The discussion revolves around the conversion of integrals from Cartesian to polar coordinates, specifically addressing the need to multiply by "r" when changing variables in double integrals. Participants are exploring the implications of this multiplication in different contexts, such as integrating with respect to dxdy versus dydz.
Some participants have provided insights into the dimensional aspects of the variables involved, suggesting that the multiplication by "r" is necessary for maintaining dimensional consistency. However, there remains uncertainty about the correct application of these principles in various integral forms, and multiple interpretations are being explored.
There is a lack of clarity regarding the specific coordinates being used in the integrals, and participants are navigating through different scenarios of integration, including surface integrals and volume integrals in cylindrical coordinates.
I seriously doubt that yo can "substitute d\thetadz" without multiplying by r but I don't understand what coordinates you are using. In converting from Cartesian, x, y, z, coordinates to cylindrical r, \theta, z the "differential of volume is r dr d\theta dz. The only kind of integral I can think of that would involve "d\theta would be on the surface of a cylinder. And even then, you would multiply by the radius of the cylinder. That is because lines of constant \theta grow farther apart as r increases.Niles said:Homework Statement
When dealing with an integral integrated with respect to dxdy, I can convert this to polar coordinates, and then integrate with respect to dr d\theta. But I have to multiply with a "r" before integrating.
If I am dealing with an integral with respect to dydz, I can substitute this with d\theta dz, but not have to multiply with "r". Why is that? (And is what I wrote even correct?)