After humanity has colonized and begun terraforming Mars, the next likely destination would be Saturn's moon Titan. One of the drawbacks of Mars is that it lacks nitrogen which is needed to grow food and also to create a breathable atmosphere. Enormous aerostats or NIFT spacecraft could harvest nitrogen and other resources from Titan's atmosphere and then transport these resources to a nuclear propulsion spacecraft. Given that Titan's gravity is only one-sixth as strong as that of the Earth's, we could use Titan's indigenous resources to construct a space elevator where nuclear propulsion spacecraft could be launched from. These spacecraft would carry their payload to Mars and, after arriving, all of that nitrogen could be deposited into Mars' atmosphere. It would also be very useful to send nitrogen to the Moon and to settlements along the asteroids in the inner-asteroid belt where nitrogen could be used for growing food. And after settling Titan, humans would likely go on to harvest helium-3 (which could be used to power nuclear fusion spacecraft) from Saturn and the other gas giants in the solar system which will be used to power nuclear powered spacecraft to the Kuiper belt, the Oort cloud, and perhaps even to the stars..

If water is being poured into a cone at a constant rate, what is the rate-of-change of the height of the water inside of the cone with respect to time? To answer this question, we'll need to use the chain rule.

According to Newtonian classical mechanics, if you knew the initial conditions of the entire universe (meaning that you know the initial positions and momenta of every particle in the universe) you could predict the entire past and future of the whole universe with infinite precision. But during the nineteenth-century, mathematicians studying the three-body problem—a problem which alluded Newton—concluded that the behavior of the solar system becomes completely unpredictable after sufficiently long periods of time. The reason why this was the case was because of something known as sensitivity to initial conditions. In the 1950s, an MIT professor named Edward Lorenz discovered that the weather, like the solar system, loses all predictability after a sufficient length of time (about one month). This lead to the birth of chaos theory. But, despite the apparent randomness of chaotic phenomena, it turns out that there is an underlying geometric order to such phenomena called fractal geometry. We shall introduce chaos theory and fractal geometry in this lesson.

In previous lessons, we learned how the derivative \(f'(x)\) gives us the steepness at each point along a function \(f(x)\). In this lesson, we'll discuss how using the concept of a partial derivative we can find the steepness at each point along a surface \(z=f(x,y)\). To find the partial derivative we treat one of the variables as a constant and then take the ordinary derivative of \(f(x,y)\). Using this concept, we can specify how steep a surface \(f(x,y)\) is along the \(x\) direction and along the \(y\) direction at each point along the surface. In other words, for every point along the surface, there is a steepness of the surface associated with both the \(x\) and the \(y\) directions at that point.

In this article, we'll discuss the history behind the discovery of one of the most exotic and remarkable objects ever discovered in science—a quasar. Quasars consist of an accretion disk made of ultra-hot gas and dust surrounding a supermassive black hole; two immense strands of super-heated plasma extended in a direction perpendicular to the disk for millions of the light-years through the mostly empty void of intergalactic space. Quasars were ubiquitous in the early, young universe and were located at the center of most galaxies; but today, most quasars are gone because all of the matter comprising the accretion disk eventually got gobbled up by the super massive black hole. For example, our home galaxy—the Milky Way—now only has the left-over remnant of a quasar at its center—a super massive black hole.

In this lesson, we'll discuss how by using the concept of a definite integral one can calculate the volume of something called an oblate spheroid. An oblate spheroid is essentially just a sphere which is compressed or stretched along one of its dimensions while leaving its other two dimensions unchanged. For example, the Earth is technically not a sphere—it is an oblate spheroid. To find the volume of an oblate spheroid, we'll start out by finding the volume of a paraboloid . (If you cut an oblate spheroid in half, the two left over pieces would be paraboloids.) To do this, we'll draw an \(n\) number of cylindrical shells inside of the paraboloid; by taking the Riemann sum of the volume of each cylindrical shell, we can obtain an estimate of the volume enclosed inside of the paraboloid. If we then take the limit of this sum as the number of cylindrical shells approaches infinity and their volumes approach zero, we'll obtain a definite integral which gives the exact volume inside of the paraboloid. After computing this definite integral, we'll multiply the result by two to get the volume of the oblate spheroid.

Undoubtedly, if our wisdom and foresight rises to be commensurate with our science and technology, the future of humanity in the 21st century is Utopian. We will have re-engineered the surface of the Earth with cities, transportation and communication systems, and new energy infrastructure which are designed and constructed to have optimal efficiency according to known science. We will have also spread across much of the solar system and, perhaps, have sent robotic spacecraft off to the nearest star system, Alpha Centauri. Aside from re-engineering the Earth and other worlds in our solar system, we will also likely re-engineer ourselves as we merge with our technology and machines. This will be the subject of discussion in this article.

To find the gravitational force exerted by a disk on a particle a height \(h\) above the center of the disk, we must use Newton's law of gravity and the concept of a definite integral.

A Shkadov thruster is a type of megastructure which involves constructing a gargantuan orbital mirror next to a star. In this lesson, we'll start off by discussing how such a stellar engine works. The orbital mirror is placed in a position next to a star where it acts as a statite: that is, the star's gravity acting on the mirror is canceled out by the star's radiation pressure acting on the mirror. This allows the mirror to stay in a position that is stationary relative to the star's surface. The mirror bounces some of the star's light back at itself; when that reflected light collides with the star, it exerts a thrust on the star which causes it to accelerate and move. This will bring us to the second main focus of this lesson: namely, what are the possible uses of a Shkadov thruster? As we'll discuss, since a Shkadov thruster can move the star and since all of the planets, moons, comets, and asteroids in the star system is gravitationally bound to the star, not only does the star move but the entire solar system moves away also. In the distance future, our Sun will eventually die. But we might be able to use a Shkadov thruster to move the Earth to another solar system, but this would take many millions of years.

The law of reflection had been well known as early as the first century; but it took longer than another millennium to discover Snell's law, the law of refraction. The law of reflection was readily observable and could be easily determined by making measurements; this law states that if a light ray strikes a surface at an angle \(θ_i\) relative to the normal and gets reflected off of the surface, it will be reflected at an angle \(θ_r\) relative to the normal such that \(θ_i=θ_r\). The law of refraction, however, is a little less obvious and it required calculus to prove. The mathematician Pierre de Fermat postulated the principle of least time: that light travels along the path which gets it from one place to another such that the time \(t\) required to traverse that path is shorter than the time required to take any other path. In this lesson, we shall use this principle to derive Snell's law.