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Why Does dxdy = rdrd(theta)? Why Not Just drd(theta)?

Date: 12/29/2009 at 14:48:08
From: scott
Subject: why  dxdy = rdrd(theta) from x=rcos(theta) and y=rsin(theta)

In transforming from rectangular form to polar form, I know that

  x = rcos(theta) 

and

  y = rsin(theta)

But why does dxdy = rdrd(theta)?  Why the additional r term?  Why 
isn't dxdy equal to just drd(theta)?

I have always memorized this without understanding it.  I would like 
to know how to prove this.

Thanks in advance for your help.



Date: 12/29/2009 at 17:50:21
From: Doctor Jerry
Subject: Re: why  dxdy = rdrd(theta) from x=rcos(theta) and y=rsin
(theta)

Hello Scott,

Thanks for writing to Dr. Math.

Look at this figure:

  

It shows the area corresponding to an increase in theta of d(theta) 
and an increase in r of dr.  The small figure with sides of dr and r*d
(theta) is very nearly a rectangle, and has area r*dr*d(theta).

Please feel free to write back if you have questions.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 12/30/2009 at 12:00:11
From: scott
Subject: Thank you (why  dxdy = rdrd(theta) from x=rcos(theta) and 
y=rsin(theta))

Thank you very much. You guys are awesome. I really appreciate it.
Associated Topics:
High School Coordinate Plane Geometry

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