Proof of Green's Theorem

Proof of Green's Theorem

For a vector field \(\vec{F}(x,y)\) defined at each point \((x,y)\) within the region \(R\) and along the continuous, smooth, closed, piece-wise curve \(c\) such that \(R\) is the region enclosed by \(c\), we shall derive a formula (known as Green’s Theorem) which will allow us to calculate the line integral of \(\vec{F}(x,y)\) over the curve \(c\).

Gravitational Force Exerted by a Sphere

Gravitational Force Exerted by a Sphere

To find the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of that sphere, we must first subdivide that sphere into many very skinny shells and find the gravitational force exerted by anyone of those shells on \(m\). We'll see, however, that finding the gravitational force exerted by such a shell is in of itself a somewhat tedious exercise. In the end, we'll see that the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of the sphere (where \(D\) is the center-to-center separation distance between the sphere and the particle) is completely identical to the gravitational force exerted by a particle of mass \(M\) on the other particle of mass \(m\) such that \(D\) is there separation distance.

Introduction to Double Integrals

Introduction to Double Integrals

In previous lessons, we learned that by taking the integral of some function \(f(x)\) we can find the area underneath that curve by summing the areas of infinitely many, infinitesimally skinny rectangles. In this lesson, we'll use the concept of a double integral to find the volume underneath any smooth and continuous surface \(f(x,y)\) by summing the volumes of infinitely many, infinitesimally skinny columns.

Colonizing and Terraforming Venus

Colonizing and Terraforming Venus

The first serious proposal in scientific literature on terraforming other worlds in the universe was about terraforming Venus. The planetary scientist Carl Sagan imagined seeding the Venusian skies with photosynthetic microbes capable of converting Venus's \(C0_2\)-rich atmosphere into oxygen. Other proposals involve assembling a vast system of orbital mirrors capable of blocking the Sun's light and cooling Venus until this hot and hellish world became very frigid and rained \(C0_2\) from its atmosphere. The solleta would also be capable of simulating an Earth day/night cycle. To create oceans and an active hydrosphere on Venus, we could hurl scores of icy asteroids from the Kuiper belt to Venus and, upon impacting the Venusian atmosphere, would rapidly disintegrate releasing enormous quantities of water vapor into the atmosphere which subsequently condense to form the first seas on Venus. Or perhaps Saturn's moon Enceladus—containing a colossal subsurface ocean dwarfing that of the Earth's—could be sacrificed towards the end of creating the first seas on Venus. But even if humans never terraform this hellish world, they could still live their—partially at least—by deploying thousands of blimps into the Venusian skies capable of supporting a long-term, human presence of perhaps over a million people. Venusian sky cities. But eventually, after many millennia of terraforming Venus, a rich ecosystem of life—including us—could live on Venus's surface.

How to Produce Water and Oxygen on Mars

How to Produce Water and Oxygen on Mars

The lack of oxygen in Mars' atmosphere and running liquid water on its surface is very inconveniant for any humans living their since oxygen and liquid water are necessary for humans to survive. Fortunatelly, there is an abundance of frozen water on Mars' surface. In this lesson, we'll discuss various techniques which can be used to extract all of this water. Once the water is obtained, by performing electrolysis on the water we can distill all of the oxygen from that water we need.

Harvesting Resources from Saturn and Titan

Harvesting Resources from Saturn and Titan

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..

Chaos and Fractals

Chaos and Fractals

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.