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Coordinate Systems - Example 2

In mathematics, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the "x"-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The most common coordinate system in modern mathematics is the Cartesian coordinate system.

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Video Transcript

welcome to the video reviewing how to convert from Cartesian coordinates to cylindrical polar coordinates. So in the previous video, we did just to polar coordinates. Now we're gonna add in this Z thing that I talked about before we have this Z axis in the vertical direction, which we can think of as multiple of these polar planes stacked up on top of each other. So remember again are variables are are Seda and Z in this case now, before we've been thinking just in the two dimensional Cartesian, though obviously we can't really go from the two to the three very well. So I'm gonna throw in our picture from the three dimensional Cartesian, which before I label the axes as X, y and Z. Okay, so those are three variables Now, I should say that this isn't necessarily the most common way to think about the orientation of of the three axes. In fact, very often. Instead, we would label them as X here. Why here and see here And in fact, this is helpful comparing it to our cylindrical polar picture here in that we have the X Y plane in which data is measured is in the same more or less in in the same plane as the cylindrical one here. So let's think about it. Uh, this way with that second set here now, thinking about how to convert them from our cylindrical pull our Cartesian toe are cylindrical polar. It's gonna look a lot like it did with E with the Cartesian thio simple polar Because all we're gonna do is we're gonna say the Z becomes Z So we're going from X. Why Z into are Data Z And as you might have guessed, Z is equal to Z. So we can use the exact same formulas that we used before where x is equal to r cosine theta. Why is equal to our science data? And then we have our is equal to the square root of X squared plus y squared and feta is equal to the inverse tangent or arc tangent of why over X. Remember, you have to be careful about which quadrant here in with that conversion. Now coming back then over here, we could say, Well, let's let's pick a point. Let's ah, pick a point on this one. We'll say we'll go on to 34 out. Okay, so we have an r of four. We'll pick an angle of 30 degrees and then a z of to or actually, I guess that sea of one in this picture. So z of one. So that being said in order to get to X, Y and Z, then what we need to do is we need to say Okay, well, I have four times the cosine of 30 degrees and then comma four times the sign of 30 degrees. And then what? And that's all you have to do in the reverse direction. If I were to pick a point here, uh, let's say I've got a point that's at, um let's see. It looks like it's maybe one in the X. And then we'll pick three in the UAE and five and the Z in order to go into our r theta Z coordinates that I would have square root of one squared, plus three squared, then the arc tangent of one of 3/1 notice. We're in the first quadrant there, so it's not a problem to just write it down like that. And then Z is five, so it looks pretty much identical to our polar coordinate conversion. All we're doing is we've added in this five where Z is equal to see now. The reason you'd want to use cylindrical coordinates instead of the spherical polar coordinates is that one. It's a little simpler to use because you have this Z equals e concept. And then also because there's a lot of things out there that actually have a cylindrical symmetry. Things like wires, um, or cylinders themselves with you thinking about a can rolling down a hill or something, which is a classic physics problem. Then you can use the cylindrical symmetry to solve for some things like the moment of inertia, or to use some of the more advanced calculus principles later on, when we get into electromagnetism.