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The Physics of Dark Energy
An exploration of a recent discovery in cosmology.

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Differential Equations

This article set explores some physical processes that involve changes over time, an obvious application for Calculus. It is possible to read and understand about 80% of the content without understanding the mathematical ideas behind the physics, but for those who want to fully grasp the presentation, I offer this summary of the mathematical notation.

Throughout the article set I will be using a consistent mathematical notation that many will recognize as that of the Calculus of differential equations. Many basic physical equations will be expressed as functions, and then the function syntax will be accented to express the idea of a derivative, like this:

f(t) = t2
A statement of a function with respect to t, defined as t squared. Let's say for this example that f(t) represents a position in space for a particular time t.
Note the added apostrophe. This represents the first derivative of the function f(t). If f(t) represents a position, then f'(t) represents a rate of change in the position with respect to time, or velocity (or speed, if no direction is provided).
This represents the second derivative of the function f(t). If f(t) represents a position, then f''(t) represents (read carefully) a rate of change in a rate of change with respect to time, or acceleration.

This notation represents an efficient way to describe a physical process, one for which mere words may not convey sufficient meaning. And it's not too strong to say that it (and mathematics in general) represents a descriptive language with its own syntax, rules and logic, a language that can express things that natural languages cannot. Nobel Prizewinner Richard Feynman said, "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature."

If you are not a math whiz, don't panic. For these articles, it's not necessary to know how to process these functional notations or convert one to another, it is sufficient to remember that f'(t) refers to a rate of change in f(t), and f''(t) refers to a rate of change in f'(t). And again, if you don't understand this notation, you will still understand most of the content because of the plain-English explanations.

Now I want to speak directly to my female readers. You may have been told that women have a problem with mathematics, that you are not intellectually suited to it, or that it's the province of geeks and nerds. People who say things like this are trying to rob you of your birthright. Throughout history there have been many spectacularly brilliant female mathematicians, people who contribute to our understanding of the world by learning nature's language and then speaking it, without shame or apology. And there is no basis for the oft-asserted claim that women are innately unsuited to scientific and technical professions. The truth is many men would rather you not learn mathematics, for the simple reason that those who understand mathematics possess greater employability and social status in the worlds of business and science.

Because of our increasingly technical world, it is not an exaggeration to say that those who understand mathematics better understand the world in which they live, and those who do not understand mathematics must have the modern world explained to them by those who do. This is why mathematical knowledge is the moral property of all educated people, without regard to gender. Do not allow yourself to be deprived of your birthright, your right to understand nature on her own terms, and remember that the most insidious forms of sexism are the kinds women accept without argument.

There is one more reason to learn mathematics, a reason that might as well be an atomic secret, and it is this: a well-conceived mathematical idea possesses a kind of transcendental beauty that cannot be explained to someone who doesn't understand mathematics. It is as if there is an epic poem that moves all its readers to tears, but that cannot be translated into everyday language. The only way to read the poem is to learn the language it's written in.

If you would like to learn a bit more mathematics, click this link for a Calculus tutorial on this site, one that will help you understand the notation described above.


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