Office Hours: Tuesdays, 2-4pm (or the day before homework is due)
Extra materials:
A summary of calculus without limits (i.e., the old-fashioned differentials): calculus.pdf
There is a rigorous approach to calculus with infinitesimals by the name of non-standard analysis. This textbook by H. Jerome Keisler is supposedly an introduction for freshmen, though when I did look into it I didn't really like it.
The best book I've come across is the delightful and short Calculus Made Easy by Silvanus Thompson.
Projects:
The history of mathematics is largely being neglected in all the math classes. For whatever topic you do, digging up some history of it would always be interesting. The expository journal Mathematical Intelligencer and the encyclopedic Princeton Companion to Mathematics have many well-written articles on various aspects of college-level mathematics, and you should have access to the former on Cornell's network. If you have spent many hours on a topic but couldn't come to grips with it, come talk to me (and don't feel obliged to switch to a different topic: Your write-up (or oral presentation if you prefer?) needs not be a complete solution; a report of whatever understanding you have achieved would be good enough.) Again, the purpose of all this is to broaden your perspectives on mathematics beyond the textbook materials (which I'm sure will also shed new lights on familiar concepts), and most importantly to have some fun with it. You are encouraged to work in groups, and feel free to do as few or many of the topics below as you like.
I will update the list as we go on. And if you know some other topic which is related to your intended field of study, go for it! You may come to me for suggestions.
Couple of other books:
Dunham, William: The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton, 2005. Or the short sample version published on American Mathematical Monthly.
Woods, R. Grant: Calculus Mysteries and Thrillers.
List of topics:
I didn't talk about parametric equations (except the example of the cycloid), but they are actually very useful for dealing with plane curves, a fertile ground for playing with infinitesimals. In each topic below, you may ask: How would you figure out the tangent at a particular point? What happens at a self-intersection (e.g., see 1 below)? How would you calculate the length of the curve? The area enclosed by the curve, if it is closed? Come up with your own questions!
One can come up with many pretty curves like $r=\sin(4\theta)$ (graphing by google, or WolframAlpha, or Grapher on your Mac). One way to compute the area is $$\frac{1}{2}\int r^2\,d\theta$$or alternatively you may easily get $x$ and $y$ as functions of $\theta$, and calculate $$\int x\, dy \qquad \text{or}\qquad -\int y\, dx$$by expressing everything in $\theta$. What interval of $\theta$ should be used? And why do these integrals work?
All the conic sections (ellipse, parabola, hyperbola) can be written beautifully in polar coordinates, with the origin at one of the foci. \[r=\frac{a}{1-e\cos\theta}\][$e$ being the eccentricity, not 2.718...] How does this equation make sense with the definition(s) of conic sections you know of? Show that lights coming from the origin will bounce off the curve to converge at (or diverge from) the other focus. Can you demonstrate that the area of the ellipse is the same as the calculation based on Cartesian coordinates: $A=\pi a b$? [The circumference of an ellipse, though, is notoriously difficult, see below.]
Along the same lines of the cycloid, what happens if a circle is rolling around another circle? How would it depend on the relative radii and what if it's rolling "on the inside''? Look up deltroid, and see if you can figure out its length and area.
There are many classical objects of study in calculus that seem to have fallen out of vogue in modern textbooks. Learn about envelopes: curves that seem to be "enveloped" by a family of straight lines or circles. One famous example is to connect each degree mark on a big full circle with twice the degree by a straight line, and all these lines together seem to make out a heart-shaped curve, known as the cardioid (Try it yourself on paper!). Figure out the points on the cardioid (a classic reasoning with infinitesimals) and try to find the length and area of it. Here is a nice video about it.
Try to read the solution to the Brachistochrone problem (pp. 657-658, or on wikipedia and the reference therein), or alternatively another classic problem of the hanging-chain curve (catenary). This is another branch of calculus, called "calculus of variations," that had an early start immediately after Newton and Leibniz but turned out to be more profound both on the mathematical side and on the physical implications. [It is sad that the quest of a rigorous foundation resulted in the neglect of an important branch of the subject.] If this seems too confusing, you may attemp to compute the time it takes to descend along the brachistochrone curve (upside down cycloid) and compare with a couple of other simpler paths that you come up with. Also check out this blog post which contains a curious fact that I didn't know before.
(Higher-dimensional geometry) If you look at the integrals for calculating the area of a circle and the volume of a sphere, it only takes a bit of imagination to speculate higher-dimensional analogues of spheres, defined as the set of points within a fixed distance from the origin: \[\{(x_1,x_2,\ldots, x_n)\in\mathbb R^n \, |\, \sum_{i=1}^n x_i^2\leq r^2\}\][The standard terminology for it is actually an $n$-ball, and the name $n$-sphere is reserved for the boundary surface of the $(n+1)$-ball.] So you can continue the sequence \[A=\pi r^2 \qquad V=\frac{4\pi}{3}r^3 \qquad V_4=?r^4 \qquad V_5=?r^5 \qquad \cdots\]and differentiating these (with respect to $r$) will give us the boundary spheres: \[C=2\pi r \qquad S=4\pi r^2 \qquad S_3=?r^3 \qquad S_4=?r^4 \qquad \cdots\]Can you find the general formulas for $V_n$ and $S_n$? One interesting fact is that, as the dimension goes up, a stack of tightly-packed spheres is leaving more room in between. See if you can stack up five (hyper)spheres in 4 dimensions, by analyzing how it works in 2 and 3 dimensions. How would one measure the density? The sphere-packing problem has a fascinating history (which is ongoing! Look up Kepler's conjecture).
(Kepler) Perhaps the single most important application of calculus, in terms of solidifying it as a cornerstone of mathematics and science, and how much it has changed the course of human history (not an exaggeration!), is Newton's derivation of Kepler's Laws of planetary motions from the law of gravitation and the three laws of mechanics. In particular, the first law of Kepler, that the planet goes around the sun in an ellipse (see above), is the most difficult. You probably can find some readable explanations on the web (if you don't feel like reading the Latin original). Also check out the article by Heckman on Mathematical Intelligencer which gives three proofs (and dispels the myth that Newton's proof was incomplete).
(Rainbow) The phenomenon of the rainbow was essentially explained by Descartes in his treatise Optics, which along with Meteorology and Geometry (yes, the one in which he laid out the coordinate system that bridges algebra and geometry) are in fact the appendices to his Discourse on the Method. Firstly, the law of refraction follows from Fermat's principle as an easy application of calculus (see p.267 of our textbook), and knowing that light travels in water about 3/4 the speed it does in air (or vacuum), we could calculate (after understanding Descartes' and Newton's explanation of the optics, if you didn't already know) the magical angle of $42^\circ$ that you see the rainbow span (from center to rim). If you see a double rainbow, the outer one is at $51^\circ$ (a similar calculation), though dimmer and reversed in colors. Only geometry and differentiation are involved.
(Fractional derivatives) It is said that Feynman invented fractional derivatives all by himself (I'd be interested to know how true this is). One formulation of fractional derivatives is by way of Riemann-Liouville integrals (should only involve a simple application of integration by parts). You may want to read here and here.
(Elliptic integrals) As you know, some integrals such as $\int e^{-x^2} dx$ turn out to be impossible to do (meaning impossible to express in terms of elementary functions). Another example comes up in calculating the circumference of an ellipse, which has inspired a significant portion of mathematics since. It gives the name to elliptic functions and elliptic curves (which are definitely NOT ellipses!), which incidentally played a crucial role in the proof of Fermat's Last Theorem (around the time you guys were born). It'd be too much for a calculus project to learn all this (I know very little myself), but the history of the beginning of it should be accessible. I will try to find a good source for it.
In the method of partial fractions, we noted that some quadratic factors, such as $x^2+1$, may not be broken further down, but if we allow complex numbers, it becomes $(x+i)(x-i)$. So we'd get \[\int\frac{dx}{x^2+1}=\frac{1}{2i}\int\left(\frac{-1}{x+i}+\frac{1}{x-i}\right)dx = \frac{1}{2i}\log\left(\frac{x-i}{x+i}\right)+C\]Seems all legit, except that it requires us to define the log of a complex number. We will mention a little more of functions of a complex variable later in the course, and there's a whole course dedicated to this subject, called complex analysis, for which this investigation would serve as a good motivation. See if you can make a sensible definition of log over the complex numbers, such that the complex-valued function $\log(x+i)$ of the real variable $x$ satisfies \[ \frac{d}{dx} \log(x+i) = \frac{1}{x+i}=\frac{x}{x^2+1}-i\frac{1}{x^2+1}\]What happens to this complex $\log$ function along the negative $x$-axis?
Euler was a master of infinite series (really a master of all the mathematics of his time), and two of his most famous results were featured in Prof. William Dunham's public lecture last year. They are \[\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots = \log \log \infty \qquad\text{and}\qquad 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots = \frac{\pi^2}{6}\] (The first being the reciprocals of all the prime numbers, the second being the reciprocals of all the squares.) Try to follow the steps, and maybe you could improve on some of them.
(Continued Fractions) To start, consider the expression \[1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}\] which continues forever. What does that even mean?
A nice application of geometric series, similar to the one that you saw on Prelim II, is the calculation of the area of the "Koch snowflake," defined as an infinite iterative process (see figure). More surprising is the fact that its "circumference" is infinite! Try to figure it our on your own before looking it up. [In fact, this boundary curve is a perfect example of a continuous, but nowhere differentiable curve, as long as we can mathematically describe this curve.]
