Calculating the volume of a sphere after switching to spherical coordinates?
up vote
5
down vote
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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 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
asked 16 hours ago
sonicboom
1283
add a comment |
up vote
5
down vote
favorite
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 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
asked 16 hours ago
sonicboom
1283
add a comment |
up vote
5
down vote
favorite
up vote
5
down vote
favorite
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 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
asked 16 hours ago
sonicboom
1283
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 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
asked 16 hours ago
sonicboom
1283
edited 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
asked 16 hours ago
sonicboom
1283
edited 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
edited 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
edited 16 hours ago
Αλέξανδρος Ζεγγ
3,8021927
3,8021927
asked 16 hours ago
sonicboom
1283
asked 16 hours ago
sonicboom
1283
asked 16 hours ago
sonicboom
1283
1283
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
7
down vote
accepted
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 15 hours ago
Henrik Schumacher
47.2k466134
add a comment |
up vote
5
down vote
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 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
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
up vote
7
down vote
accepted
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 15 hours ago
Henrik Schumacher
47.2k466134
add a comment |
up vote
7
down vote
accepted
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 15 hours ago
Henrik Schumacher
47.2k466134
add a comment |
up vote
7
down vote
accepted
up vote
7
down vote
accepted
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 15 hours ago
Henrik Schumacher
47.2k466134
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 15 hours ago
Henrik Schumacher
47.2k466134
edited 9 hours ago
answered 15 hours ago
Henrik Schumacher
47.2k466134
answered 15 hours ago
Henrik Schumacher
47.2k466134
answered 15 hours ago
Henrik Schumacher
47.2k466134
47.2k466134
add a comment |
add a comment |
up vote
5
down vote
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 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
add a comment |
up vote
5
down vote
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 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
add a comment |
up vote
5
down vote
up vote
5
down vote
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 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
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 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
edited 15 hours ago
answered 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
answered 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
answered 15 hours ago
Αλέξανδρος Ζεγγ
3,8021927
3,8021927
add a comment |
add a comment |
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