Finding new region for double integral

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Discussion Overview

The discussion revolves around determining the new region of integration for a double integral after a change of variables from the original coordinates (u, v) to the new coordinates (z, t). The context includes aspects of calculus, specifically related to double integrals and variable transformations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant describes the original region of integration as the first quadrant defined by ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}## and seeks to find the corresponding region in the new (z, t) coordinates.
  • Another participant suggests drawing a sketch and expressing the boundaries u=0 and v=0 in the new coordinates to aid in visualizing the transformation.
  • Further contributions reiterate the importance of step-by-step analysis, indicating that if u=0, then z=v, and if v=0, then z=u, but caution against assuming these are the only options.
  • There is a clarification regarding the notation, where one participant mistakenly refers to x and y instead of u and v, which is corrected in subsequent posts.
  • Another participant notes that if both u and v are zero, then z also equals zero, suggesting this information could help define the integration limits.

Areas of Agreement / Disagreement

Participants generally agree on the need to analyze the transformation step-by-step, but there is no consensus on the complete characterization of the new region of integration in the (z, t) coordinates.

Contextual Notes

Limitations include the potential for missing assumptions about the nature of the transformation and the need for clarity on the boundaries of the new region in the (z, t) coordinate system.

Mr Davis 97
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I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the zt-coordinate system?
 
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Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?
 
mfb said:
Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?
Will I know that in uv coordinates the region is the first quadrant.

If u=0, then z=v.
If v=0, then z=u.

Does this help me in any way?
 
Mr Davis 97 said:
If u=0, then z=v.
If v=0, then z=u.
These are not the only options, and you are trying to do two steps at the same time. Do it step by step.
 
mfb said:
These are not the only options, and you are trying to do two steps at the same time. Do it step by step.
If ##x=0,y \ne 0##, then ##z=y,t=0##
If ##y=0,x \ne 0##, then ##z=x, t=1##.

Is this correct?
 
What are x and y?
 
mfb said:
What are x and y?
Sorry, replace x with u and y with v.
 
And if u=v=0 then z=0.
That looks good and it should tell you what to integrate over.
 

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