Why You Should STOP Using (x, y, z) Coordinates (in certain scenarios) - Polar Coordinates, Parth G

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Many of us will be familiar with the concept of using coordinates to represent positions in space. In 2 dimensions, we most commonly use x and y coordinates. These are coordinates that are perpendicular to each other (orthogonal) and always point in the same direction. When we study 3D systems, we add a third z coordinate which is perpendicular to x and y. This coordinate system is known as the Cartesian coordinate system, named after Descartes.

However in some situations, the Cartesian coordinate system is not the most convenient one to use. In this video we see how in two dimensions, a polar coordinate system made up of a radial and angular / azimuthal coordinate can equally validly represent any point on a flat plane. The radial coordinate is formed using the length of the vector between the origin and the point we are describing, and the angular coordinate is the angle between this vector and the positive x axis. This polar coordinate system (circular polars) is best used to describe systems with circular symmetry such as a spinning disc, or the rings of Saturn.

We can then extend this to 3 dimensions by simply adding in the z axis from the Cartesian coordinate system. This way, the x and y coordinates are replaced with the radial (r or rho) and angular (phi) coordinates, while the z coordinate remains the same. This coordinate system (cylindrical polar) is best for describing systems with cylindrical symmetry. This doesn't just mean perfect cylinders, but rather any system where rotating about just one axis (z axis) leads to invariant quantities (or no change being seen). Hence the new coordinates are (r, phi, z) rather than (x, y, z). And we also see how the coordinates are always orthogonal.

Another coordinate system, known as the spherical polar coordinate system, best describes systems with spherical symmetry (such as the electric field generated by a point charge). With cylindrical polars, the radial coordinate now represents the distance between the point being described and the origin in any direction, not just the z = 0 plane. The angle phi remains the same, which is the angle between the vector's projection in the z = 0 plane, and the x axis. And a new angle, theta, is defined to be the angle between the r vector and the z axis. Hence the new coordinates are (r, theta, phi).

In this video we also look at two problems with the circular, cylindrical, and spherical polar coordinate systems. The first one is an easy one to solve. It involves the periodicity (repetition) seen when the value of phi or theta exceeds 360 degrees. In other words, there are multiple valid values for each angle even when describing the same point in space. However this can be fixed by restricting the value of theta of phi between 0 and 360 degrees. Sometimes this isn't even necessary as certain physical systems need there to be flexibility past 360 degrees.

The second problem is harder to solve. In these new coordinate systems, the value of phi or theta is not uniquely defined at the origin. In other words, when r = 0 the value of any of these angles could be absolutely anything. And there is no easy way to get around this problem. However the polar coordinate systems are much better at describing circular, cylindrical, and spherical systems so we deal with the problem as it's more convenient than using Cartesian coordinates here.

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Timestamps:
0:00 - 3D Cartesian Coordinate Grid
2:40 - A message from our sponsor, Wren - check out the link below!
4:31 - Circular Polar Coordinates (2D)
8:20 - Cylindrical Polar Coordinates (3D)
9:47 - Spherical Polar Coordinates (3D) + Electric Field
12:25 - Problems with Polar Coordinates

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