What's the deal witih Calculus?

[rcwlhk]

Hey guys,

I understand a key requirement for Econ PhD programs is Calculus and specifically multivariable calculus.

Question:

-How much calculus do you need (I'm talking about "cook book" calculus here and not real analysis)?

-Are there any applications to:
(i) Double and triple integrals
(ii) Green's Theorem and Stokes' Theorem
(iii) Surface Integrals
(iv) Divergence Theorem
(v) Various other Calc III - IV concepts?

Almost all of my Calculus classes so far only talks about applications of these concepts in physics (i.e. surface integrals for flux, double and triple integrals for volume, etc.)

Thanks!

[YoungEconomist]

Originally Posted by rcwlhk

Almost all of my Calculus classes so far only talks about applications of these concepts in physics (i.e. surface integrals for flux, double and triple integrals for volume, etc.)

I've wondered about this as well. In my Calc 3 class we talked a lot about vector calculus. I kept thinking to myself, I doubt I'll be using this in Econ grad school. Someone correct me if I'm wrong.

I'm not in grad school yet, so take this with a grain of salt. I imagine there's many things in learned calculus that we won't use in grad school. You might ask, "then why do we need so much calc?" My answer to that would be two fold. First, there's always the signalling value of being able to succeed in pure math courses. Second, although it's probably true that we don't use everything from the calc series, I imagine that we still use a significant amount of it in grad school. Furthermore, it's important to be fairly comfortable with the calc that is used (as it will come up very frequently). Once again, I hope somebody will correct me if I'm wrong.

[Antichron]

Originally Posted by rcwlhk

-Are there any applications to:
(i) Double and triple integrals
(ii) Green's Theorem and Stokes' Theorem
(iii) Surface Integrals
(iv) Divergence Theorem
(v) Various other Calc III - IV concepts?

I once did a triple integral as an RA, and I very recently saw a paper that used line integrals, but I really think most of the concepts in multivariable calculus are more useful for physicists and engineers than for economists.

[YoungEconomist]

Originally Posted by Antichron

but I really think most of the concepts in multivariable calculus are more useful for physicists and engineers than for economists.

LOL! This is all coming from someone at the most mathematical/technical econ program on the planet. I guess it's a safe assumption that I'll feel the same way in grad school.

[andrem]

I think at least for undergrad studies there a lot of non-useful math out there. Can't really say about grad studies although.

It depends a little on the field. A eminent Brazilian economist, J.A Sheickmann, who's now professor at Princeton always say in interviews that many of his recent ideas appeared after sharing thought with physicians, since he works with complex systems.

I really can't say a thing about his research since it's too advanced for me, but it clearly shows that very advanced math is necessary at some points depending on the field you are in.

Since grad economics is often much more advanced than undergrad econ, I prefer to believe that is really necessary since ever each program gives a lot of value to math, sometimes more than they give to econ studies before grad school.

[rcwlhk]

Then this really makes me wonder about a lot of things.

Namely, I understand that the whole point of "MULTIvariable" calculus is for you to learn partial derivatives (and maybe integrals?). But these concepts were long introduced to me back in Calc I and II. And similarily, I know Lagrange multiplier is important and even that, I learned back in Calc II (in single variable, but it isn't that hard to extend to the multivariable case). The Calc III sequence here is all about computing, among other things of course, double and triple integrals. Calc IV is basically just vector calculus, from paramaterization of planes and surfaces to Divergence Theorem.

Here's my two cents on this whole calculus gig (disclaimer: I'm not in econ grad school yet). Let's face it --- of all the math courses that we do in undergrad and beyond, you need to be able to "think" mathematically. That is, at the very least, you shouldn't be afraid of math symbols and equations. Calculus does provide this foundation very well. So, if an applicant / student can't even do Calculus (again, "cook book" style Calculus), it makes you wonder how well can he fair in other more advanced courses.

I don't necessarily agree that doing well in Calc I - IV (which is typically "cook book" style vs. Real Analysis) means you will be good in doing pure maths. For instance, in the Calc I - IV, when we talk about sequences and series, you have a bunch of tests and definitions for convergence and divergence --- without ever defining what is supremum / infimum and how it is related to convergence / divergence.

Originally Posted by YoungEconomist

I've wondered about this as well. In my Calc 3 class we talked a lot about vector calculus. I kept thinking to myself, I doubt I'll be using this in Econ grad school. Someone correct me if I'm wrong.

I'm not in grad school yet, so take this with a grain of salt. I imagine there's many things in learned calculus that we won't use in grad school. You might ask, "then why do we need so much calc?" My answer to that would be two fold. First, there's always the signalling value of being able to succeed in pure math courses. Second, although it's probably true that we don't use everything from the calc series, I imagine that we still use a significant amount of it in grad school. Furthermore, it's important to be fairly comfortable with the calc that is used (as it will come up very frequently). Once again, I hope somebody will correct me if I'm wrong.

[unitroot]

What you really need to know from multi-variable calculus for economics is

- vectors and planes in R^n.
- the functions and their derivatives in R^n, chain rule, product rule, including a working knowledge of matrix notation of derivatives.
- constrained and unconstrained optimization, including problems subject to inequality constraints and the second order conditions.
- implicit function theorem in R^n, comparative statics.

Unfortunately, most of this is NOT taught in a multi-variable calculus course, even though the adcoms really want to see that course on your transcript. A typical sophomore calculus course is designed for engineers, and so you spend a while learning things like polar coordinates, greens and strokes theorem, line integrals, etc, usually just in R2 or R^3.

In addition, it is useful to have a good intuition to work with sequences, limits, convergence, continuity, closed and open sets, compactness, norms, Cartesian products of sets, etc in R^n (again, this stuff if taught at all is usually taught in a real analysis course).

[TruDog]

I just finished working on a game theory problem that required the use of double integrals (involving whether contestants should take a second spin or not on the American game show The Price is Right). With that being said, I've only needed to use multiple integrals a handful of times all year.

I haven't used any of the other topics at all this year, so Antichron is right on as usual.

[cooper]

Originally Posted by andrem

It depends a little on the field. A eminent Brazilian economist, J.A Sheickmann, who's now professor at Princeton always say in interviews that many of his recent ideas appeared after sharing thought with physicians, since he works with complex systems.

Surely you mean physicists not physicians unless he is having conversations about the human body as a complex system--no pun intended .

As far as computational calculus, meaning computing derivatives, integrals, etc, integration by parts pops up in auctions (allthough this is really calc 2) and I remember seeing this other places too but can not remember. As others have mentioned, in macro, multivariable differentiation should be second nature. Allthough its not necessary to get through first semester Micro (using Mas-Collel et. al.) you should understand gradients to build intuition when utility functions are differentiable.

[Antichron]

Upon re-reading my post and the responses, I thought that I should clarify my previous post a bit, in line with other comments. Whereas most of the multivariable calculus topics related to integration do not come up often in my experience, topics related to differentiation (partial derivatives and constrained optimization) come up all the time, of course. :-)