Mathematica Asked on March 29, 2021
I want to draw the surface of the upper half of the sphere $z = sqrt {4 x – x^2 – y^2}$ inside the cylinder $(x – 1)^2 + y^2 = 1$ .
reg = ImplicitRegion[
z == Sqrt[4 x - x^2 - y^2] && ((x - 1)^2 + y^2 <= 1), {x, y, z}];
Region[%, PlotRange -> All]
Region[DiscretizeRegion[
ImplicitRegion[
z == Sqrt[4 x - x^2 - y^2] && ((x - 1)^2 + y^2 < 1), {x, y, z}]],
PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}]
But the above code can’t draw this surface. What should I do?
But the following code can get the desired result:
Region[DiscretizeRegion[
ImplicitRegion[z^2 == 4 x - x^2 - y^2, {x, y, z}]],
PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}]
ClearAll["Global`*"]
ImplicitRegion[
z == Sqrt[4 x - x^2 - y^2] && (x - 1)^2 + y^2 < 1 &&
4 x - x^2 - y^2 > 0 && z > 0, {x, y, z}],
MaxCellMeasure -> "Length" -> 0.01],
PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}]
I want to know why the first method can’t get the desired results?
However, using numerical method I can get the correct result. Maybe this is a bug?
Next, calculate the second-kind surface integral $I=iint_{Sigma} y z(y-z) mathrm{d} y wedge mathrm{d} z+z x(z-x) mathrm{d} z wedge mathrm{d} x+x y(x-y) mathrm{d} x wedge mathrm{d} y$ on this surface.
Integrate[{y*z (y - z), z*x (z - x),
x*y (x - y)}.(Normalize[
D[z - Sqrt[4 x - x^2 - y^2], {{x, y, z}}]]), {x, y, z} ∈
reg]
DiscretizeRegion[
ImplicitRegion[
z == Power[4 x - x^2 - y^2, 1/4], {{x, -2, 2}, {y, -2, 2}, {z, 0,
2}}], AccuracyGoal -> 10]
Plot3D[Sqrt[4 x - x^2 - y^2], {x, -2, 2}, {y, -2, 2},
RegionFunction -> Function[{x, y}, (x - 1)^2 + y^2 <= 1],
PlotPoints -> 80, Mesh -> None,
PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}, BoxRatios -> 1]
Or
ContourPlot3D[
z^2 == 4 x - x^2 - y^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
Mesh -> {{0}}, MeshStyle -> None,
MeshFunctions -> {Function[{x, y, z}, (x - 1)^2 + y^2 - 1],
Function[{x, y, z}, z]},
MeshShading -> {{None, None}, {Red, None}}, BoundaryStyle -> None,
PlotPoints -> 80]
reply the questioner's comment
reg = ImplicitRegion[
z^2 == 4 x - x^2 - y^2 && (x - 1)^2 + y^2 <= 1 && z >= 0 , {x, y,
z}];
Region[%, PlotRange -> All]
Region[DiscretizeRegion[
ImplicitRegion[
z^2 == 4 x - x^2 - y^2 && (x - 1)^2 + y^2 < 1, {x, y, z}],
MaxCellMeasure -> "Length" -> 0.01],
PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}]
Update-2
Replace the power index 1/2 to 0.5 and using Cylinder
domain we can draw the same surface.
reg = ImplicitRegion[
z == (4 x - x^2 - y^2)^0.5 && {x, y, z} [Element]
Cylinder[{{1, 0, 0}, {1, 0, 2}}, 1], {x, y, z}]
RegionPlot3D[DiscretizeRegion@reg,
PlotRange -> {{-2, 2}, {-2, 2}, {0, 2}}]
Answered by cvgmt on March 29, 2021
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