For simple sets, we have geometric length, area and volume. But how can we establish these dimensions for complicated curves, areas and volumes. Integral calculus provides a powerful tool for answering such questions. The area $latex {A}&fg=000000$ between the curve $latex {y=y(x)}&fg=000000$ and the $latex {x}&fg=000000$-axis is $latex \displaystyle A = \int_{x_1}^{x_2} y(x) \mathrm{dx} \,. &fg=000000$ … Continue reading Vertical or Horizontal Slices? Riemann and Lebesgue Integration. →

A function $latex {f(x)}&fg=000000$ that is differentiable at a point $latex {x}&fg=000000$ is continuous there, and if differentiable on an interval $latex {[a, b]}&fg=000000$, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable … Continue reading CND Functions: Curves that are Continuous but Nowhere Differentiable →

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it ``Continuous Calculus'' but indicates that the name ``Topological Calculus'' is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)]. Motivation Students embarking on a university programme in mathematics … Continue reading Topological Calculus: away with those nasty epsilons and deltas →

Many of us have struggled with the vector differential operators, grad, div and curl. There are several ways to represent vectors and several expressions for these operators, not always easy to remember. We take another look at some of their properties here. We consider a vector $latex {\mathbf{V} = (u, v, w)^{\mathrm{T}}}&fg=000000$ which may be … Continue reading Curl Curl Curl →

The Approximating Functions It is simple to define a mapping from the unit interval $latex {I := [0,1]}&fg=000000$ into the unit square $latex {Q:=[0,1]\times[0,1]}&fg=000000$. Georg Cantor found a one-to-one map from $latex {I}&fg=000000$ onto $latex {Q}&fg=000000$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor's map was not continuous, but … Continue reading Space-Filling Curves, Part II: Computing the Limit Function →

We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be … Continue reading Space-Filling Curves, Part I: “I see it, but I don’t believe it” →

It is a simple matter to post a paper on arXiv.org claiming to prove Goldbach's Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored. Mathematicians establish their claims … Continue reading De Branges’s Proof of the Bieberbach Conjecture →

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is $latex {\cos \pi/3 = 0.5}&fg=000000$. More generally, for an N-gon the ratio is easily shown to be $latex {\cos \pi/N}&fg=000000$. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular … Continue reading Circles, polygons and the Kepler-Bouwkamp constant →

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds … Continue reading Differential Forms and Stokes’ Theorem →

Mathematicians have traditionally dealt with convergent series and shunned divergent ones. But, long ago, astronomers found that divergent expansions yield valuable results. If these so-called asymptotic expansions are truncated, the error is bounded by the first term omitted. Thus, by stopping just before the smallest term, excellent approximations may be obtained. Astronomical Series Many of … Continue reading Divergent Series Yield Valuable Results →

If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don't be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole … Continue reading The Wonders of Complex Analysis →

Which is greater, $latex {x^y}&fg=000000$ or $latex {y^x}&fg=000000$? Of course, it depends on the values of x and y. We might consider a particular case: Is $latex {e^\pi > \pi^e}&fg=000000$ or $latex {\pi^e > e^\pi}&fg=000000$? We assume that $latex {x}&fg=000000$ and $latex {y}&fg=000000$ are positive real numbers, and plot the function $latex \displaystyle z(x,y) = … Continue reading Which is larger, e^pi or pi^e? →