Calculus: math or voodoo?

1. Aug 20, 2006

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

2. Aug 20, 2006

chroot

Staff Emeritus
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

3. Aug 20, 2006

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.

4. Aug 20, 2006

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.

5. Aug 20, 2006

matt grime

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

6. Aug 20, 2006

chroot

Staff Emeritus
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

7. Aug 20, 2006

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.

8. Aug 21, 2006

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

9. Aug 21, 2006

Daverz

Last edited: Aug 21, 2006
10. Aug 21, 2006

Daverz

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

That's not the case if you're talking about the usual
Code (Text):

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

definition of the derivitive.

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

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

Last edited: Aug 21, 2006
11. Aug 21, 2006

Office_Shredder

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

12. Aug 21, 2006

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.

13. Aug 21, 2006

Mickey

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

14. Aug 22, 2006

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?

15. Aug 22, 2006

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.

Last edited: Aug 22, 2006
16. Aug 22, 2006

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.

17. Aug 23, 2006

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?

18. Aug 23, 2006

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.

19. Aug 23, 2006

nocturnal

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 text books 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.

20. Aug 24, 2006

loom91

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

21. Aug 24, 2006

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 fucntion 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.

22. Aug 24, 2006

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.

23. Aug 24, 2006

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.

24. Aug 24, 2006

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.

25. Aug 25, 2006

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?