The Clear Science Staff has never liked this description of relativity, because it literally uses gravity as a metaphor to explain gravity. There are two ways to think of force fields (like gravity or electrostatics): either there is a field of forces at every point in space, or there’s no field but space is bent and distorted in an equivalent way.

There is a really great book about this by science fiction writer Rudy Rucker (or Rudolf von Bitter Rucker) called Geometry, Relativity and the Fourth Dimension

(Source: pizzzatime, via freshphotons)

The unit circle shows you all the angles from 0° to 360°. You can also refer to angles as segments of the unit circle circumference from 0 to 2π radians. If you imagine sweeping angles around the unit circle like a radar screen, it makes a series of triangles. The hypotenuse is (depending on how you look at it) either the longest side in the triangle or the radius of the unit circle.

A sine wave is what you get if you plot the triangle’s opposite side versus the angle. Sine waves are important in science and engineering, but also in music. When you’re playing a guitar the frets are all placed where they are to make certain sine waves. Light is a sine wave. Waves are sine waves. The motions of springs are sine waves. It goes on and on!

A circle is intimately related to angles. Everyone knows as you swing from 0 degrees all the way around to 360 degrees, you make a circle. The points on our unit circle figure show angles. Up above we drew bigger and bigger angles moving around the circle.

People talk about angles several different ways. If you’ve ever noticed the DRG button on a calculator, that stands for DEGREES RADIANS GRADIANS, which are three different ways to refer to angles. Our unit circle figure shows both degrees and radians. Radians are based on the how long the curvy part of a unit circle is that the angle hits. When you run all the way around 360 degrees, the curvy part is the whole circumference or 2π times the radius. So 360 degrees is 2π radians. At 90 degrees you get a quarter of that curvy part, so 90 degrees is π/2 radians.

This is an extremely useful diagram. Angles in degrees and radians, and their cosines and sines.

This is actually the kind of stuff I should be posting. Fractals are cool, but this is helpful.

The Clear Science staff is going to take a crack at convincing everyone this figure is an essential and beautiful component of everything. And by everything we mean not only science, engineering, and math but also art, music, nature, and pretty much anything else.

(Source: nxte)

We talked about how integrals will tell you the “area under a curve.” Look up above where we drew a trapezoid to find the area under the curve. A little bit of the trapezoid is actually over the curve though, so it’s not exactly the right area. But it’s close. If we want to get closer, why not use two trapezoids? Then we can add their areas together. Great, how about four trapezoids? Even better, but still a little bit off because the top of a trapezoid is straight, and the curve is curved.

With an integral it’s like you make the trapezoid width Δx infinitesimally small, in which case it becomes dx. Then you add up an infinite amount of them, so you’ve used the smallest trapezoid size possible. This gives you the true area under the curve.

When we were worried about a heat transfer problem, we spent a couple posts on derivatives and what they tell you. Integration is the inverse of differentiation. If you do calculus you end up doing those two things over and over again: taking the derivative and taking the integral.

Up above we have a curve plotted, which is y = 1/x. (By the way, if you ever want to know what a curve looks like, try fooplot. Type in 1/x there and hit enter and you’ll see it.)

Note the shaded area we’ve drawn, which is the area under the curve between x = 0.5 and x = 2. If you want to know the area (A) of that, you multiply the curve by dx and integrate from 0.5 to 2. So basically integrals tell you area.

Okay, so let’s think about that. What is dx again?

Trying to find the temperature decay from a flame, we came up with two equations:

1. a heat flux balance, which we derived
2. Fourier’s law, which we said was more or less a fundamental law

Now, we combine them to get a differential equation (step 1). Step by step we work through eliminating the derivatives with integration. At the end we get an answer with two constants C1 and C2 in it (step 7). This always happens solving differential equations because each integration produces a constant. Then we take the boundary conditions we specified at the top and solve for the constants (step 8). The answer is linear (step 9)! So there’s the answer: the temperature decay is linear.

But there are a couple things to consider here: first this is one-dimensional. We did that to make the math easy. With a real flame, the heat could go in any direction outward, basically in a spherical shape. Also we’ve specified that 20 cm away from the flame the temperature is 20 °C. This is often the case in the real world, because air is free to move around, and some distance from a flame there will always be cool air to be a heat sink. Problems like this can get really complicated if we want to calculate that distance from first principles, but it can be done!

The Clear Science staff is in the middle of administering a Clear Science final exam. You’ve probably heard that internet company Yahoo bought Tumblr for \$1.1 billion. Let’s talk some science about the number 1.1 billion.

In science, when you want to talk about big numbers, you use prefixes. Kilo means “times ten to the third power” or “times one thousand.” Every three more powers of ten gets another prefix, and they go like kilo, mega, giga, tera, etc. That means Yahoo bought Tumblr for 1.1 gigadollars

People don’t usually talk about money with these scientific prefixes. But let’s ask ourselves, Clear Scientists: why not? We think it sounds cool.

(PS we’re still answering a heat transfer question, which we’ll come back to pretty soon.)

Finding the temperature decay from a heat source (like a flame) got us talking about heat flux. Heat flux is the movement of heat, and heat is going to flux away from the flame. Heat moves from high temperatures to low temperatures, and wherever heat goes it increases the temperature.

We called heat flux in the x-direction qx. Let’s draw a little box and call it a “system" and do what’s called a "heat flux balance" for the system.

1. In English: The heat flux into the box equals the heat flux out of the box.
2. In math: (qx at x) minus (qx at x+Δx) equals zero. Now divide both sides by Δx. Now take the limit as Δx goes to zero, which means the system width becomes infinitesimally small.
3. When you start saying “infinitesimal" you know you’re doing calculus. This is the definition of a derivative, and our balance ends up with “the negative derivative of heat flux in the x-direction equals zero.”

If you translate 3 back to English it says “heat flux is the same at every value of x.”

So a derivative is like picking two points on a graph and calculating the difference in the y-values and dividing by the difference in the x-values … when the two points you pick are infinitesimally close. Up above we’ve drawn a plot of temperature T versus distance x, and we’ve shown the derivative at three points.

The way you can picture a derivative is this: If you draw a straight line that barely touches the curve at one point only, then that line is called a tangent. And the derivative at a point tells you the slope of the tangent. Where the curve is steep, the slope is high (3) and where it’s not steep slope is low (1/3). (The -ve signs are because temp goes down as x increases.)