vector calculus integration identity problem The Next CEO of Stack Overflow$LaTeX$ format copy...
Lucky Feat: How can "more than one creature spend a luck point to influence the outcome of a roll"?
Cannot shrink btrfs filesystem although there is still data and metadata space left : ERROR: unable to resize '/home': No space left on device
What does "shotgun unity" refer to here in this sentence?
Strange use of "whether ... than ..." in official text
Point distance program written without a framework
What is the process for cleansing a very negative action
TikZ: How to fill area with a special pattern?
Does Germany produce more waste than the US?
Is it okay to majorly distort historical facts while writing a fiction story?
Reshaping json / reparing json inside shell script (remove trailing comma)
What is the difference between "hamstring tendon" and "common hamstring tendon"?
how one can write a nice vector parser, something that does pgfvecparse{A=B-C; D=E x F;}
Film where the government was corrupt with aliens, people sent to kill aliens are given rigged visors not showing the right aliens
Is there a reasonable and studied concept of reduction between regular languages?
vector calculus integration identity problem
Are the names of these months realistic?
"Eavesdropping" vs "Listen in on"
Can this note be analyzed as a non-chord tone?
Man transported from Alternate World into ours by a Neutrino Detector
What CSS properties can the br tag have?
Does higher Oxidation/ reduction potential translate to higher energy storage in battery?
Reference request: Grassmannian and Plucker coordinates in type B, C, D
How to find image of a complex function with given constraints?
Is there such a thing as a proper verb, like a proper noun?
vector calculus integration identity problem
The Next CEO of Stack Overflow$LaTeX$ format copy problemIs it possible to do vector calculus in Mathematica?Dipolar magnetic field lines inside a cylinderComparing unit normal definition in calculus with FrenetSerretSystemManipulating curl and div of a vector in spherical coordinatesIntegration with a matrix as the the integrandGet the vector Norm without absolute values?matrix calculus with types (similar to matrixcalculus.org)How do I verify a vector identity using Mathematica?Einstein summation convention for symbolic vector calculusVector calculus with index notation
$begingroup$
This is a follow up from another post . I was using the integration symbol available in the Basic Math Assistance available in Wolfram Mathematica.
I am new to vector calculus operations. There is a known identity found in the textbooks
$$int _{4 pi }hat{s} (hat{s}cdot A) d omega=frac{4 pi}{3}A$$
I have no idea how to do this type of integration. This is what I tried but return a dissaster
Integrate[s*(Dot[s, A]), s, {0, 4 [Pi]}]
Also , without success
Integrate[{Sin[[Theta]],
Cos[[Theta]]}*(Dot[{Sin[[Theta]], Cos[[Theta]]}, {a1,
a2}]), [Theta], {0, 4 [Pi]}]
It is obviosu that I am doing something fundamentally not correct. I go to WM documentation on Vector Calculus but does not offer much substance or examples. How will you enter the equation above in order to return the identity in the right?
UPDATE 1
In respond to comment, here is a copy of the text. This is from page 10 Optical-Thermal Response of Laser-Irradiated Tissue ISBN 9789048188307
$$w$$ is the surface area of a sphere in solid angle steradian. s is the directional vector of a pencil of radiation located inside the sphere
vector-calculus
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
$endgroup$
|
show 3 more comments
$begingroup$
This is a follow up from another post . I was using the integration symbol available in the Basic Math Assistance available in Wolfram Mathematica.
I am new to vector calculus operations. There is a known identity found in the textbooks
$$int _{4 pi }hat{s} (hat{s}cdot A) d omega=frac{4 pi}{3}A$$
I have no idea how to do this type of integration. This is what I tried but return a dissaster
Integrate[s*(Dot[s, A]), s, {0, 4 [Pi]}]
Also , without success
Integrate[{Sin[[Theta]],
Cos[[Theta]]}*(Dot[{Sin[[Theta]], Cos[[Theta]]}, {a1,
a2}]), [Theta], {0, 4 [Pi]}]
It is obviosu that I am doing something fundamentally not correct. I go to WM documentation on Vector Calculus but does not offer much substance or examples. How will you enter the equation above in order to return the identity in the right?
UPDATE 1
In respond to comment, here is a copy of the text. This is from page 10 Optical-Thermal Response of Laser-Irradiated Tissue ISBN 9789048188307
$$w$$ is the surface area of a sphere in solid angle steradian. s is the directional vector of a pencil of radiation located inside the sphere
vector-calculus
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
$endgroup$
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
2
$begingroup$
Here's my guess:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ]--- or this:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]
$endgroup$
– Michael E2
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago
|
show 3 more comments
$begingroup$
This is a follow up from another post . I was using the integration symbol available in the Basic Math Assistance available in Wolfram Mathematica.
I am new to vector calculus operations. There is a known identity found in the textbooks
$$int _{4 pi }hat{s} (hat{s}cdot A) d omega=frac{4 pi}{3}A$$
I have no idea how to do this type of integration. This is what I tried but return a dissaster
Integrate[s*(Dot[s, A]), s, {0, 4 [Pi]}]
Also , without success
Integrate[{Sin[[Theta]],
Cos[[Theta]]}*(Dot[{Sin[[Theta]], Cos[[Theta]]}, {a1,
a2}]), [Theta], {0, 4 [Pi]}]
It is obviosu that I am doing something fundamentally not correct. I go to WM documentation on Vector Calculus but does not offer much substance or examples. How will you enter the equation above in order to return the identity in the right?
UPDATE 1
In respond to comment, here is a copy of the text. This is from page 10 Optical-Thermal Response of Laser-Irradiated Tissue ISBN 9789048188307
$$w$$ is the surface area of a sphere in solid angle steradian. s is the directional vector of a pencil of radiation located inside the sphere
vector-calculus
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
$endgroup$
This is a follow up from another post . I was using the integration symbol available in the Basic Math Assistance available in Wolfram Mathematica.
I am new to vector calculus operations. There is a known identity found in the textbooks
$$int _{4 pi }hat{s} (hat{s}cdot A) d omega=frac{4 pi}{3}A$$
I have no idea how to do this type of integration. This is what I tried but return a dissaster
Integrate[s*(Dot[s, A]), s, {0, 4 [Pi]}]
Also , without success
Integrate[{Sin[[Theta]],
Cos[[Theta]]}*(Dot[{Sin[[Theta]], Cos[[Theta]]}, {a1,
a2}]), [Theta], {0, 4 [Pi]}]
It is obviosu that I am doing something fundamentally not correct. I go to WM documentation on Vector Calculus but does not offer much substance or examples. How will you enter the equation above in order to return the identity in the right?
UPDATE 1
In respond to comment, here is a copy of the text. This is from page 10 Optical-Thermal Response of Laser-Irradiated Tissue ISBN 9789048188307
$$w$$ is the surface area of a sphere in solid angle steradian. s is the directional vector of a pencil of radiation located inside the sphere
vector-calculus
vector-calculus
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
edited 45 mins ago
J. M. is slightly pensive♦
98.8k10311467
98.8k10311467
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
asked 2 hours ago
Jose Enrique CalderonJose Enrique Calderon
1,058718
1,058718
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
2
$begingroup$
Here's my guess:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ]--- or this:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]
$endgroup$
– Michael E2
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago
|
show 3 more comments
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
2
$begingroup$
Here's my guess:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ]--- or this:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]
$endgroup$
– Michael E2
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
2
2
$begingroup$
Here's my guess:
With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ] --- or this: With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]$endgroup$
– Michael E2
1 hour ago
$begingroup$
Here's my guess:
With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ] --- or this: With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]$endgroup$
– Michael E2
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
1
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Here's my guess:
With[{s = {x, y, z},
A = {A1, A2, A3}}, Integrate[s (s.A), s ∈ Sphere[]] ]
(* {(4 A1 π)/3, (4 A2 π)/3, (4 A3 π)/3} *)
--- or this:
With[{s = {x, y, z}, A = {A1, A2, A3}},
Integrate[s (s.A), s ∈ Sphere[]] == 4 Pi/3 A ]
(* True *)
answered 1 hour ago
Michael E2Michael E2
150k12203482
$endgroup$
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
$begingroup$
@Jose The syntax{s, 0, 4 Pi}already implies one-dimensionalsfrom Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.
$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can useRegionIntersection[]withSphere[]and eitherConicHullRegion[]orHalfSpace[].
$endgroup$
– J. M. is slightly pensive♦
47 mins ago
|
show 2 more comments
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "387"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathematica.stackexchange.com%2fquestions%2f194347%2fvector-calculus-integration-identity-problem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here's my guess:
With[{s = {x, y, z},
A = {A1, A2, A3}}, Integrate[s (s.A), s ∈ Sphere[]] ]
(* {(4 A1 π)/3, (4 A2 π)/3, (4 A3 π)/3} *)
--- or this:
With[{s = {x, y, z}, A = {A1, A2, A3}},
Integrate[s (s.A), s ∈ Sphere[]] == 4 Pi/3 A ]
(* True *)
answered 1 hour ago
Michael E2Michael E2
150k12203482
$endgroup$
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
$begingroup$
@Jose The syntax{s, 0, 4 Pi}already implies one-dimensionalsfrom Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.
$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can useRegionIntersection[]withSphere[]and eitherConicHullRegion[]orHalfSpace[].
$endgroup$
– J. M. is slightly pensive♦
47 mins ago
|
show 2 more comments
$begingroup$
Here's my guess:
With[{s = {x, y, z},
A = {A1, A2, A3}}, Integrate[s (s.A), s ∈ Sphere[]] ]
(* {(4 A1 π)/3, (4 A2 π)/3, (4 A3 π)/3} *)
--- or this:
With[{s = {x, y, z}, A = {A1, A2, A3}},
Integrate[s (s.A), s ∈ Sphere[]] == 4 Pi/3 A ]
(* True *)
answered 1 hour ago
Michael E2Michael E2
150k12203482
$endgroup$
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
$begingroup$
@Jose The syntax{s, 0, 4 Pi}already implies one-dimensionalsfrom Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.
$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can useRegionIntersection[]withSphere[]and eitherConicHullRegion[]orHalfSpace[].
$endgroup$
– J. M. is slightly pensive♦
47 mins ago
|
show 2 more comments
$begingroup$
Here's my guess:
With[{s = {x, y, z},
A = {A1, A2, A3}}, Integrate[s (s.A), s ∈ Sphere[]] ]
(* {(4 A1 π)/3, (4 A2 π)/3, (4 A3 π)/3} *)
--- or this:
With[{s = {x, y, z}, A = {A1, A2, A3}},
Integrate[s (s.A), s ∈ Sphere[]] == 4 Pi/3 A ]
(* True *)
answered 1 hour ago
Michael E2Michael E2
150k12203482
$endgroup$
Here's my guess:
With[{s = {x, y, z},
A = {A1, A2, A3}}, Integrate[s (s.A), s ∈ Sphere[]] ]
(* {(4 A1 π)/3, (4 A2 π)/3, (4 A3 π)/3} *)
--- or this:
With[{s = {x, y, z}, A = {A1, A2, A3}},
Integrate[s (s.A), s ∈ Sphere[]] == 4 Pi/3 A ]
(* True *)
answered 1 hour ago
Michael E2Michael E2
150k12203482
answered 1 hour ago
Michael E2Michael E2
150k12203482
answered 1 hour ago
Michael E2Michael E2
150k12203482
answered 1 hour ago
Michael E2Michael E2
150k12203482
150k12203482
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
$begingroup$
@Jose The syntax{s, 0, 4 Pi}already implies one-dimensionalsfrom Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.
$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can useRegionIntersection[]withSphere[]and eitherConicHullRegion[]orHalfSpace[].
$endgroup$
– J. M. is slightly pensive♦
47 mins ago
|
show 2 more comments
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
$begingroup$
@Jose The syntax{s, 0, 4 Pi}already implies one-dimensionalsfrom Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.
$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can useRegionIntersection[]withSphere[]and eitherConicHullRegion[]orHalfSpace[].
$endgroup$
– J. M. is slightly pensive♦
47 mins ago
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
$begingroup$
Why it simply does not work with limits of integration {s,0,4Pi}
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
1
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Jose, the author was being lazy with the limits (basically, shorter than saying "integrate over the whole area of the unit sphere"). It is fine to be lazy in mathematics, but not so much when programming.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
$begingroup$
@J.M. is slightly pensive Ok.. but why Mathematica function proposed in the answear does not work with With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), {s,0,4Pi}] ]
$endgroup$
– Jose Enrique Calderon
58 mins ago
1
1
$begingroup$
@Jose The syntax
{s, 0, 4 Pi} already implies one-dimensional s from Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.$endgroup$
– J. M. is slightly pensive♦
55 mins ago
$begingroup$
@Jose The syntax
{s, 0, 4 Pi} already implies one-dimensional s from Mathematica's view, while in the "abuse of notation" used in your reference, $hat{s}$ is implied to be a vector.$endgroup$
– J. M. is slightly pensive♦
55 mins ago
1
1
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can use
RegionIntersection[] with Sphere[] and either ConicHullRegion[] or HalfSpace[].$endgroup$
– J. M. is slightly pensive♦
47 mins ago
$begingroup$
@Jose: the easiest way is that you have to switch to spherical coordinates if you need to integrate across arbitrary angles. If you insist on keeping yourself to regions, you can use
RegionIntersection[] with Sphere[] and either ConicHullRegion[] or HalfSpace[].$endgroup$
– J. M. is slightly pensive♦
47 mins ago
|
show 2 more comments
Thanks for contributing an answer to Mathematica Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathematica.stackexchange.com%2fquestions%2f194347%2fvector-calculus-integration-identity-problem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
What are $s$ and $omega$ supposed to be? It might be helpful if you can give an example of the textbook with the formula.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
2
$begingroup$
Here's my guess:
With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] ]--- or this:With[{s = {x, y, z}, A = {A1, A2, A3}}, Integrate[s (s.A), s [Element] Sphere[]] == 4 Pi/3 A ]$endgroup$
– Michael E2
1 hour ago
$begingroup$
@Michael, yes, that does seem to be it. This is why people should always define what their variables mean in their formulae.
$endgroup$
– J. M. is slightly pensive♦
1 hour ago
$begingroup$
@Michael E2 please post it as an answear for upvote
$endgroup$
– Jose Enrique Calderon
1 hour ago
1
$begingroup$
I've never seen this author's notation. My guess is that $int_{4pi}cdots$ means the integral over the sphere of measure $4pi$, i.e., the unit sphere.
$endgroup$
– Michael E2
51 mins ago