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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?

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.)

We wondered what a derivative is. Imagine you have a graph with temperature on the y-axis and x on the x-axis. If you pick two points on the graph you can calculate the difference in their y values and the difference in their x values. Dividing those, you would get ΔT/Δx.

In the top-left graph we pick two points far apart. Going from the first point to the second we move 3.1 spaces down on the y-axis, so that ΔT is -3.1. We move 5.5 spaces on the x-axis so that Δx is 5.5. Doing the math it’s -0.56.

But look, if we pick different points we get different values. In the top-right we get -1.67, and in the bottom-left we get -0.36. It depends on what two points we pick.

Now this is a derivative: what if we say the two points we pick are zero distance apart so essentially they are the same point? That is dT/dx, shown in the bottom-right. Each point on the graph will have a different dT/dx value, which is the derivative at that point.

This is now calculus btw, because we talked about two points zero distance apart. (Or an “infinitesimal distance apart” which means infinitely close together.)

Asking about the temperature near a hot flame we brought up an important equation called Fourier’s law. The heat flux (q) away from a flame is a constant (k) times the negative of the temperature gradient. And we symbolized the temperature gradient with an upside down triangle in front of T.

That upside down triangle is called “del” and if we’re only worried about one dimension (the left-right dimension in the picture, which we’ll call the x-direction), this “del T” is the derivative of temperature with respect to that dimension. You write it dT/dx. You usually say it “dee-T dee-x” or “dee-T by dee-x.”

If you know calculus, you’ll recognize that is what we’re doing. It’s not really that complicated a concept though. So: What is a derivative, really?

Fourier’s law was first formulated by Jean Baptiste Joseph Fourier, whose name is pronounced like “Foo-ree-ay.” The Fourier transform and the Fourier series are important concepts in math.

(Awesome animated math gifs if you click those, FYI.) 

The Clear Science staff was going to answer the question “Is there a decay rate in heat at distance from a flame/heat source?” To do that let’s consider one way that heat transports from one place to another: conduction.

Heat is energy. Say we have a flame on the left and no flame on the right. The flame is there because some chemical reaction is happening: chemical bonds are breaking and their energy is being liberated. Because of this the temperature of the flame is high, like 1500 degrees. On the right temperature is only room temperature or 20 degrees.

Heat moves by conduction from high temperatures to low ones. This is a basic property of the universe, and it is described by Fourier’s law. Written above in “math language,” what it says in English is “heat flux is proportional to the negative of the temperature gradient.” Or: heat fluxes from high temp to low.