2D Cartesian integral to polar integral

In summary, the conversation discusses the calculation of an integral in polar coordinates using a small infinitesimal area in the form of a circle or disk. The integral is found by taking the double integral of log(1/r) over the area, with the area being represented as rdrd\theta. The conversation also mentions the difference between a circle and a disk in this context.
  • #1
Somefantastik
230
0
Hey everybody,

[tex] \int_{B(0,\epsilon)} log \frac{1}{r} \ dxdy \ = \ \int^{2\pi}_{0} d\theta \ \int^{\epsilon}_{0} log \frac{1}{r} \ rdr [/tex]

when r = r(x,y)
and B is a small ball with radius [tex] \epsilon [/tex]

Is this right? I haven't done this in forever and I need to be sure.

Thanks!
 
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  • #2
Looks right to me. If we have a small infinitesimal area in polar coordinates (in 2d) it would be roughly a box and have dimensions [tex]rd\theta \times dr[/tex] I believe, leading to the [tex]rdrd\theta[/tex] "volume" element you're using.
 
  • #3
Yeah, looks ok, if by "ball" you actually meant circle.
 
  • #4
Yeah yeah a ball in the 2d world...

thanks for the input ;)
 
  • #5
Defennder said:
Yeah, looks ok, if by "ball" you actually meant circle.

To be even more pedantic, it should be "disk", not "circle"!
 

What is a 2D Cartesian integral?

A 2D Cartesian integral is a mathematical tool used to find the area under a curve on a 2-dimensional Cartesian coordinate system. It is represented by the integral symbol and the limits of integration are typically the x and y coordinates of the curve.

What is a polar integral?

A polar integral is a mathematical tool used to find the area under a curve on a polar coordinate system. Unlike a 2D Cartesian integral, the limits of integration are represented by angles rather than coordinates.

How do you convert a 2D Cartesian integral to a polar integral?

To convert a 2D Cartesian integral to a polar integral, you need to use the appropriate conversion formulas. The x and y coordinates in the integrand should be replaced with the corresponding polar coordinates, and the limits of integration should be adjusted accordingly.

Why would someone want to convert a 2D Cartesian integral to a polar integral?

There are several reasons why someone may want to convert a 2D Cartesian integral to a polar integral. One reason is that certain curves are easier to represent and integrate in polar coordinates, such as circles and spirals. Additionally, polar integrals can be used to solve problems that involve symmetry or circular motion.

Are there any limitations to using a polar integral over a 2D Cartesian integral?

Yes, there are limitations to using a polar integral over a 2D Cartesian integral. One limitation is that polar coordinates cannot represent certain shapes and curves, such as straight lines and parabolas. Additionally, the conversion process can be more complicated and time-consuming compared to using a 2D Cartesian integral.

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