Findind Area element in Cylindrical Coordinate System

[chessmath]

Hi
I would like to know is there any way except using graph to find area element in cylindrical ( or Spherical) coordinate system?
Thanks.

 

[HallsofIvy]

Cylindrical and spherical coordinate systems are three dimensional so you would have to say what two dimensional object you want to find the area of before an area element can be given. However, we can say this- any area element is Cartesian coordinates can be written as Adxdy+ Bdxdz+ Cdydz for some A, B, C, which may be functions of x, y, and z, depending on the surface.

In cylindrical coordinates, we have $x= r cos(\theta)$, $y= r sin(\theta)$, and $z= z$ so that $dx= cos(\theta)dr- r sin(\theta)d\theta$, $dy= sin(\theta)dr+ rcos(\theta)d\theta$, $dz= dz$. From that, we can compute, remembering that the "wedge product" of differentials is skew-commutative,
$$dxdy= r cos^2(\theta)drd\theta- r sin^2(\theta)d\theta dr= r cos^2(\theta)drd\theta+ r sin^2(\theta)drd\theta= r dr d\theta$$
$$dxdz= cos(\theta)drdz- r sin(\theta)d\theta dz$$
$$dydz= sin(\theta)drdz+ rcos(\theta)d\theta dz$$
and, of course, changing the variables in A(x,y,z), B(x,y,z), C(x,y,z) to r, $\theta$, and z.

 

[chessmath]

Thanks I know rdrdθ is a valid statement but what about other area element, you got something that both depends on cosθ and sin but we know area element in the θ direction is just drdz. how can I calculate those?
Thank you.