| Some notes on trigonometry. Surely you already
know trig (and you won't learn it from this page,
anyway). But if you don't know why it's called a cosine
or
an arctan you might find this interesting. |
Some
Trigonometry |
| On this page, we
find the value of π directly from its definition, using
nothing more sophisticated than Pythagoras's theorem. |
Pi
from
Pythagoras |
| Continuing in the
same vein, we find the area of a circle, the volume of a
sphere, and the surface area of a sphere using simple
geometric arguments. |
Volume
of
a Sphere |
The common proof that the square root of 2
is irrational works by contradiction, and is unsatisfying
(to me, anyway). On this page we present what I
think is a more intuitive proof that the square root of 2,
along with all non-integer roots of integers, is
irrational.
|
Irrational
Roots
|
| Calculus of a
single variable. In this collection of pages, I
present visual arguments to motivate a number of the basic
facts about calculus, which are often presented only with
"symbol pushing" arguments and little actual explanation. |
Calculus |
| Some basic facts about the hyperbolic functions, and
what they mean for special relativity. |
Hyperbolic
Rotations |
| The edge of a slice through a cone forms a circle,
ellipse, parabola, or hyperbola. This page contains
a proof of that assertion. |
Conic
Sections |
| A point-by-point comparison of some basic properties of
the circular trig functions with the hyperbolic trig
functions. It doesn't add much of anything but I
found it entertaining to write; maybe someone will find it
entertaining to read. |
Circular
versus Hyperbolic Functions |
| Solid angles:
Definition of steradians, and derivation of the
formula to find the solid angle subtended by a cone. |
Solid
Angles |
| Integers, rational
numbers, and real numbers. The axioms for all of
them are presented and we develop models for all of them.
(This is not really "basic" material but it fits
better here than anywhere else on the website.) |
Numbers |
| A general
discussion of ideal lenses and mirrors, some aspects of
real lenses and mirrors, the brightness of an image in a
camera, in your eye, and when viewed through a telescope,
and, possibly, some other items as they come up. |
Simple
Optics |
This includes a handful of results which I
think are really kind of cool which can be derived just
from the second law of thermodynamics. In order to
get to the results we also include a very brief
introduction of the first two laws of thermodynamics and a
brief introduction to the concept of a blackbody.
This page uses almost no mathematics; the results
are derived by simple arguments and a few diagrams.
|
Blackbodies
|
| A very brief introduction to Lagrangian mechanics |
Lagrangians |
| A very brief essay on the dangers of "rigid rods" and
similar nearly-physical connectors. |
Rigid
Rods |
| Do you find the fact that the traveling twin ages more
than the stay-at-home twin paradoxical, since from the
point of view of the traveler, the stay-at-home twin is
the one who moved? Simple explanations of time
dilation obscure more than they explain. |
Against
Gamma |
