Polar coordinate integration in different planes?

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Homework Help Overview

The discussion revolves around the application of polar coordinates in different planes, specifically the xy, xz, and yz planes. Participants are exploring how to express Cartesian coordinates in terms of polar coordinates in these various contexts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning how polar coordinates translate to the xz and yz planes, comparing these to the established xy plane definitions. There is a focus on understanding the definitions of r and θ in these different contexts.

Discussion Status

The discussion is ongoing, with participants offering different perspectives on the relationship between the planes and the definitions of polar coordinates. Some are suggesting that the transition between planes may be straightforward, while others are seeking clarification on the definitions and implications of using polar versus spherical coordinates.

Contextual Notes

There is a mention of the need for spherical coordinates in three-dimensional problems, indicating a potential limitation in using polar coordinates alone for certain analyses.

theBEAST
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I know that when you are integrated over dA in the xy plane, for your polar coordinates, x = rcosθ and y = rsinθ. However what about in the xz and yz plane?

I noticed in one of the textbook problems, where the integration is over an area in the xz plane, x = rcosθ and z = rsinθ. How did the solution know to use that?
 
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What do you mean? Why would xy and xz-planes be any different? You can get xz-plane from xy-plane by just a simple rotation
 
clamtrox said:
What do you mean? Why would xy and xz-planes be any different? You can get xz-plane from xy-plane by just a simple rotation

If you integrate over some polar area in the xy plane you find that x = rcosθ and y = rsinθ. but let's say instead of integrating over the xy plane, we integrate over the yz plane then what are y and z in terms of r and θ?
 
theBEAST said:
If you integrate over some polar area in the xy plane you find that x = rcosθ and y = rsinθ. but let's say instead of integrating over the xy plane, we integrate over the yz plane then what are y and z in terms of r and θ?

Well, how do you define r and θ? I'm guessing r is the distance, so r=√(x2+z2) and θ is some angle, but what exactly is it?
 
r, theta polar coordinates are for the plane only. If you have a three-dimensional problem to analyze, then spherical coordinates (r, theta, phi) would be called for.

The conversion of polar coordinates (r, theta) to Cartesian (x, y), where x = r cos theta, y = r sin theta, is just simple trigonometry.
 

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