Why learn integration techniques?

[FrogPad]

Blanket statements that say that numerical solutions do not offer insight or obviously wrong

I agree with you. I should not have said that numerical solutions do not offer insight. However, I think it was obvious that I was making a point... if someone doubts the need and utility of numerical methods, then we've got a problem.

I don't think we cannot underestimate the need and utility of integration techniques though. They support the basic foundation and seem to be useful as a tool for teaching engineering.

I wanted to speak on a few of your points individually:

Case 2:
Toy problems are important. Toy problems can capture the same complicated ideas as non-toy problems, but in a much easier to understand format. If you had a solution to a toy problem that was analytic would you use the numerical one?

Case 3:
Referring to:

...meaningful results are best done numerically.

You can't say that meaningful results are best done numerically in research. Obviously this really depends on the problem. This is just as much as a blanket statement as the one I made. But your point is there, and I can't say I disagree.

 

[DavidWhitbeck]

I'm not discussing calculus as a set of tools for engineers. I couldn't care less about the importance or otherwise of integration per se to engineers. I'm pointing out that you learn more in a mathematics class than just a technique to solve an integral, but learn a valuable lesson in how to actually think.

Sorry for misrepresenting you.

 

[epenguin]

Limiting to the question actually asked, I had hesitantly expressed the same doubt before.*
I suspect this will go the same way as whether you are allowed to use calculators in schools for arithmetic. Maybe if you have learned one way, you find it hard to imagine not doing that can be sound.

Isn't it true that most people never forget how to differentiate, but a year after a course on integration you will have forgotten it mostly unless you have used it fairly constantly?

Maybe for many courses in physics teachers could stop pretending they are solving equations, and just demonstrate the solutions?
At least I recommend to students if it is at all heavy, go straight to the solutions given, check that they are solutions, that may make it more obvious how they were solved.

And whatever the arguments, even if it is better to know than not to, you have to set that against the time and effort taken away from other things it would be desirable to know. I suspect that there are so many of these that in the long run this technique of no great cultural value will not be able to hold its place.


*"In a sense the only way to solve a differential equation or integration is to know the answer, or at least enough of its shape to be able to hammer it till it fits. I admit to subversive questionings about whether therefore it is a valid thing to teach or give so much emphasis on it and integrations, given that it will be used only by real professionals, and for understanding physics you only need those answers, which you can check."

 

[Ben Niehoff]

I'll throw in my two cents here...

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution. When talking of simple, one-dimensional integrals, the reasons are not so apparent; you can, after all, plot a graph of the solution either way. However, my recent course on Jackson's electrodynamics makes a good example of the value of analytical solutions, because vector fields and potentials in 3-dimensional space are not so easy to visualize (and even harder in higher-dimensional spaces!).

The best example I remember from the class is the problem of a (positive) point charge in the region exterior to a (positively) charged, conducting sphere. What does the field look like? What is the behavior of the system when the point charge is in the regions A) very close to the surface of the sphere, B) near the sphere, but not extremely close, C) far away from the sphere, and D) in the limit at infinity?

With a numerical solution, the best you can do is plot values and draw a picture. If you know the right plots to make, you might learn something. But making the right plots takes a bit of guesswork. You can gain a much more general understanding by deriving the analytical formula and taking the appropriate limits. You would discover:

A) Very near the sphere, the point charge is actually attracted (despite being of the same sign), as though the sphere were an infinite, uncharged conducting plane. That is, the force felt is toward the sphere, and constant w.r.t. \alpha, where \alpha is a small compared to R, the radius of the sphere.

B) Just outside the attractive region, the point charge and the sphere repel with a force that goes as the inverse cube of the distance between them.

C) Farther away, the usual inverse square repulsion is felt.

D) At infinity, the force goes to zero.

These are all easy to derive with the formula. How would you get them with a numerical solution?

Good luck with A. You would probably find that there is a region of attractive force, but unless you guessed correctly to plot things in terms of \alpha, you would miss the valuable insight that any conducting surface "looks like" an infinite plane up close.

B would also throw you for a loop. How would you find the region of inverse-cube variation, except by choosing to do power law fitting on that portion of the graph? Certainly that's a waste of computer resources, when a few minutes on paper can get an exact solution.

C you could probably guess, if you know anything about electrostatics.

And for D, finding the limits at infinity are always tricky with numerical solutions. You can graph something that looks like it should go to zero at infinity. But you can't prove it goes to zero at infinity unless you use a formula and take limits.Overall, I would say the ability to take limits is the primary advantage of analytic solutions. It allows you to derive definite trends in the solution, valid in different regions, and you can get a lot of insight on the problem without actually having to calculate anything. Also, being able to see how the mathematics works, symbolically, is what allows you to make connections between different bodies of mathematics, and extend theories in a universal, logical way.

For example, how could you use numerical solutions to demonstrate the equivalence of Newtonian and Lagrangian mechanics? You could write some programs to implement each, and you would find that they gave the same numerical answers, but you could never prove, deductively, that they are the same; all you would have is empirical evidence from your programs. This is no way to conduct a logical proof. The same goes for demonstrating the equivalence of Schrodinger's, Heisenberg's, and the path integral formulations of quantum mechanics.

Not to mention the fact that in order to have numerical methods in the first place, someone has to write the code for them, and that person has to have an understanding of the underlying mathematics in order to implement it correctly and efficiently. These things don't just spring into being, fully-formed. And how could the numerical programs be checked for correctness? By comparing them to known analytical solutions, of course!If all you wish to do is engineering, then for the most part you can get away with numerical solutions. All you really need are correct answers. But if you want to do anything more theoretical, or get an understanding of the problem without having to look up tables of results, then there is value in learning how to do things analytically.

 

[DavidWhitbeck]

I'll throw in my two cents here...

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution.

This is cute because you then proceeded to give an example where you examined qualitative features without reference to the complete analytic solution. You did not make your point with that example!

With a numerical solution, the best you can do is plot values and draw a picture. If you know the right plots to make, you might learn something. But making the right plots takes a bit of guesswork.

What guess work are you talking about? Plot |F| vs |r|. That's the only thing you were talking about in this problem.

These are all easy to derive with the formula.

That's overkill, you don't need the full equation. Everything you stated could be deduced from Gauss' Law without the need for a full analytic solution.

you would miss the valuable insight that any conducting surface "looks like" an infinite plane up close.

That's common sense, a surface is locally flat because it's smooth. That has nothing to do with the analytic solution.

B would also throw you for a loop. How would you find the region of inverse-cube variation, except by choosing to do power law fitting on that portion of the graph?

Well I have an idea... perhaps you could simply note that in the multipole expansion it would be the first term that doesn't vanish. Hey check that out, I didn't even know the formula, and I still saw the truth! Funny that.

And for D, finding the limits at infinity are always tricky with numerical solutions.

This property is again obvious from the de, you don't need the solution to uncover the asymptotes, in fact that is a standard elementary calculus problem to find asymptotes from a de without solving it.

Bottomline: You are confusing the ability to mathematically analyze a problem with finding an analytic solution. Nothing stops you from figuring out qualitative features and using that to find a solution either analytically or numerically. And a good physicist can figure out the qualitative features of the solution without needing to extract the full solution using either method.

 

[Ben Niehoff]

This is cute because you then proceeded to give an example where you examined qualitative features without reference to the complete analytic solution. You did not make your point with that example!

If you wouldn't selectively ignore the important parts of my argument, you would see that my example does indeed make my point.

Bottomline: You are confusing the ability to mathematically analyze a problem with finding an analytic solution. Nothing stops you from figuring out qualitative features and using that to find a solution either analytically or numerically. And a good physicist can figure out the qualitative features of the solution without needing to extract the full solution using either method.

Here you are agreeing with me that you need analytical methods to gain actual insight about the problem. The fastest way to analyze this particular problem is, in fact, to derive the solution using the method of images, and then take limits. Attempting to find facts such as (B) would be extremely difficult from Maxwell's equations alone, if you insisted on never actually solving them.

You also neglected to address my point about who writes the numerical solution algorithms in the first place, and how they are checked for correctness.

 

[DavidWhitbeck]

Here you are agreeing with me that you need analytical methods to gain actual insight about the problem.

So what? You were debating the value of analytic solutions vs numerical solutions. What, did you actually think that I would reply to your strawman fallacy? Let me requote the thesis of your argument to remind you--

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution.

Now that we've taken care of that let's move right along.

You also neglected to address my point about who writes the numerical solution algorithms in the first place, and how they are checked for correctness.

I was doing you a favor, that supports my argument, not yours. It cleanly debunks the silly image you constructed of a numerical analyst that goes problem --> blackbox --> answer. Coming up with an algorithm, implementing, and extracting results from the solution requires just as much thought, if not more, than using analytic methods. From the bottom of my heart, I thank you for making the point for me.

 

[FrogPad]

Coming up with an algorithm, implementing, and extracting results from the solution requires just as much thought, if not more, than using analytic methods

ahhhhh...?



Got a question for you?
How would you teach a course in E&M? Would you use only numerical methods?


I'm kind of getting lost in your arguments here. Are you saying that we should not teach symbolic integration techniques? Or are you saying something more like, in education we should teach these integration techniques, but for real world engineering problems it is pointless?

And you didn't answer my question about toy problems. If you have an analytic solution to a toy problem, would you use it? Or would you do something more along the lines of show the differential equation (as an example), and just present numerical "answers" to it?

 

[DavidWhitbeck]

ahhhhh...?

If you can find an analytic solution you're done. Coming up with and using a numerical scheme involves (a) the functions involved satisfy all the properties needed for the algorithm to work, (b) the method needs to be not only convergent, but efficient, how do you find the balance? (c) how do you estimate the error? when do you stop the program, and why?

Many times in research the problems that you are trying to solve will be sufficiently complex and pathological that simple methods like Runge-Kutta and Gaussian Quadrature would make for poor choices. It's not as if you can just copy and paste code from Numerical Recipes and call it a day.

Got a question for you?
How would you teach a course in E&M? Would you use only numerical methods?

Thanks, another strawman, wonderful. Look, no we were discussing solutions not methods. You and your cohorts were arguing that analytic solutions offer insight, and numerical solutions do not. I can quote you and others if I need to. I'm getting really tired of it though. I am also sick and tired of having my view misrepresented as the complete antithesis of what you said.

I'm kind of getting lost in your arguments here. Are you saying that we should not teach symbolic integration techniques?

No, and in fact in my very first post on this thread, I said that the real value in teaching differential and integration techniques is that it shores up students' ability to reason and think algebraically.

Or are you saying something more like, in education we should teach these integration techniques, but for real world engineering problems it is pointless?

Yes that is what I'm saying. The complexity of physics/chemistry/engineering research problems far exceed what you see in elementary calculus. You need to understand calculus for all of those fields, but you won't be solving textbook calculus problems on a regular basis.

And you didn't answer my question about toy problems. If you have an analytic solution to a toy problem, would you use it? Or would you do something more along the lines of show the differential equation (as an example), and just present numerical "answers" to it?

I wouldn't necessarily do either. And if you carefully read my reply to Ben, you will understand why. Let me give you an example from when I am teaching physics. A common mechanics problem is to identify the terminal velocity given free fall in the presence of a drag force that varies as some function of velocity. Now, I know perfectly well the complete analytic solution to the problem. Do I use it? Do I show it to my students? No. Why? You don't need it, and it doesn't offer physical insight that is not present in Newton's 2nd Law. It was the same with Ben's example. You did not actually need the solution to deduce the properties that he mentioned. Now before you say this, because God knows you will... I'm not saying that you never have to solve the equations you write down.

The entire point of one of my earlier posts was that one should not confuse the method with the solution. You and Ben are treating them as interchangable, and that is why you are so clearly confused and so poorly misrepresenting my posts. Analyzing the physics of a problem is not necessarily the same as solving an equation.

 

[DavidWhitbeck]

I am going to try one very last time to state my points.

(1) Doing elementary integrals is not a routine aspect of physical science and engineering at the research level.

(2) However all of that tedious drilling in calculus does help students shore up their ability to think and reason algebraically.

(2) At any level in your education, you can find insight in either analytic solutions or numerical solutions.

(3) Analyzing physics problems and gaining insight does not necessarily imply a need for a complete solution.

(4) Challenging problems in research are solved numerically more often than not. But insight is still readily gained from their solutions.

(5) And I in no way suggested that numerical methods should be taught to the exclusion of anything else, thank you very much!

Alright I'm done.

 

[maze]

I agree with DavidWhitbeck here.

 

[FrogPad]

I am going to try one very last time to state my points.

(1) Doing elementary integrals is not a routine aspect of physical science and engineering at the research level.

(2) However all of that tedious drilling in calculus does help students shore up their ability to think and reason algebraically.

(2) At any level in your education, you can find insight in either analytic solutions or numerical solutions.

(3) Analyzing physics problems and gaining insight does not necessarily imply a need for a complete solution.

(4) Challenging problems in research are solved numerically more often than not. But insight is still readily gained from their solutions.

(5) And I in no way suggested that numerical methods should be taught to the exclusion of anything else, thank you very much!

Alright I'm done.

Hey man, don't get all feisty here. I'm very glad you brought your argument on the table, because it was needed. I again will say, I agree that saying:

"Analytic solutions offer insight. Numerical solutions do not" ---> IS WRONG.

However, I want to remind you that this thread started with "why learn integration techniques". All of my arguments were from more of an educational standpoint. I misinterpreted you. I honestly thought, that you were saying that symbolic integration techniques should be disregarded. I didn't understand this, because I couldn't dream of how you would teach engineering classes... I mean you have to have a foundation, and to think algebraically before getting into numerical routines (in my opinion) is very important. Personally, I remember those toy problems very well and like to build up from there. So having an analytic solutions gives me an idea of a very simple (and NON-REAL) solution and offers tremendous insight. As an example, in Control Theory a very powerful example is looking at controlling the dynamics of a car via a basic cruise control scheme.

What's the first thing you do with a problem like that? Neglect. Neglect. Neglect. It becomes unreal very fast. However, when learning the ideas is it really that important to model wind with extremely complicated models? nahh... the very basic control ideas can be taught without resorting to numerical methods. So for the foundation, learning the analytic methods is very important (in my opinion). However, once you get past the elementary stuff, numerical methods pick up.

If you were arguing the points (1) - (5) that you maid I agree with you.However, I do not agree with this part of what you said: (well I do... but not completely).

If you can find an analytic solution you're done. Coming up with and using a numerical scheme involves (a) the functions involved satisfy all the properties needed for the algorithm to work, (b) the method needs to be not only convergent, but efficient, how do you find the balance? (c) how do you estimate the error? when do you stop the program, and why?

Finding the analytic solution is far from trivial (we are not just talking about integration techniques anymore right?).

I can't remember the exact quote but it was something to do with the Russians and the United States during the space-race times. It was something like, "when solving a challenging problem, the US throws more computers at it, while the Russians just invent new math".

I totally butchered the quote, but I hope the point remains. Obviously neither are trivial. Writing a numerical routine is complicated, and solving a problem analytically is also complicated. I just want to restate that solving analytically can be extremely and impossible.

 

[Ben Niehoff]

I was doing you a favor, that supports my argument, not yours. It cleanly debunks the silly image you constructed of a numerical analyst that goes problem --> blackbox --> answer. Coming up with an algorithm, implementing, and extracting results from the solution requires just as much thought, if not more, than using analytic methods. From the bottom of my heart, I thank you for making the point for me.

Nonsense. If you're going to split hairs over "methods" vs. "solutions", then a numerical solution IS a blackbox answer to a problem!

The method of writing a program to give an efficiently-convergent answer requires both extensive knowledge of how algorithms are implemented on computers (e.g., what sort of calculations are fast), and strong skills in traditional, pencil-and-paper mathematics, in order to take the mathematical methods and break them down into steps that can be efficiently calculated on a computer.

Remember the original question of this thread: "Why learn integration techniques?"

I'm pretty sure you are not actually arguing against the teaching of integration techniques anyway, so the rest of this is moot. No one could ever write a program to do integration without some prior knowledge of integration techniques, which is a point you've emphatically agreed to.

As to whether people in professional positions actually do symbolic calculus in their day-to-day duties, it really depends on the demands of the profession. Engineers, experimentalists, financial analysts, I'm sure use numerical solutions almost exclusively. But theorists and mathematicians need to work with analytical methods. They might use a symbolic package such as Mathematica to save themselves some tedium, but that is not strictly a "numerical" method anyway.