How to Work Through Spivak's Calculus?

[zooxanthellae]

I took BC Calculus this school year, but since I want a deeper grasp of Calculus [and since the course I hope to take next semester uses Spivak] I've decided to do a little independent study this summer.

My question is over the process of actually doing this: I've seen suggestions of doing every single problem, but that seems a bit ridiculous. The first chapter has some 100 problems!

Also, when the problem asks me to prove; how formal should I be? Should each line have a definition/theorem/corollary next to it?

Thanks!

 

[aber_leider]

Put your Spivak back in your bookshelf for a while. Find a text or a website on "How to write and understand mathematical proofs" and read it completely. Maybe you will not work with Spivak at all before your next course, but the time spent understanding how rigorous mathematical thinking is done will pay off.

 

[FieldDuck]

When I read the textbook for a class prior to starting the class, I don't intend to read it so that I have completely mastery of the material, but rather to read it to get an overall idea of the course so that I could keep in mind where the class is heading to help make it all make sense.

I agree with aber that it would be beneficial to read a introduction to mathematical proofs book if you are not comfortable with how to write a proof and the logic associated with such task. Practicing that now will greatly aid your endeavor in understanding the more subtle beauty in mathematics.

Also, if you choose to keep working on Spivak (which I think you should), then don't focus on solving all problems. Like I stated earlier, I personally don't go into reading a book I will use next semester to solve and understand every completely. So with that in mind, my method would be to read the chapter, write down main ideas, look at each problem and consider each problem. (This is to say that in my mind I think "ok this is the problem, if I were to solve this I would try this method and say this and that.") If a problem isn't obvious to me or seems like it is interesting, then I sit down and actually try to solve it. Other times, I do every 4 problems or so to check and see if I am truly understanding what I am reading. (For example, in Spivak Chapter 1, there's a list of inequalities, I pick a few of what seem like the tougher ones and I try to prove them, if I can then I feel that I understood what I wrote. If they turn out to be tougher than expected, then I go back and reread what Spivak wrote and/or try more problems.)

After I am done with chapter j I go onto chapter j+1, and read it. I then try to see how chapter j+1 uses information from chapter j and chapter j-1. I"m personally really big into mapping out relationships visually, so I would say, something like oh in chapter j he introduced this inequality and used it in this theorem and then he used this theorem to prove something in chapter j+1. That's me personally, you may not be a visual person like that, but I think it's important to see how everything connects.

Well, hope this helps.

 

[zooxanthellae]

So does anyone want to say whether or not I should be writing a theorem beside after step of a proof? Or is that just a question which shows how badly I need to consult a proof-writing guide?

 

[thrill3rnit3]

So does anyone want to say whether or not I should be writing a theorem beside after step of a proof?

Sorry, that didn't make any sense (to me at least).

 

[zooxanthellae]

Sorry, that didn't make any sense (to me at least).

Sorry about that. I'll try giving an example:

a + (b + c) = (a + b) + c -------- Associative Property

Should I have something like the bolded justification for every single step?

 

[FieldDuck]

I think what he is getting at is how they teach people to write proofs in geometry classes in the US. The structure follows like this.

1) Statement - Given
2)Next Statement - by this theorem
3)Next Statement - By this other theorem
4)Conclusion - because of this and this other theorem then this theorem implies the conclusion.

I hope that makes sense (i'm a bit to lazy to give a real example.)

No you shouldn't write a proof like that, it should be more fluid. Google "proof writing guide" and the first link should be a PDF. Give it a read and it might give you a more basic understanding of how to write a real proof. I'll say this though. In the first few chapters of Spivak, many of the proofs by nature are simply algebraic. There you might want to write a proof like
1)x^2 - y^2
2)x^2 -xy + xy ^y^2 by P#
etc

Just because there isn't much else you can do, but in general a proof shouldn't be written like that.

 

[zooxanthellae]

I think what he is getting at is how they teach people to write proofs in geometry classes in the US. The structure follows like this.

1) Statement - Given
2)Next Statement - by this theorem
3)Next Statement - By this other theorem
4)Conclusion - because of this and this other theorem then this theorem implies the conclusion.

I hope that makes sense (i'm a bit to lazy to give a real example.)

No you shouldn't write a proof like that, it should be more fluid. Google "proof writing guide" and the first link should be a PDF. Give it a read and it might give you a more basic understanding of how to write a real proof. I'll say this though. In the first few chapters of Spivak, many of the proofs by nature are simply algebraic. There you might want to write a proof like
1)x^2 - y^2
2)x^2 -xy + xy ^y^2 by P#
etc

Just because there isn't much else you can do, but in general a proof shouldn't be written like that.

That's pretty impressive, since:

a) I'm an American student
b) What I'm referring to is how my Geometry class taught us to write proofs.

I'll take a look at that guide. Thanks for the help.

 

[zooxanthellae]

I'm looking at this guide right now. I'll be heading here in the Fall, so I figure this style is something I should get used to. Thoughts?

http://www.math.uchicago.edu/~eugenia/proofguide/proofguide.pdf

EDIT: On further review, the author of this guide no longer teaches at the university, but I still ask if this is more the type of proof I should be doing. I looked at the PDF from Google and had some trouble with it since I'm not well-versed in logical symbols.

 

[thrill3rnit3]

Ah, yeah I completely understand now.

And no, in higher mathematics, proofs aren't usually written in a numbered format like the "proofs" done in geometry class. Most of them are written in paragraph form. Also proofs in geometry seem choppy (to me at least). Applying that format to more advanced proofs might leave some gaps within the proof that need justification, which would make the proof less rigorous than what it should be.

 

[wisvuze]

writing proofs always seemed pretty natural to me. It's like writing formal expository rhetoric targeted at someone who knows absolutely nothing about what you're talking about (hence, you go for a "take no prisoners" approach, citing all theorems, proving your statements.. et c)

 

[rasmhop]

I'm looking at this guide right now. I'll be heading here in the Fall, so I figure this style is something I should get used to. Thoughts?

The guide has some good advice on the errors beginners often make, however the style of proof it uses is NOT standard. I suspect it uses this format for there to be no ambiguity, however in real math proofs we use English, but we use it carefully. "For all real x there exists a unique integer n such that $n\leq x < n+1$." is a good way to state what could also be written:
$$\forall x \in\mathbb{R}\,\exists n \in \mathbb{Z}\, \forall m \in \mathbb{Z} \left[m \leq x < m+1 \Leftrightarrow n=m \right]$$
if we insist on avoiding English. The guide seems to settle on some kind of middle-ground, but real proofs use less logical symbolism. One reason to use this much logical symbolism is to force yourself to think in a logically meaningful manner.

Try taking a look at one of the proofs in Spivak. That is how a real proof looks. Universities do not have their own "proof notation". Everyone has their own style and people can read other people's style of proof.

To give an example of the kind of proof I would expect from someone who has completed their a course doing proofs (something very similar to this is probably an exercise in Spivak):

Theorem (AM-GM inequality in two variables): For all non-negative real numbers x and y we have
$$\frac{x+y}{2} \geq \sqrt{xy}$$
with equality if and only if x=y.
Proof:
Let x and y be distinct non-negative real numbers. Then $\sqrt{x} \not= \sqrt{y}$ so $\sqrt{x} - \sqrt{y} \not= 0$. This implies $(\sqrt{x}-\sqrt{y})^2 > 0$. Expanding using the rule $(a-b)^2 = a^2 + b^2 - 2ab$ we get:
$$0 < \sqrt{x}^2 + \sqrt{y}^2 - 2\sqrt{x}\sqrt{y} = x + y - 2\sqrt{xy}$$
Rearranging the terms and dividing by 2 we get:
$$\frac{x+y}{2} > \sqrt{xy}$$
We have now proven that if x=y is false, then the inequality holds, but without equality. It now only remains to show that if x=y, then:
$$\frac{x+y}{2} = \sqrt{xy}$$
But this is immediate by the following calculation:
$$\frac{x+y}{2} = \frac{x+x}{2} = x = \sqrt{xx} = \sqrt{xy}$$

Note that at several points I just assumed results without referencing them since these are basic (what is basic of course depends on your level and when you write expository work: your audience's level). For instance I wrote $\sqrt{x}$, but does x actually have a square root? It does because I assumed that x is non-negative. If you were a bit unsure about this, then there would be nothing wrong with adding:
Since x and y are non-negative they both have unique square roots.
I also assumed that since x and y are distinct we know that $\sqrt{x}$ and $\sqrt{y}$ are distinct. This is well-known, but you cannot simply accept it on faith. There are plenty of other functions for which it does not hold in general. For instance if a and b are distinct real numbers I can't deduce that a^2 and b^2 are distinct since we may for instance have a=1 and b=-1. I have used plenty of other small facts that needs proofs, but I assume they have already been given (I use $\sqrt{x}^2 =x$ and $\sqrt{x}\sqrt{y} =\sqrt{xy}$)

The level of detail, and what details you choose to focus on depends on your level. At your level it would probably be prudent to add notes such as "it's well-known that if x and y are distinct, then so are their square roots." I believe Spivak actually proves this, and until that point you can't reference it. As you evolve as a mathematician so will your proof style. If the same proof were given in a graduate class it would probably just read:
"Follows trivially from (x-y)^2 \geq 0 with equality if and only if x=y."
or more likely:
"Exercise for the reader" or "Trivial".
because at this level people have so much experience that they can easily reconstruct the proof, probably even just do it in their head. If there is any step where you are not sure exactly why the next part follows, then you have not provided enough details. The authors of your books may get away with shorter proofs than you, but that's because they are more experienced and can easily see how the proof works even if you leave out a couple of simple steps. Also a gap in a proof in a book is often taken as an extra exercise.

You may be at a level where you can't really see when we use commutativity, associativity or distributivity, and in that case you should add "By associativity we see ..." and statements like that, but at some point this should become trivial for you to work out in your head so you may write:
Using the definition of squaring and the usual arithmetic rules we get:
(a+b)^2 = (a+b)(a+b) = aa + ab + ba + bb = a^2 + 2ab + b^2
Once such manipulations become simple you may simply write:
(a+b)^2 = a^2 + 2ab+b^2
since it's obvious that to check it we just multiply out and collect terms.

Last quarter I was enrolled in an algebra course using the text Algebra by Hungerford, and our professor commented that Hungerford often follows up a theorem with what he calls a "proof sketch", but despite the convention to call these arguments proof sketches we should consider them proofs. Basically Hungerford targeted an audience a little less experienced than us so his "proof sketches" had gaps large enough that he felt that the reader should fill in the details, however a somewhat more experienced reader would easily be able to fill in the gaps and consider the proof sketches perfectly acceptable proofs.

Another thing I would like to note is that a proof is an expository work meant for other humans to read. Therefore there is no shame in helping them a bit. If your proof becomes a bit cluttered try to clean it up, and up sentences explaining what you are intending.
1) "We shall start by proving (2)"
2) "Let us first start by proving the statement with the extra assumption that n is a prime number".
are both perfectly acceptable parts of a proof.

Anyway I think I have rambled enough, just take a look at the proofs by Spivak to get a feel for what proofs are. Hope this makes sense.

 

[fasterthanjoao]

Also, when the problem asks me to prove; how formal should I be? Should each line have a definition/theorem/corollary next to it?

This is why things like this are perhaps best left to wait until you're taking lectures on the subject. It can be difficult to learn to do things in a certain way, then be told a totally different process - even if it is picking and choosing which parts to write. The way your lecturers in future teach you to do it, is the way they will expect to see it done.

Otherwise, calculus certainly is about solving problems. Do as many problems as you need to! If that means you need to do all 100 problems, then go for it. If you tackle 25, then can do them without even having to write anything down, you're ready to move to the next section. Calculus is also about giving yourself refreshes. Keep your solutions so that you may look at them later down the line when your calculus skills start to wane