Schools did not understand Calculus until Grad School

[symbolipoint]

This has been stated in a few topics in some places in physicsforums; that people did not really understand Calculus until they studied analysis or advanced calculus in graduate school. So what did such people really accomplish when they studied their first-year Calculus and their semester of Intermediate/Multivariable Calculus? How is it that students did not really understand it in their undergrad situations but did understand it as graduate students? What made the difference, specifically?

As a possible comparison, one may say, "I really understood Algebra much better once I finished my three semesters of Calculus"; but what are the details which makes the undergrad Calculus to graduate level Calculus work?

 

[tacosareveryyum]

Thisi s a good question. I would like to know as well.

 

[Brad Barker]

i believe when people say such a thing, they are using the word "understand" in a very particular context.

in your undergraduate calculus studies, you learn the basic concepts of calculus and how to solve problems.

in analysis courses, you look more rigorously at the mathematical foundations of calculus. you get to see exactly the properties of mathematical structures that you only saw the surface of during the first run through of calculus. in that sense, you have gone beneath the veneer of calculus and see it for what it really, truly is, and that is what they mean by understanding calculus.

so you can "understand" everything in your undergrad calc sequence, sure. that's just not the same kind of understanding, though, that you get from an analysis sequence.

it is absolutely unlike saying that you understand algebra better after a calculus sequence, since that would mean that the understanding is achieved solely through strenuous use of that subject. it's more like saying you understand algebra better after taking a class where you learn about field axioms, the real number system, etc.

 

[arunma]

Uh, are we speaking from a physicist's perspective, or a mathematician's? Because if a physics major didn't understand the mechanics of calculus in undergrad, he'd probably never get into grad school. If, however, we're talking about math majors, then I assume that "understanding" refers to the formal theory behind calculus, in which case I see what you mean.

But as for physics grad students who can't do calculus...well, let's just say I have trouble believing that such an indivudal exists.

 

[animalcroc]

Uh, are we speaking from a physicist's perspective, or a mathematician's? Because if a physics major didn't understand the mechanics of calculus in undergrad, he'd probably never get into grad school. If, however, we're talking about math majors, then I assume that "understanding" refers to the formal theory behind calculus, in which case I see what you mean.

But as for physics grad students who can't do calculus...well, let's just say I have trouble believing that such an indivudal exists.

A physics major must understand calculus as well as a math major. Of course a physics major must understand calculus to get into a grad program, just as a math major would. I think the original poster's usage of the word understand is different from your and most engineering student's usage of the word. Engineers usually think they understand calculus just because they can evaluate integrals and do integration by parts. But to understand why it works at the most fundamental level etc is probably what the original poster meant. That is something harder to do.

 

[mathwonk]

ones understanding deepens with experience. it is amusing to hear it is thought possible to understand calculus deeply the first time from a standard undergrad treatment aimed at physicists.

physicists who understand the gauss and divergence theorems physically do have a better chance at intuiting several variable calculus than those without that background.

but the topological subtleties (the dedekind and archimedean properties) of real numbers that are taken for granted in beginning calculus courses are often not even mentioned there, so some physics grad students may know nothing of them, much less the concept of everywhere continuous but nowhere differentiable functions, or non constant continuous functions with derivative almost everywhere zero.

Or how to define the derivative even of a nowhere continuous function as an operator on smooth functions, using integration by parts.

I myself obtained admittance to a PhD program in math without much deep understanding of calculus, and only began to get it from reading Spivak's book as a third year grad student. At the same time, I had studied and received good marks in, several advanced real analysis courses in undergrad school, including banach and hilbert spaces, and abstract measure and integration, including spectral theory in hilbert space! But I did not at all understand these things.

so the title of this thread certainly applies to me.

Hence I assume the discussion here is foundering on a difference of interpretation. Physics students indeed must excel at understanding the intuitive meaning and uses of calculus, but may understand little of the theoretical underpinnings and exotic extensions of it.

Advanced physics students however will learn also at least the exotic extensions, as they help solve differential equations and should have applications to sub atomic physics.

 

[animalcroc]

I believe most grad students understand that a derivative is a rate of change and integration is finding the "parent" function and subtracting the change in values at different points. This is the crux of calculus. I bet the people you referred to understand this but
were talking about delving more into calculus.
Physicists understand calculus. Look at Newton. He was a physicsist who co-invented calculus to put physics on another level. Because of Newton's work as a physicist, mathematicians get to play with integrals.

 

[mathwonk]

integration is not finding antiderivatives, that is antidifferentiation.

integration is a process of averaging. the two concepts do not even agree for finding areas of rectangles. i.e. a function which equals 1 between 0 and 1, and equals 2 between 1 and 3, does not have a differentiable parent function, but the graph is a rectangle, hence it has an integral. i.e. an area. still there is a piecewise linear parent function, namely the moving area function for the (discontinuous) graph.

this is exactly the sort of "understanding" I am talking about, on a very elementary level.

on a more sophisticated level, the cantor function is continuous on [0,1], and has derivative equal to zero at almost all points, yet equals 0 at 0 and equals 1 at 1. Hence its derivative is essentially zero, and yet it is not constant.

This means it cannot be recovered by integrating its derivative. I.e. this function is apparentkly the parent function of the zero function, but does not comoute the integral of the zero function.

the point is a riemann integrable function does not have to be continuous, as shown by riemann himself on the next page after he defined the integral. The function only has to be continuous off a set of measure zero.

But an integrable function does have a continuoius "parent" function, obtained by integrating it from a to x, i.e. the indefinite integral/ This funtion does evaluate the integral by subtracting its values at a and b, but how does one recognize such a parent function?

The parent function will have derivatiuve equal to the original function oiff a et of measure zero, but this is not enough, one also needs to require the parent function to be lipschitz continuous, in order to rule out counter examples like the cantor function.

the easy rule that the integral of can be computed by a function whose derivative equals f, is only useful when f is continuous, since other integrable functions will not have such antierivatives.

very few beginning calculus students, especially physics majors, have mastered these details of the calculus, and yet i taught this in my first semester honors calc course a few years back.

however Newton indeed knew some of this, including that inegrable functions may not have antiderivatives, since he proved, well before riemann that all monotone functions have integrals, even though such functions may have infinitely many discontinuities.

in advanced physics it seems to me that discontinuous functions are encountered, hence one needs a deeper grasp of the underlying mathematics, e.g. in quantum mechanics.

i love the suggestion that since Newton understood calculus, so do (all?) physicists. i thought we were talking about typical physics students, not geniuses.

i thought the issue here was whether the average student understands calculus well before grad school, not whether Newton did so.

 

[mathwonk]

i skipped calc in college, or did not study, same thing, and took then banach space calc ala loomis sternberg. as a first year grad student i was facing prelims in analysis and admitted to jerry kazdan i did not know how to integrate X^(-2).

He looked at me aghast and said, "you haVE NO CHANCE of passing!"

But i knew better, as I said inwardly, "oh yeah? they aren't going to ask that!"

and they didn't, they asked about the meASURE OF THE GRaPH OF A FUNCTION OF BOUNDED VARIATION AND OTHER ESOTERIC MAFTERS WHICH I HANDLED APpARENTLY WELL ENOUGH, As I PASSED.I always thought passing tests is an art, largely unrelated to understanding the material. As a teacher I try now to contradict that philosophy.

I.e. I try to make my tests unpassable by students who (as I was) are merely clever or smart and creative, but only by ones who actually know the material.

e.g. a student who is clever but untutored cannot answer : "state the radon nikodym theorem", but can possibly answer: "prove that a lipschitz continuous function whose derivative is zero a.e. is constant on intervals."

 

[mathwonk]

well to be honest, i try to pay attention to both groups of students. i.e. an ignorant student who can construct a proof of a significant theorem will probably pass my test. but i do try to make it also passable by an average student who has merely learned the material.

i.e. just show me something! either raw power or diligence.

 

[Emmanuel114]

i skipped calc in college, or did not study, same thing, and took then banach space calc ala loomis sternberg. as a first year grad student i was facing prelims in analysis and admitted to jerry kazdan i did not know how to integrate X^(-2).

He looked at me aghast and said, "you haVE NO CHANCE of passing!"

But i knew better, as I said inwardly, "oh yeah? they aren't going to ask that!"

and they didn't, they asked about the meASURE OF THE GRaPH OF A FUNCTION OF BOUNDED VARIATION AND OTHER ESOTERIC MAFTERS WHICH I HANDLED APpARENTLY WELL ENOUGH, As I PASSED.

 

[threetheoreom]

I am just working for an equilibrium of both qualities

 

[animalcroc]

Of course differentiability,etc are one of a hundred ways to determine a function's integral. That's common knowledge.

Newton's and Leibniz calculus was primitive by today's standards and yet his idea of differentiation and integration is still considered to be the crux of calculus. Newton did not know about Cantor function and today's extensions and modifications but did that stop him and leibniz from showing the world that instantaneous rates of changes and antiderivatives exist? No. So how did Newton understand calculus if all he knew was primitive by today's standards?
I don't believe today's extensions change the fundamental concept of calculus.
I agree that *most* physicists don't need to know calculus as well as mathematicians, but they nevertheless aren't clueless about it. They know *much* more than Newton and Leibniz did.

 

[Howers]

Isnt calculus just breaking things into infinitely small parts and finding their gradient?

 

[animalcroc]

arunma:
I did not mean to quote your post. I agree with the "formal understanding" part.

 

[trinitron]

No..

 

[mathwonk]

To return to the OP's original question, I would say the main difference for me in understanding calc as a grad student, compared to the rudimentary grasp of an undergrad, was precisely getting away from the idea that integration means the same as antidifferentiation.

For computing easy elementary integrals, all an undergrad needs to know is to find the antiderivative. To understand more difficult integrals he needs to return to the meaning of the integral as a limit of sums.

The clear sign of someone who does not have an advanced grasp of calculus, is he thinks integration MEANS antidifferentiation. To have a deeper grasp, one needs to understand the two processes as different things, and be able to investigate the relation betwen them.

This is crucial e.g. in understanding path integrals in complex analysis, and in general understanding the integral of non conservative vector fields, i.e. those which are not gradients and hence have no antiderivative.

geometrically, integration is about finding paths through vector fields.

 

[Nolen Ryba]

The clear sign of someone who does not have an advanced grasp of calculus, is he thinks integration MEANS antidifferentiation.

How is this even possible? When the integral is first introduced one knows the integral of a function is a number (area under curve) and the antiderivative of a function is a function.