MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Post Made Community Wiki by S. Carnahan
3 added 20 characters in body

I taught a problem solving class for advanced first year students at my university some time ago with the aim of showing them interesting aspects of university mathematics. It basically turned into a baby manifold course - with the goal of understanding the concept of de Rham cohomology. The point is that using just a little more machinery than the theory they already knew from calculus (e.g., div, curl, derviatives), it was possible to still prove some quite interesting results like the Brouwer fixed point theorem.

I felt having this 'goal' in the course to be quite effective: the students always had some kind of idea where the course was going and most importantly, they knew why new concepts were introduced. There are a lot of motivating examples and questions to give them, like, 'How can one detect the shape of a space'? This made it easier to motivate new concepts, like vector tangent spaces, differentials,etc.

Even though your students would probably benifit from learning the standard material in calculus (you say they are not math majors), I think you should be able to incorporate some interesting examples like the above into your course. For example, when talking about Taylor series, you could give the nice proof of the irrationality of $e$ or $\pi$. It would certainly make it more fun for you to teach, and probably not take too much of your time.

For reference, here are some of the topics we covered in our course:

Polynomials, cubic equations, symmetric functions, Vieta's relations, special integrals and series, irrationality of $e$ and $\pi$, number theory in finite fields, basic group theory, $\mathbb{RP}^2$, stereographic projection, conics, Bertrand's postulate, divergence of $\sum_{primes}\frac1p$, generating functions,differentible manifolds, vector spaces, tangent spaces, differential forms, Stokes theorem, de Rham cohomology, Brouwer's fixed point theorem, Fundamental theorem of algebra, the fundamental group.

There are plenty of good books you could take a look at for finding interesting examples, e.g., I found the following books helpful.

Proofs from the Book by M. Aigner, G. Ziegler

From calculus to cohomology by Ib Madsen, Jørgen Tornehave

The fundamental theorem of algebra by B. Fine, G. Rosenberger.

I taught a problem solving class for advanced first year students at my university some time ago with the aim of showing the students them interesting aspects of university mathematics. It basically turned into a baby manifold course - with the goal of understanding the concept of de Rham cohomology. The point is that using just a little more machinery than the theory they already knew from calculus (e.g., div, curl, derviatives), it was possible to still prove some quite interesting results like the Brouwer fixed point theorem.

I felt having this 'goal' in the course to be quite effective: the students always had some kind of idea where the course was going and most importantly, why. There are a lot of motivating examples and questions to give them, like, 'How can one detect the shape of a space'? This made it easier to motivate new concepts, like vector spaces, differentials,etc.

Even though your students would probably benifit from learning the standard material in calculus (you say they are not math majors), I think you should be able to incorporate some interesting examples like the above into your course. For example, when talking about Taylor series, you could give the nice proof of the irrationality of $e$ or $\pi$. It would certainly make it more fun for you to teach, and probably not take too much of your time.

For reference, here are some of the topics we covered in our course:

Polynomials, cubic equations, symmetric functions, Vieta's relations, special integrals and series, irrationality of $e$ and $\pi$, number theory in finite fields, basic group theory, $\mathbb{RP}^2$, stereographic projection, conics, Bertrand's postulate, divergence of $\sum_{primes}\frac1p$, differentible generating functions,differentible manifolds, vector spaces, tangent spaces, differential forms, vector spacesStokes theorem, de Rham cohomology, Brouwer's fixed point theorem, Fundamental theorem of algebra, the fundamental group.

There are plenty of good books you could take a look at for finding interesting examples, e.g., I found the following books helpful.

Proofs from the Book by M. Aigner, G. Ziegler

From calculus to cohomology by Ib Madsen, Jørgen Tornehave

Excursions in calculus

The fundamental theorem of algebra by R.YoungB. Fine, G. Rosenberger.

1