![]() ![]() We can visualize this easily using a globe. This thinks of the world as sets of spheres inside one another. These guys are kind of tricky but not because they are hard, only because we don’t really use them that often. Now for the third step it says, how far up is the point P from the shadow? First, it says, for my point P in space, I am going to project P on to the xy plane (think of the “shadow” of P on the plane) Now it says, just like in polar I’ll find the angle between the x axis and this shadow, followed by the length I have to travel to get out to it. And you’ll find, if you understand polar, there is almost nothing to understanding this! It thinks of the world as a set of concentric circles (the xy plane) and there are infinitely many layers of these concentric circle sets (the z plane) Since it’s in 3D, it takes 3 steps. This is an extension of polar coordinates to 3D. This is a very smart system and very intuitive and I kick myself thinking I used to be so fearful of it! It thinks of the world as a set of concentric circles and says, if I want a “map” to a point, I can do two steps: say what angle the point is at from the x axis, and then say how far out from the origin at that angle I would have to travel to get to the point. Don’t think the “grid” is the only way to do things! In fact, sometimes there are much easier ways to describe things!Īnother 2D system. Growing up using this over and over we almost get convinced that this is the only reality, but this is just one of infinitely many ways of describing points in space. It looks at the world as a grid, and it says, I can describe a point by 2 steps: going over to the right x units (which means if x is negative we’re going left), and going up y steps (going down then when we have a negative y). This system to describe a point in 2D (the plane) is very natural because we have grown up on it. ![]() There are so many ways I can do this, but there are certain ways that are more natural than others and so they are commonly employed in mathematics. The only thing that we might change is what map maker we employ to make the maps that describe P. Does it’s orientation in space change? No. But similarly in reverse, if any one else in the world is trying to make a map for P, as long as they follow the map maker, they will come up with an identical map. ![]() Think of this description as a “map” and the coordinate system as the “map maker”. What do I mean by this? I mean, if I have a point P, hanging around in space somewhere, I want to describe it uniquely. I suppose I didn’t entirely understand what exactly these coordinate systems were, as I had spent my whole life exposed only to the common, rectangular (cartesian) one (go x distance to the left, go y distance up)Īnd then I finally buckled down and studied it, and I realized this is all these is to it:Ī coordinate system establishes a way to uniquely define a point in space. If something said “polar coordinates” I would just shy away, afraid it was that “trig stuff” I was so weary of. The conversion tables below show how to make the change of coordinates.I used to be very daunted by the different coordinate systems out there. Spherical coordinates would simplify the equation of a sphere, such as, to. The paraboloid would become and the cylinder would become. Cylindrical coordinates can simplify plotting a region in space that is symmetric with respect to the -axis such as paraboloids and cylinders. A change in coordinates can simplify things. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.ĭefining surfaces with rectangular coordinates often times becomes more complicated than necessary. (b) In cylindrical coordinates, a charge per unit length. Remember to immediately save it in your own home directory. (a) In spherical coordinates, a charge Q uniformly distributed over a spherical shell of radius a. When you hit enter, you can then choose MA1024 and then choose the worksheet Coords_start_B13.mw You canĬopy that worksheet to your home directory by going to your computer's Start menu and choose run. To assist you, there is a worksheet associated with this lab thatĬontains examples and even solutions to some of the exercises. ![]()
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