Spherical Coordinates

Spherical Coordinates - Spherical Coordinates In this...

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Spherical Coordinates In this section we will introduce spherical coordinates. Spherical coordinates can take a little getting used to. It’s probably easiest to start things off with a sketch. Spherical coordinates consist of the following three quantities. First there is . This is the distance from the origin to the point and we will require . Next there is . This is the same angle that we saw in polar/cylindrical coordinates. It is the angle between the positive x -axis and the line above denoted by r (which is also the same r as in polar/cylindrical coordinates). There are no restrictions on . Finally there is . This is the angle between the positive z -axis and the line from the origin to the point. We will require . In summary, is the distance from the origin of the point, is the angle that we need to rotate down from the positive z- axis to get to the point and is how much we need to rotate around the z -axis to get to the point. We should first derive some conversion formulas. Let’s first start with a point in spherical coordinates and ask what the cylindrical coordinates of the point are. So, we
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and what to find . Of course we really only need to find r and z since is the same in both coordinate systems. We will be able to do all of our work by looking at the right triangle shown above in our sketch. With a little geometry we see that the angle between z and is and so we can see that, and these are exactly the formulas that we were looking for. So, given a point in spherical coordinates the cylindrical coordinates of the point will be, Note as well that, Or, Next, let’s find the Cartesian coordinates of the same point. To do this we’ll start with the cylindrical conversion formulas from the previous section . Now all that we need to do is use the formulas from above for
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Spherical Coordinates - Spherical Coordinates In this...

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