Re: Area Between Contours in ContourPlot
- To: mathgroup at smc.vnet.net
- Subject: [mg39656] Re: [mg39624] Area Between Contours in ContourPlot
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Thu, 27 Feb 2003 00:33:54 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
John, It may be helpful to think about the example of computing the area of the "polar rectangle" bounded by f(x,y) = r^2 < f(x,y) = R^2, g(x,y) = m1 < g(x,y) = m2 where f(x,y) = x^2 + y^2 and g(x,y) = y/x. The region is the image of the rectangle G = { (u,v) : r^2 < u < R^2, m1 < v < m2 } , under the transformation T(u,v) = ( Sqrt[u] v/Sqrt[1+v^2], Sqrt[u]/Sqrt[1+v^2]) (Get that by solving u == f(x,y), v == g(x,y), for x and y.) The area of the region is the integral over G of the absolute value of the determinant of the Jacobian of T, which here turns out to be 1/(1+v^2). Integrate[ 1/(2(1+v^2)), {u,r^2,R^2}, {v,m1,m2}] The result is 1/2 (R^2 - r^2)(ArcTan[m2] - ArcTan[m1]), which coincides with the usual formula. More generally, if you want the area of a "patch" defined by f(x,y) = c1 < f(x,y) = c2, g(x,y) = c3 < g(x,y) = c4, solve (if possible ) u == f(x,y), v == g(x,y), for x and y, compute the Jacobian determinant in terms of u and v, then integrate its absolute value over c1 < u < c2, c3 < v < c4. A sometimes convenient alternative method for getting the Jacobian determinant is to take the reciprocal of the Jacobian determinant of the inverse transformation Tinv(x,y) = (f(x,y), g(x,y)), expressed in terms of u and v. Hope this helps... ---- Selwyn Hollis On Wednesday, February 26, 2003, at 02:43 AM, John.Hornbuckle at csiro.au wrote: > Hi All > > I am new to Mathematica and am wondering if someone on the list would > be > kind enough to help me out. I have a contourplot of a function and > wish to > find the area between certain contours contained within the plot. Is > there a > standard function in Mathematica which will do this? If not I would > appreciate any suggestions on how I should go about doing this with > Mathematica, I'm using version 4.2. > > Thanks in advance for your time and help. > > Cheers John > > John Hornbuckle > CSIRO Land and Water > PMB No. 3, Griffith, > NSW, 2680 > Australia > > Tel. (02) 69601500 > Fax. (02) 69601600 > >