a high school syllabus? Also, is it only single variable real or multi-variate also?

 

[HallsofIvy]

The definition of $\frac{dy}{dx}$ is:
$$\frac{dy}{dx}= \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$
so it is not a fraction; but it is a limit of a fraction.

Since we can "go back before the limit, use fraction properties, then take the limit", we can always treat the derivative like a fraction.
The main reason for defining differentials, dx and dy, separately (as opposed to the derivative) is to make that notationaly correct.

 

[mathwonk]

spivak has two calculus books, one called calculus is one variable. the other called calculus on manifolds is several variable. the second is very short, but what is there is excellent. there is no taylor series in the second one.

 

[nocturnal]

Does Spivak have difficult to crack problems that are within a high school syllabus? Also, is it only single variable real or multi-variate also?

Within a high school syllabus? I would say no! Many of the problems in Spivak are quite challenging and require you to prove something. The challenging problems vastly outnumber the straightforward computational ones that most high schoolers are accustomed to and even the "straightforward" problems are "not overly simple" to state it in the author's own words.

However, it is definately worth a trip to the library to see for yourself if it's the right book for you. Note that while his writing is fairly gentle and very insightful, much of the meat of Calculus is contained in the problems.

His book only deals with real analysis of a single variable however he has a couple of chapters devoted to Complex functions. There is also a wonderful chapter that constructs the reals from the rationals and then goes on to prove that R is the only complete ordered field up to isomorphism.

edit: This only describes the single variable book, not the mulitivariable one mentioned by mathwonk above.

 

[loom91]

Within a high school syllabus? I would say no! Many of the problems in Spivak are quite challenging and require you to prove something. The challenging problems vastly outnumber the straightforward computational ones that most high schoolers are accustomed to and even the "straightforward" problems are "not overly simple" to state it in the author's own words.

However, it is definately worth a trip to the library to see for yourself if it's the right book for you. Note that while his writing is fairly gentle and very insightful, much of the meat of Calculus is contained in the problems.

His book only deals with real analysis of a single variable however he has a couple of chapters devoted to Complex functions. There is also a wonderful chapter that constructs the reals from the rationals and then goes on to prove that R is the only complete ordered field up to isomorphism.

edit: This only describes the single variable book, not the mulitivariable one mentioned by mathwonk above.

By high-school I meant topic-wise, not difficulty wise. Since it's mostly single-variable real I guess that criteria is satisfied. One of the greatest obstructions to quality education where I live is the total absence of libraries of even modest quality, particularly when it comes to textbooks, even that foreign. I was just discussing with my chemistry a few days ago how a good library can make a world of difference.

If I want to read a textbook, I would either have to spend a lot of money buying it new or spend hours or even days hunting for it among the mammoth bazaar of used books that is College Street. Anyway, I think I will look for Spivak.

Molu

 

[loom91]

Can anyone tell me who publishes/distributes this book in India? It's proving to be a very rare book, clearly none of the shopkeepers have ever heard the name.

 

[neutrino]

Spivak? Don't think so (I could be wrong).You can find the two-volume calc books by Apostol from Wiley, though. (~ Rs. 650) I assume that you're from Calcutta. There must be a British Library (Dep. High. Comm.) there. Check it out.

 

[loom91]

You mean Spivak's book is not distributed in India? Is Apostol good?

 

[neutrino]

You mean Spivak's book is not distributed in India? Is Apostol good?

I'm not sure, at least I couldn't find anything on the 'net. You can contact these guys http://www.mathpop.com/ for further info. https://www.amazon.com/dp/0471000051/?tag=pfamazon01-20's supposed to be one of the good 'pure-maths-type' intro. I have borrowed the book once from a library, but being 'infintely impaired' , I couldn't get beyond a few pages.

 

[mathwonk]

any book you can read is a goodbook for you. then afterwards progress to another book.

 

[z-component]

What makes James Stewart rubbish? I think the fifth edition is very robust.