calculus and analysis - Can Mathematica help me evaluate an integral over disjoint disks $I = int_{D_1} int_{D

calculus and analysis - Can Mathematica help me evaluate an integral over disjoint disks $I = int_{D_1} int_{D_2} log|x-y| dy dx$?

I want to evaluate an integral that involves two disjoint unit disks $D_1$ and $D_2$. $D_1$ is centered at $(-2,0)$ and $D_2$ is centered at $(0,2)$. The integral I want to compute is

$$I = \int_{D_1} \int_{D_2} \log|x-y| dy dx.$$

I looked at the in-built Python integration methods and also the quadpy library but although they have lots of options for integration over a single disk, I couldn't find anything that can help me with integrating over two disjoint disks.

Is it possible to evaluate this integral in Mathematica? I don't need an optimum method, I just need to obtain the value of this integral.

Answer

NIntegrate[
 Log[Norm[{x1, x2} - {y1, y2}]], 
 {x1, x2} ∈ Disk[{-2, 0}, 1], 
 {y1, y2} ∈ Disk[{2, 0}, 1]
 ]

13.6822

(the actual result being 13.682176919165677),

In order to enter ∈, just type Esc e l Esc.

In order to increase the precision, use the option PrecisionGoal.

Edit

Another possibility that relieves Mathematica from the need to discretize the disks and that allows her to use higher-order quadrature formulas is to employ polar coordinates on each of the disks:

NIntegrate[
 Log[Norm[{r1 Cos[θ1] - 2,r1 Sin[θ1]} - {r2 Cos[θ2] + 2,r2 Sin[θ2]}]] r1 r2,

 {r1, 0, 1}, {θ1, -Pi, Pi},
 {r2, 0, 1}, {θ2, -Pi, Pi},
 Method -> "LocalAdaptive"
 ]

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

Blog

Search This Blog