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Help with vector calculus integration identity proof
Using divergence theorem on sphereTrouble understanding a common vector calculus example2 calculus questions with integration - check meIntegration of bundle-valued differential formsHow do I calculate the solid angle of the intersection between a right circular cone and a 'square' cone?Help with limits of integration in spherical coordinatesIntegration by parts with surface integralsArea of a surface using integration. Confusion with aspect of formal definition.2D Line Integral of a Polar FunctionHow to calculate solid angle of nonspherical surface?
$begingroup$
I have limited knowlledge of how integration of a vector is correctly apply. In the following example, a vector is multiply to a vector dot product with an area. The integration is around the area of the sphere 4 Pi steradian.
I am interested to see how finding proof of this vector integral. Not sure how to get started. I have found this identity in textbooks but cant see how this integration results in 1/4 the surface area of a sphere.
$$int _4 pi oversetrightharpoonup s (oversetrightharpoonup scdot A) d omega=frac13 (4 pi ) A$$
Any good reference?
integration
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
add a comment |
$begingroup$
I have limited knowlledge of how integration of a vector is correctly apply. In the following example, a vector is multiply to a vector dot product with an area. The integration is around the area of the sphere 4 Pi steradian.
I am interested to see how finding proof of this vector integral. Not sure how to get started. I have found this identity in textbooks but cant see how this integration results in 1/4 the surface area of a sphere.
$$int _4 pi oversetrightharpoonup s (oversetrightharpoonup scdot A) d omega=frac13 (4 pi ) A$$
Any good reference?
integration
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
add a comment |
$begingroup$
I have limited knowlledge of how integration of a vector is correctly apply. In the following example, a vector is multiply to a vector dot product with an area. The integration is around the area of the sphere 4 Pi steradian.
I am interested to see how finding proof of this vector integral. Not sure how to get started. I have found this identity in textbooks but cant see how this integration results in 1/4 the surface area of a sphere.
$$int _4 pi oversetrightharpoonup s (oversetrightharpoonup scdot A) d omega=frac13 (4 pi ) A$$
Any good reference?
integration
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
I have limited knowlledge of how integration of a vector is correctly apply. In the following example, a vector is multiply to a vector dot product with an area. The integration is around the area of the sphere 4 Pi steradian.
I am interested to see how finding proof of this vector integral. Not sure how to get started. I have found this identity in textbooks but cant see how this integration results in 1/4 the surface area of a sphere.
$$int _4 pi oversetrightharpoonup s (oversetrightharpoonup scdot A) d omega=frac13 (4 pi ) A$$
Any good reference?
integration
integration
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
asked Apr 1 at 2:29
Jose Enrique CalderonJose Enrique Calderon
1177
1177
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$newcommandbbx[1],bbox[15px,border:1px groove navy]displaystyle#1,
newcommandbraces[1]leftlbrace,#1,rightrbrace
newcommandbracks[1]leftlbrack,#1,rightrbrack
newcommandddmathrmd
newcommandds[1]displaystyle#1
newcommandexpo[1],mathrme^#1,
newcommandicmathrmi
newcommandmc[1]mathcal#1
newcommandmrm[1]mathrm#1
newcommandpars[1]left(,#1,right)
newcommandpartiald[3][]fracpartial^#1 #2partial #3^#1
newcommandroot[2][],sqrt[#1],#2,,
newcommandtotald[3][]fracmathrmd^#1 #2mathrmd #3^#1
newcommandverts[1]leftvert,#1,rightvert$
Your question is NOT so clear but I guess that the answer is something like
beginalign
int_4pivecsparsvecscdotvecAddomega & =
sum_i,j in bracesx,y,zhate_iA_jint_4pis_is_j,ddomega =
sum_i,j in bracesx,y,z
hate_iA_j,delta_ijint_4pis_i^2,ddomega
\[5mm] & =
pars1 over 3int_4pi overbracesum_k in bracesx,y,zs_k^2^ds= 1
,ddomega
overbracesum_i in bracesx,y,zhate_iA_i^ds= vecA
\[5mm] & =
pars1 over 3 overbraceint_4piddomega^ds= 4pivecA =
bbx1 over 3pars4pivecA
endalign
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
$endgroup$
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$newcommandbbx[1],bbox[15px,border:1px groove navy]displaystyle#1,
newcommandbraces[1]leftlbrace,#1,rightrbrace
newcommandbracks[1]leftlbrack,#1,rightrbrack
newcommandddmathrmd
newcommandds[1]displaystyle#1
newcommandexpo[1],mathrme^#1,
newcommandicmathrmi
newcommandmc[1]mathcal#1
newcommandmrm[1]mathrm#1
newcommandpars[1]left(,#1,right)
newcommandpartiald[3][]fracpartial^#1 #2partial #3^#1
newcommandroot[2][],sqrt[#1],#2,,
newcommandtotald[3][]fracmathrmd^#1 #2mathrmd #3^#1
newcommandverts[1]leftvert,#1,rightvert$
Your question is NOT so clear but I guess that the answer is something like
beginalign
int_4pivecsparsvecscdotvecAddomega & =
sum_i,j in bracesx,y,zhate_iA_jint_4pis_is_j,ddomega =
sum_i,j in bracesx,y,z
hate_iA_j,delta_ijint_4pis_i^2,ddomega
\[5mm] & =
pars1 over 3int_4pi overbracesum_k in bracesx,y,zs_k^2^ds= 1
,ddomega
overbracesum_i in bracesx,y,zhate_iA_i^ds= vecA
\[5mm] & =
pars1 over 3 overbraceint_4piddomega^ds= 4pivecA =
bbx1 over 3pars4pivecA
endalign
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
$endgroup$
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
add a comment |
$begingroup$
$newcommandbbx[1],bbox[15px,border:1px groove navy]displaystyle#1,
newcommandbraces[1]leftlbrace,#1,rightrbrace
newcommandbracks[1]leftlbrack,#1,rightrbrack
newcommandddmathrmd
newcommandds[1]displaystyle#1
newcommandexpo[1],mathrme^#1,
newcommandicmathrmi
newcommandmc[1]mathcal#1
newcommandmrm[1]mathrm#1
newcommandpars[1]left(,#1,right)
newcommandpartiald[3][]fracpartial^#1 #2partial #3^#1
newcommandroot[2][],sqrt[#1],#2,,
newcommandtotald[3][]fracmathrmd^#1 #2mathrmd #3^#1
newcommandverts[1]leftvert,#1,rightvert$
Your question is NOT so clear but I guess that the answer is something like
beginalign
int_4pivecsparsvecscdotvecAddomega & =
sum_i,j in bracesx,y,zhate_iA_jint_4pis_is_j,ddomega =
sum_i,j in bracesx,y,z
hate_iA_j,delta_ijint_4pis_i^2,ddomega
\[5mm] & =
pars1 over 3int_4pi overbracesum_k in bracesx,y,zs_k^2^ds= 1
,ddomega
overbracesum_i in bracesx,y,zhate_iA_i^ds= vecA
\[5mm] & =
pars1 over 3 overbraceint_4piddomega^ds= 4pivecA =
bbx1 over 3pars4pivecA
endalign
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
$endgroup$
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
add a comment |
$begingroup$
$newcommandbbx[1],bbox[15px,border:1px groove navy]displaystyle#1,
newcommandbraces[1]leftlbrace,#1,rightrbrace
newcommandbracks[1]leftlbrack,#1,rightrbrack
newcommandddmathrmd
newcommandds[1]displaystyle#1
newcommandexpo[1],mathrme^#1,
newcommandicmathrmi
newcommandmc[1]mathcal#1
newcommandmrm[1]mathrm#1
newcommandpars[1]left(,#1,right)
newcommandpartiald[3][]fracpartial^#1 #2partial #3^#1
newcommandroot[2][],sqrt[#1],#2,,
newcommandtotald[3][]fracmathrmd^#1 #2mathrmd #3^#1
newcommandverts[1]leftvert,#1,rightvert$
Your question is NOT so clear but I guess that the answer is something like
beginalign
int_4pivecsparsvecscdotvecAddomega & =
sum_i,j in bracesx,y,zhate_iA_jint_4pis_is_j,ddomega =
sum_i,j in bracesx,y,z
hate_iA_j,delta_ijint_4pis_i^2,ddomega
\[5mm] & =
pars1 over 3int_4pi overbracesum_k in bracesx,y,zs_k^2^ds= 1
,ddomega
overbracesum_i in bracesx,y,zhate_iA_i^ds= vecA
\[5mm] & =
pars1 over 3 overbraceint_4piddomega^ds= 4pivecA =
bbx1 over 3pars4pivecA
endalign
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
$endgroup$
$newcommandbbx[1],bbox[15px,border:1px groove navy]displaystyle#1,
newcommandbraces[1]leftlbrace,#1,rightrbrace
newcommandbracks[1]leftlbrack,#1,rightrbrack
newcommandddmathrmd
newcommandds[1]displaystyle#1
newcommandexpo[1],mathrme^#1,
newcommandicmathrmi
newcommandmc[1]mathcal#1
newcommandmrm[1]mathrm#1
newcommandpars[1]left(,#1,right)
newcommandpartiald[3][]fracpartial^#1 #2partial #3^#1
newcommandroot[2][],sqrt[#1],#2,,
newcommandtotald[3][]fracmathrmd^#1 #2mathrmd #3^#1
newcommandverts[1]leftvert,#1,rightvert$
Your question is NOT so clear but I guess that the answer is something like
beginalign
int_4pivecsparsvecscdotvecAddomega & =
sum_i,j in bracesx,y,zhate_iA_jint_4pis_is_j,ddomega =
sum_i,j in bracesx,y,z
hate_iA_j,delta_ijint_4pis_i^2,ddomega
\[5mm] & =
pars1 over 3int_4pi overbracesum_k in bracesx,y,zs_k^2^ds= 1
,ddomega
overbracesum_i in bracesx,y,zhate_iA_i^ds= vecA
\[5mm] & =
pars1 over 3 overbraceint_4piddomega^ds= 4pivecA =
bbx1 over 3pars4pivecA
endalign
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
edited Apr 1 at 16:46
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
answered Apr 1 at 5:38
Felix MarinFelix Marin
68.9k7110147
68.9k7110147
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
add a comment |
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
Why was my question not so clear? I am new to subject. Your respond hit right on the bulls eye that I was looking after. Thanks for your response
$endgroup$
– Jose Enrique Calderon
Apr 1 at 17:40
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
$begingroup$
@JoseEnriqueCalderon Well, you didn't explicitly say what was $displaystylevecs$. Anyway, I guess it was a vector position inside the sphere. Don't worry: Everything is fine. Thanks.
$endgroup$
– Felix Marin
Apr 1 at 23:56
add a comment |
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