Our focus is on the region under the circle above the x-axis. The question is: what is the maximum area that a triangle inside this region can have?
It may occur to you that there is a reasonable `quick’ answer, but the point of the problem is to reason it out carefully so you more or less have a proof that you do indeed get a maximum area. Since the region is concave, the vertices of a triangle cannot be so that one is too close to the far right while another vertex close to the far top left (or else the triangle would not be fully inside the region).
In the diagram shown we see a big circle of radius R that is tangent to both the x and y axes. What is the radius of the little circle to its southwestern corner? Express its radius in terms of R. (The little circle is also required to be tangent to the axes. The diagram is draw for a circle of radius 2 but we’re working with a general radius R.)
After having solved this problem, the next question will be this:
You can easily see that we can repeat the process by looking at the southwestern circles for ever and ever because there will always be a gap, and you get smaller and smaller circles with small radii. What if you add up all their radii? Do they add up? If so to what number do they add up, in terms of the original radius R. If they don’t add up, why don’t they?
The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of K.
One of the amazing facts about this radiation is that it almost perfectly matches Planck’s radiation formula (discovered in 1900) for a black body:
In this formula, is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and is the energy density as a function of frequency .
The other variables are: is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), is the speed of light in vacuum, is Planck’s constant, and is Boltzmann’s constant from statistical mechanics.
If you plot the graph of this energy density function (against ) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:Here you see various plots of Planck’s function for different temperatures . The horizontal axis labels the frequency , and the vertical gives the energy density per frequency. (Please ignore the rising black dotted curve.)
You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.
To those of us who know calculus, we can actually compute what frequency gives maximum density and give our answer in terms of the temperature . All we do is compute the derivative of with respect to and set it to zero and solve the resulting equation for . You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:
(The equal sign here is an approximation!)
The is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency gets, and from Planck’s energy equation , so also does the energy of the radiation drop.
Now plug in the observed value for the temperature of the background radiation, which is (degrees Kelvin), and working it out we get (approximately)
This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)
Far back in time, when photons were released from their collision `trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.
Homework Question: what was the max-frequency at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! ;) )
(It isn’t hard as it can be figured from the data above.)
Anyway, I thought working these out was fun.
The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength and frequency for electromagnetic waves we use the simple formula
where c is the speed of light (in vacuum).
And that’s the end of our little story for today!
Cheers, Sam Postscript.
The sacred physical constants:
Speed of light .
Suppose you have a list of n numbers .
Their sum is often shorthanded using the Sigma notation like this
which is read “sum of from k=1 to k=n.” This letter k that varies from 1 to n is called an ‘index’.
Vectors. You can think of a vector as an ordered list of numbers
If you have two vectors and their dot product is defined by multiplying their respective coordinates and adding the result:
Using our summation notation, we can abbreviate this to
While working thru his general theory of relativity, Einstein noticed that whenever he was adding things like this, the same index k was repeated! (You can see the k appearing once in A and also in B.) So he thought, well in that case maybe we don’t need a Sigma notation! So remove it! The fact that we have a repeating index in a product expression would mean that a Sigma summation is implicitly understood. (Just don’t forget! And don’t eat fatty foods that can help you forget!)
With this idea, the Einstein summation convention would have us write the above dot product of vectors simply as
In his theory’s notation, it’s understood that the index k here would vary from 1 to 4, for the four dimensional space he was working with. That’s Einstein’s index notation where 1, 2, 3, are the indices for space coordinates (i.e., ), and k=4 for time (e.g., ). One could also write space-time coordinates using the vector where t is for time.
(Some authors have k go from 0 to 3 instead, with k=0 corresponding to time and the others to space coordinates.)
I used ‘k’ because it’s not gonna scare anyone, but Einstein actually uses Greek letters like instead of the k. The convention is that Greek index letters range over all 4 space-time coordinates, and Latin indices (like k, j, m,etc) for the space coordinates only. So if we use instead of k the dot product of the two vectors would be
So if we write it means we understand that we’re summing these over the 4 indices of space-time. And if we write it means that we’re summing these over the 3 indices of space only. More specifically,
There is one thing that I left out of this because I didn’t want to complicate the introduction and thereby scare readers! (I already may have! Shucks!) And that is, when you take the dot product of two vectors in Relativity, their indices are supposed to be such that one index is a subscript (‘at the bottom’) and the other repeating index is a superscript (‘at the top’). So instead of writing our dot product as it is written as
(This gets us into covariant vectors, ones written with subscripts, and contravariant vectors, ones written with superscripts. But that is another topic!)
How about we promote ourselves to Tensors? Fear not, let’s just treat it as a game with symbols! Well, tensors are just like vectors except that they can involved more than one index. For example, a vector such as in the above was written , so it involves one index . What if you have two indices? Well in that case we have a matrix which we can write . (Here, the two indices are sitting side by side like good friends and aren’t being multiplied! There’s an imaginary comma that’s supposed to separate them but it’s not conventional to insert a comma.)
The most important tensor in Relativity Theory is what is called the metric tensor written . It describes the distance structure (metric = distance) on a curved space-time. So much of the rest of the geometry of space, like its curvature, how to differentiate vector fields, curved motion of light and particles, shortest path in curved space between two points, etc, comes from this metric tensor .
The Einstein ‘gravitational tensor’ is one such tensor and is written . Tensors like those are called rank 2 tensors because they involve two different indices. Another good example of a rank 2 tensor is the energy-momentum tensor often written as . This tensor encodes the energy and matter distribution in spaces that dictate its geometry — the geometry (and curvature) being encoded in the Einstein tensor . (If you’ve read this far, you’re really getting into Relativity! And I’m very proud of you!)
You could have a tensors with 3, 4 or more indices, and the indices could be mixed subscripts and superscripts, like for example and .
If you have tensors like this, with more than 1 or 2 indices, you can still form their dot products. For example for the tensors D and F, you can take any lower index of D (say you take and set it equal to an upper index of F — and add! So we get a new tensor when we do this dot product! You get
where it is understood that since the index is repeated, you are summing over that index (from 1 to 4) (as I’ve written out on the right hand side). Notice that the indices that remain are . So this dot product gives rise to yet another tensor with these indices – let’s give the letter C:
This process where you pick two indices from tensors and add their products along that index is called ‘contraction‘ – even though it came out of doing a simple idea of dot product. Notice that in general when you contract tensors the result is not a number but is in fact another tensor. This process of contraction is very important in relativity and geometry, yet it’s based on a simple idea, extended to complicated objects like tensors. (In fact, you can call the original dot product of two vectors a contraction too, except it would be number in this case.)
Here is a short, sweet, and quick summary of some of Albert Einstein’s philosophy and religious views which I thought were interesting enough to jot down while I have that material fresh in mind. (I thought it’s good to read all these various views of Einstein’s in one fell swoop to get a good mental image of his views.) These can be found in most biographies on Einstein, but I included references  and  below for definiteness. (Throughout this note, ‘he’ refers, of course, to Einstein.) Let’s begin!
- Einstein began to appreciate and identify more with his Jewish heritage in later life (as he approached 50).
- He had profound faith in the order and discernible laws in the universe, which he said was the extent to which he calls himself ‘religious.’
- God had no choice but to create the universe in the way He did.
- He believed in something larger than himself, in a greater mind.
- He called nationalism an infantile disease.
- He received instruction in the Bible and Talmud. He is a Jew, but one who is also enthralled by “the luminous figure of the Nazarene.”
- He believed Jesus was a real historical figure and that Jesus’ personality pulsates in every word in the Gospels.
- He was not an atheist, but a kind of “deist.”
- He did not like atheists quoting him in support of atheism.
- He believed in an impersonal God, who is not concerned with human action.
- His belief in an impersonal God was not disingenuous in order to cover up an underlying ‘atheism’.
- He was neither theist nor atheist.
- He did not believe in free will. He was a causal determinist. (Not even God has free will! :-) )
- Though he did not believe in free will, nevertheless he said “I am compelled to act as if free will existed.”
- He liked Baruch Spinoza’s treatment of the soul and body as one.
- He did not believe in immortality.
- He believed that the imagination was more important than knowledge.
- He believed in a superior mind that reveals itself in world of experience, which he says represents his conception of God.
- He believed in a “cosmic religious feeling” which he says “is the strongest and noblest motive for scientific research.”
- “Science without religion is lame, religion without science is blind.”
There you have it, without commentary! ;-)
 Albert Einstein, Ideas and Opinions.
 Walter Isaacson, Einstein: His Life and Universe. (See especially chapter 17.)
Albert Einstein’s views on religion and on the nature and existence of God has always generated interest, and they continue to. In this short note I point out that he expressed different views on whether he subscribed to “pantheism.” It is well-known that he said, for instance, that he does not believe in a personal God that one prays to, and that he rather believes in “Spinoza’s God” (in some “pantheistic” form). Here are two passages from Albert Einstein where he expressed contrary views on whether he is a “pantheist”.
“I’m not an atheist and I don’t think I can call myself a pantheist. We are in the position of a little child entering a huge library filled with books in many different languages.…The child dimly suspects a mysterious order in the arrangement of the books but doesn’t know what it is. That, it seems to me, is the attitude of even the most intelligent human being toward God.” (Quoted in Encyclopedia Britannica article on Einstein.)
Here, Einstein says that he does not think he can call himself a pantheist. However, in his book Ideas and Opinions he said that his conception of God may be described as “pantheistic” (in the sense of Spinoza’s):
“This firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of experience, represents my conception of God. In common parlance this may be described as “pantheistic” (Spinoza)” (Ideas and Opinions, section titled ‘On Scientific Truth’ – also quoted in Wikipedia with references)
It is true that Einsteins expressed his views on religion and God in various ways, but I thought that the fact that he appeared to identify and not identify with “pantheism” at various stages of his life is interesting. Setting aside that label, however, I think that his general conceptions of God in both these quotes — the mysterious order in the books, a universe already written in given languages, a superior mind that reveals itself in such ways — are fairly consistent, and also consistent with other sentiments he expressed elsewhere.
Here’s an image of something I wrote down, took a photo of, and posted here for you. It’s a little game you can play with any irrational number. I took as an example.
You just learned about an important math concept/process called continued fraction expansions.
With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish.
As an example, if you truncate the above last expansion where the 292 appears (so you omit the “1 over 292” part) you get the rational number 335/113 which approximates to 6 decimal places. (Better than 22/7.)
You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers.
Exercise: for the square root of 2, show that the sequence you get is
1, 2, 2, 2, 2, …
(all 2’s after the 1). For the square root of 3 the continued fraction sequence is
1, 1, 2, 1, 2, 1, 2, 1, 2, …
(so it starts with 1 and then the pair “1, 2” repeat periodically forever).