Calculating the volume of a sphere after switching to spherical coordinates?
I used the code in the second answer on this page to switch from Cartesian to Spherical coordinates and integrate a function over the sphere. I would like to use this code to calculate the volume of a sphere. However if I set $ f(x, y, z) = 1 $, I get the following output:
$ -frac{1}{3}r^3 varphicos(theta) $
Here is the code I used:
Needs["VectorAnalysis`"]
JacobianDeterminant[Spherical[r, θ, ϕ]]
f[x_, y_, z_] := 1
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[
Spherical @@ #] &@{r, θ, ϕ},
r, θ, ϕ]
How can I make the code return the volume of a sphere which is $frac{4}{3}pi r^3$?
calculus-and-analysis coordinate-transformation
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
add a comment |
I used the code in the second answer on this page to switch from Cartesian to Spherical coordinates and integrate a function over the sphere. I would like to use this code to calculate the volume of a sphere. However if I set $ f(x, y, z) = 1 $, I get the following output:
$ -frac{1}{3}r^3 varphicos(theta) $
Here is the code I used:
Needs["VectorAnalysis`"]
JacobianDeterminant[Spherical[r, θ, ϕ]]
f[x_, y_, z_] := 1
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[
Spherical @@ #] &@{r, θ, ϕ},
r, θ, ϕ]
How can I make the code return the volume of a sphere which is $frac{4}{3}pi r^3$?
calculus-and-analysis coordinate-transformation
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
add a comment |
I used the code in the second answer on this page to switch from Cartesian to Spherical coordinates and integrate a function over the sphere. I would like to use this code to calculate the volume of a sphere. However if I set $ f(x, y, z) = 1 $, I get the following output:
$ -frac{1}{3}r^3 varphicos(theta) $
Here is the code I used:
Needs["VectorAnalysis`"]
JacobianDeterminant[Spherical[r, θ, ϕ]]
f[x_, y_, z_] := 1
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[
Spherical @@ #] &@{r, θ, ϕ},
r, θ, ϕ]
How can I make the code return the volume of a sphere which is $frac{4}{3}pi r^3$?
calculus-and-analysis coordinate-transformation
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
I used the code in the second answer on this page to switch from Cartesian to Spherical coordinates and integrate a function over the sphere. I would like to use this code to calculate the volume of a sphere. However if I set $ f(x, y, z) = 1 $, I get the following output:
$ -frac{1}{3}r^3 varphicos(theta) $
Here is the code I used:
Needs["VectorAnalysis`"]
JacobianDeterminant[Spherical[r, θ, ϕ]]
f[x_, y_, z_] := 1
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[
Spherical @@ #] &@{r, θ, ϕ},
r, θ, ϕ]
How can I make the code return the volume of a sphere which is $frac{4}{3}pi r^3$?
calculus-and-analysis coordinate-transformation
calculus-and-analysis coordinate-transformation
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
edited Dec 12 '18 at 12:25
Αλέξανδρος Ζεγγ
4,0441928
4,0441928
asked Dec 12 '18 at 11:48
sonicboom
1383
asked Dec 12 '18 at 11:48
sonicboom
1383
asked Dec 12 '18 at 11:48
sonicboom
1383
1383
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
Employing most of your own code;
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ},
{r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
]
(4 π R^3)/3
That is, you just have to add the integral boundaries.
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
add a comment |
Try this
transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},
rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];
jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];
jdet f @@ coordi /. rules
]
f[x_, y_, z_] := 1
Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]
(4 π r^3)/3
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Employing most of your own code;
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ},
{r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
]
(4 π R^3)/3
That is, you just have to add the integral boundaries.
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
add a comment |
Employing most of your own code;
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ},
{r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
]
(4 π R^3)/3
That is, you just have to add the integral boundaries.
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
add a comment |
Employing most of your own code;
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ},
{r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
]
(4 π R^3)/3
That is, you just have to add the integral boundaries.
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
Employing most of your own code;
Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ},
{r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
]
(4 π R^3)/3
That is, you just have to add the integral boundaries.
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
edited Dec 12 '18 at 19:02
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
answered Dec 12 '18 at 12:41
Henrik Schumacher
48.8k467139
48.8k467139
add a comment |
add a comment |
Try this
transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},
rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];
jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];
jdet f @@ coordi /. rules
]
f[x_, y_, z_] := 1
Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]
(4 π r^3)/3
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
add a comment |
Try this
transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},
rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];
jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];
jdet f @@ coordi /. rules
]
f[x_, y_, z_] := 1
Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]
(4 π r^3)/3
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
add a comment |
Try this
transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},
rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];
jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];
jdet f @@ coordi /. rules
]
f[x_, y_, z_] := 1
Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]
(4 π r^3)/3
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
Try this
transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},
rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];
jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];
jdet f @@ coordi /. rules
]
f[x_, y_, z_] := 1
Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]
(4 π r^3)/3
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
edited Dec 12 '18 at 12:41
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
answered Dec 12 '18 at 12:40
Αλέξανδρος Ζεγγ
4,0441928
4,0441928
add a comment |
add a comment |
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