Why are there so many difficult problems in math and physics textbooks?

[Alpharup]

This is one of the major questions which is asked by the students. Whenever we read mathematics or physics book(generally), we see that many problems are given at the end of the chapter. These eventually test our understanding and application of concepts.
I have few questions here. Who creates these problems? Is it the author of the text-book or is it copied from some of the other textbooks? I want to ask these because reading and grasping some derivations or concepts take time. After that, when we solve the problems( mainly the twisted ones) on our own, we may think in different angles and we may apply different manipulations. Some can be solved in two minutes with different thinking and others may require larger time(For example,I have taken three hours for solving one!).Even when we can't solve it, we may refer to other resources(in extreme cases, even solutions)! Those who have very high intelligence may solve all of them.

What I want to say is that solving tough problems itself takes time. How could he(say the author) invent it? Inventing a problem and solving it may take tremendous amount of time. The inventor of these problems may also refer it to his colleagues who may cross-check it. This may also involve some time.

So please explain me the process of writing the text-book and also creating the problems. I am a bit curious.

 

[member 392791]

I noticed that a lot of textbooks have similar problems, the concept is the same but the question is obviously worded differently to not be plagiarized, yet the premise of the problem is identical and it is solved the exact same way. I think some problems in physics are so popular that they wind up in nearly every textbook. It is kind of like how every physics class does a lot of experiments that are identical. They are just replications of famous experiments.

 

[Alpharup]

I noticed that a lot of textbooks have similar problems, the concept is the same but the question is obviously worded differently to not be plagiarized, yet the premise of the problem is identical and it is solved the exact same way. I think some problems in physics are so popular that they wind up in nearly every textbook. It is kind of like how every physics class does a lot of experiments that are identical. They are just replications of famous experiments.

Yes, I agree with you. Creating math problems, solving them and verifying them is little easy(I think so) because the materials of proving or solving is mostly done on paper and paper is not costly. But however,for new mechanics problems, the experts may agree on particular solution. But, you need experimental verification of the solution. Won't it be costly to conduct such experiments?

 

[Lavabug]

Creating problems that illustrate and teach physics concepts without becoming abhorrently difficult and long is an art. Most books on any given subject I've seen cover a few "classic" problems and give it their own spin or flavor, often forcing you to really think, but not have to do any more than 3-4 pages of calculation (with small handwriting) to get to an answer in my experience. In general, there is only so much you can do in a course/book.

Most of the prof's I had would make up original problems on tests or homework assignments and it showed. Trying to make something "original" often results in something ridiculously tedious, like a rigid body dynamics problem with a rotating solid cone with an offset C.O.M, spinning about a pole attached via a spring (weirdest toy I've ever seen). 4 natural coordinates, have fun getting those Euler-Lagrange equations... Good fun, but not on a timed test!

 

[Alpharup]

Most of the prof's I had would make up original problems on tests or homework assignments and it showed. Trying to make something "original" often results in something ridiculously tedious, like a rigid body dynamics problem with a rotating solid cone with an offset C.O.M, spinning about a pole attached via a spring (weirdest toy I've ever seen). 4 natural coordinates, have fun getting those Euler-Lagrange equations... Good fun, but not on a timed test!

You must be lucky in having such prof's. Such thinking profs are difficult to find these days. The problem you gave drives anybody crazy. I think that the problems given in physics books were developed this way. Some thinker(i think mostly profs) long ago would have made these problems and we are enjoying the fruits by solving them.

 

[Lavabug]

I do feel lucky in that it was a good learning experience, especially in my optics, mechanics and QM courses. But I still recall going into those exams feeling pretty terrified, along with the rest of my class.

One of the old QM lecturers at my uni published a book based on past exam problems... save for about 3-4 that were pulled shamelessly from Cohen-Tannoudji (with credit, of course), they're all original and stupid, STUPID hard and long, compared to what you find in most textbooks. If anyone has the money and inclination to put themselves through them, here's the book: http://www.casadellibro.com/libro-problemas-resueltos-de-fisica-cuantica/9788484918684/1623527

Also, if you're curious about my CM, this is an old collection of problems and exams from a former prof of mine: http://vdelgado.webs.ull.es/mecanica/ (the problem I mentioned is the 4th sheet on the right panel)

Some of my profs "apologized" to the class on a few occasions, often stating "there's not much else I can come up with". :P

 

[WannabeNewton]

There are many mechanics and EM books out there with extremely clever problems which have little to no pointless calculations but require ingenuity to setup the problem, with the solution then following immediately. These are the problems I like and learn best from. I find the long calculation problems tedious and rarely instructive -glances over at copy of Wald's "General Relativity"- xP. Beyond mechanics and EM I have to search harder to find books that do the same but certainly they exist. I have personally found problems in upper-level physics texts (e.g. Goldstein) to be noticeably easier than the ones in difficult lower level physics texts (e.g. Morin or Purcell) in the sense described above with one consistent exception, on a personal level, being Landau and Lifgarbagez; this may be because mechanics and EM allow for a lot more "toy" problems (a professor/textbook willing to assign the same kind of "toy" problems in a general relativity class would surely be a madman and sadist!). On the other hand, as far as upper level math texts go, the level of ingenuity required to solve problems (as opposed to annoying explicit computations and/or computational proofs) tends to stay strong.

 

[retro10x]

It should be noted that creating these hard problems is often a longer/more difficult process than solving them. Sure, anyone can throw together many constraints, external forces, etc. but most of the time they aren't solvable, or aren't solvable using the course material.

It can take textbook authors YEARS to write a textbook, and one of my profs who co-authored a textbook (long outdated now) said about 10% of the time was actually writing the book, and the other 90% was creating the problems/editing (He had to keep going back and editing problems over and over again for clarity or because they were constructed incorrectly).

We would complain about the difficult problem sets and he would retort how it took him at least twice as long to make the problem that it should be to solve it.

Also as others have mentioned, many problems are "inspired" from other textbooks and many are famous enough that they are in every textbook of that subject. Ex. the ladder in the barn problem in SR

 

[Lavabug]

Ex. the ladder in the barn problem in SR

We called it the relativistic pole-vaulter in a barn.

 

[WannabeNewton]

I wonder what the most "famous" of ubiquitously used problems is/are in a given subject. Certainly the classic "Hello World" problem from K&R has been reused to death in every introductory programming text/class ever so that should top the list :p

 

[retro10x]

I have personally found problems in upper-level physics texts (e.g. Goldstein) to be noticeably easier than the ones in difficult lower level physics texts (e.g. Morin or Purcell)

The problems in Morin's textbook are just insane, never have a spent a week trying to solve a single problem before. I find that the more basic the material, the more conceptually difficult the authors try to make the problems, as if to compensate. (At least in physics, I haven't read any higher-math textbooks past PDE's)

I still find my intro physics exam questions harder than 3rd year classical mechanics problems :/

 

[WannabeNewton]

The problems in Morin's textbook are just insane, never have a spent a week trying to solve a single problem before.

Certainly, I agree. It's the hardest textbook I've had to use thus far.

I find that the more basic the material, the more conceptually difficult the authors try to make the problems, as if to compensate.

This is exactly what I was thinking too. And it's harder this way, imo, because mathematics is straightforward as far as its use in physics goes, relative to the difficulty of conceptual subtleties, so those upper-level physics problems that are basically just math problems in the context of physics I find to be noticeably easier than the conceptually mind wracking problems in books like Morin, which as you say are made that way because the underlying machinery of the theory is in and of itself simple. They are elegant problems but they drive you insane.

I still find my intro physics exam questions harder than 3rd year classical mechanics problems :/

I couldn't agree more!

 

[Alpharup]

I do feel lucky in that it was a good learning experience, especially in my optics, mechanics and QM courses. But I still recall going into those exams feeling pretty terrified, along with the rest of my class.

One of the old QM lecturers at my uni published a book based on past exam problems... save for about 3-4 that were pulled shamelessly from Cohen-Tannoudji (with credit, of course), they're all original and stupid, STUPID hard and long, compared to what you find in most textbooks. If anyone has the money and inclination to put themselves through them, here's the book: http://www.casadellibro.com/libro-problemas-resueltos-de-fisica-cuantica/9788484918684/1623527

Also, if you're curious about my CM, this is an old collection of problems and exams from a former prof of mine: http://vdelgado.webs.ull.es/mecanica/ (the problem I mentioned is the 4th sheet on the right panel)

Some of my profs "apologized" to the class on a few occasions, often stating "there's not much else I can come up with". :P

I think the material in the link is in Spanish. It appears that your prof' want students to think about physics all the time. Making new hard problems would make some students to swear at them all the time. But some intelligent students may enjoy the exercise. They develop good relation with their teachers. This has a positive effect. Learning will become enjoyable for these students and they may feel good in this process. Thus, this system awards merit.


I don't have any experience in quantum mechanics but people say that it is hard. Making material hard improves the standards. It also gives confidence for students to solve any problem which they may encounter. But at the same time, prof's should also help the less intelligent ones by regular interaction and support. They should not neglect them. Neglecting them will result in low self-esteem.

I do like your system. Where is your college located?

 

[Lavabug]

Most enjoy the exercises, my class tends to actually care. There's still an awful lot of moaning but I think all who made it past the 2nd year are pretty masochist.

You can see what university it is in the 2nd link. :)

Unfortunately effort does not always translate well into grades under this system IMO, since almost all courses are evaluated almost entirely by a single written exam that is 3-5 hours long, as with most other universities in my country. Many of the really good students don't finish on time. As a consequence my grades are pretty unimpressive and it's very unlikely I'll get into grad school now, so I don't think I'll be putting all this hard physics to any good use. :/