Polar Coordinates

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This article will derive the area of a circle and volume of a sphere using integration. It builds on a previous article introducing calculus, and aims to function as a primer on polar and spherical coordinate systems.

A circle is a 2D object with extent in both x and y. Its area is, of course, A = 𝜋r². It is very telling that this well-known equation relies not on x or y but on the radius, r. This is purely for simplicity. The radius is that number which intuitively defines a circle, it is the same all the way around, while x and y change drastically. To see how r relates to x and y consider a circle of radius r centred on the origin. Any point along the circumference, like the blue point below, can be treated as making a right-angle triangles with both axes, as shown below.polar_coordinates_cartesian

Screen Shot 2017-06-30 at 15.00.36The sides are given by the trigonometrical relations (from r and the angle θ swept from the x-axis). In fact, any point can be specified using these coordinates r and θ; for example (2, 𝜋/6) refers to a point 2 units from the origin at an angle 𝜋/6 (or 30º) much like the one pictured above. θ ranges from 0 to 2𝜋 (or 360º) since it would simply repeat after one revolution anyway, while r can be any number greater than 0. This is the polar coordinate system.Screen Shot 2017-06-30 at 16.50.05Finding the area of a circle in the x-y coordinate system requires integrating with respect to x and then with respect to y. Polar coordinates do the same thing but with r and θ. Double integrals may look intimidating at first sight but they are simply performed one at a time from the inside out.Screen Shot 2017-06-30 at 17.07.31To solve in x and y is long and tedious, which is why area is commonly given in terms of r. A change to polar coordinates simplifies things, but at a cost. A conversion factor is needed, called the Jacobian. Fortunately, the change to polar requires a Jacobian of simply r; derived here. A more complex derivation follows later, so a glance at this one is advised if only to avoid the sensation that it’s being pulled from thin air. The area of a circle can thus be described as:Screen Shot 2017-06-30 at 16.23.12Where θ ranges from 0 to 2𝜋, the angle swept by a circle, and r is limited between 0 and r. This is really the same as saying r isn’t defined, other than its being greater than 0. The integral of = θ, so:Screen Shot 2017-06-30 at 16.23.28θ then assumes the values of its limits, top minus bottom, so θ = 2𝜋 – 0, where the zero can be ignored:Screen Shot 2017-06-30 at 17.16.19Then integrate r. Since it is undefined, i.e. r = r, its limits can be ignored.Screen Shot 2017-06-30 at 17.16.31Screen Shot 2017-06-30 at 17.16.45Finding the volume of a sphere requires a switch to the spherical coordinate system. Whereas polar coordinates specify a position on a circle, in spherical coordinates points are specified by their position on a sphere, using r, θ and φ. is exactly as before, it points out from the origin and gives how far away the point is, or the radius of the sphere. But now two angles are needed, θ and φ, to specify a position. These are equivalent to latitude and longitude on the Earth: the first tells you how far South of the North pole you are, which of course isn’t enough to know your position, while the second tells you how far East you are from the Greenwich meridian. Latitude, θ, goes from 0 to 𝜋 (180º, i.e. North pole to South pole) while longitude, φ, goes from 0 to 2𝜋 (360º, i.e. Greenwich to Greenwich). These are shown below.
250px-Spherical_polarA triple integral is needed to derive the volume of a sphere, and has the form:Screen Shot 2017-07-01 at 11.14.51In spherical coordinates these become drdθdφ and the Jacobian is r² sinθ, derived here. This gives the volume of a sphere as:Screen Shot 2017-07-04 at 14.03.35Integration is much as before, and can in fact be performed in any order.Screen Shot 2017-07-04 at 14.03.42Screen Shot 2017-07-04 at 14.03.49Screen Shot 2017-07-04 at 14.03.54Screen Shot 2017-07-04 at 14.04.01Screen Shot 2017-07-04 at 14.04.07Thus the volume of a sphere is given as expected:Screen Shot 2017-07-04 at 14.04.15

An Integral Quirk

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Calculus revolutionised maths and physics in the 17th century, allowing Isaac Newton to form his equations of motion. This article will lay out the physical meaning of its primary components, differentiation and integration, and show their power by, from first principles, deriving Newton’s equations of motion.

Differentiation finds the slope of a curve. In others words, by how much y changes when x is increased by 1. For any function:Screen Shot 2017-06-01 at 20.57.34The slope (or derivative) is given by:Screen Shot 2017-06-01 at 20.57.41For f(x) = x, as plotted below, the gradient f'(x) = 1, meaning that y increases the same as x, which makes sense.Screen Shot 2017-06-01 at 21.06.34Differentiation is therefore the rate of change. The rate of change of position is velocity, and the rate of change of velocity is acceleration.

Integration, on the other hand, finds the curve from the slope. It is the opposite of differentiation. For any function:
Screen Shot 2017-06-02 at 22.19.34The curve (or integral) is given by:Screen Shot 2017-06-02 at 22.23.17.pngSo for f'(x) = 1, the integral f(x) = x + c. The c is an uncertainty constant. This arises because knowing the slope isn’t enough to know where the curve starts and ends, just as knowing the velocity of a car gives no information about its position. Yet the integral of velocity is position, and the integral of acceleration is velocity. This last one is where we begin.

Define acceleration as a. From first principles the velocity v will be the integral of a with respect to t. Performing this integral returns:Screen Shot 2017-06-02 at 23.00.37Screen Shot 2017-06-02 at 23.01.50.pngIn other words, the a gains a t term (as expected) and there is an uncertainty constant u. Now what is u? Well, if t = 0, we can see that v = u, and so u is the initial velocity, before time commenced. This is one of Newton’s equations of motion, but he has another.

If we integrate again, we will arrive at position:
Screen Shot 2017-06-02 at 23.09.38Screen Shot 2017-06-02 at 23.12.59Where, again, x₀ is the uncertainty constant and is equal to x at t = 0. Thus it is the initial position. By redefining xx₀, (i.e. distance between final and starting positions) as the displacement s, the second of Newton’s equations of motion is derived.Screen Shot 2017-06-02 at 23.16.01