Can someone check if my proof is correct?

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I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region
$$int_S fractextbfr.nr^2 dS, = int_H fracdHr^2 ,$$



My approach towards this question was that r = xi +yj + zk then if we try to find n it is equal to



$$fracgradphimid gradphimid = frac(1,1,1)sqrt3$$



Therefore if we open dS, we would end up with the following



$$int_S' fractextbfr.nr^2fracdS'textbfn.k ,$$



If we take k to be equal to (0,0,1), we would end up with the following equation,



$$int_S' fractextbfr.(1,1,1)r^2fracdS'1 ,$$



Then r could be changed to (x,y,z)



$$int_S' frac(x,y,z).(1,1,1)r^2dS' ,$$



Therefore



$$int_S' frac(x,y,z)r^2dS' ,$$



Then changing for region H



$$iiint_H frac(x,y,z)r^2dxdydz ,$$



is equal to



$$int_H fracdHr^2 ,$$



Can somebody please check if my approach is correct?




surfaces surface-integrals




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  • Proof-verification tag is to be used when you provide a proof of something.
    – Arnaud Mortier
    12 hours ago














up vote
-3
down vote

favorite












I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region
$$int_S fractextbfr.nr^2 dS, = int_H fracdHr^2 ,$$



My approach towards this question was that r = xi +yj + zk then if we try to find n it is equal to



$$fracgradphimid gradphimid = frac(1,1,1)sqrt3$$



Therefore if we open dS, we would end up with the following



$$int_S' fractextbfr.nr^2fracdS'textbfn.k ,$$



If we take k to be equal to (0,0,1), we would end up with the following equation,



$$int_S' fractextbfr.(1,1,1)r^2fracdS'1 ,$$



Then r could be changed to (x,y,z)



$$int_S' frac(x,y,z).(1,1,1)r^2dS' ,$$



Therefore



$$int_S' frac(x,y,z)r^2dS' ,$$



Then changing for region H



$$iiint_H frac(x,y,z)r^2dxdydz ,$$



is equal to



$$int_H fracdHr^2 ,$$



Can somebody please check if my approach is correct?




surfaces surface-integrals




share|cite|improve this question





















  • Proof-verification tag is to be used when you provide a proof of something.
    – Arnaud Mortier
    12 hours ago












up vote
-3
down vote

favorite









up vote
-3
down vote

favorite











I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region
$$int_S fractextbfr.nr^2 dS, = int_H fracdHr^2 ,$$



My approach towards this question was that r = xi +yj + zk then if we try to find n it is equal to



$$fracgradphimid gradphimid = frac(1,1,1)sqrt3$$



Therefore if we open dS, we would end up with the following



$$int_S' fractextbfr.nr^2fracdS'textbfn.k ,$$



If we take k to be equal to (0,0,1), we would end up with the following equation,



$$int_S' fractextbfr.(1,1,1)r^2fracdS'1 ,$$



Then r could be changed to (x,y,z)



$$int_S' frac(x,y,z).(1,1,1)r^2dS' ,$$



Therefore



$$int_S' frac(x,y,z)r^2dS' ,$$



Then changing for region H



$$iiint_H frac(x,y,z)r^2dxdydz ,$$



is equal to



$$int_H fracdHr^2 ,$$



Can somebody please check if my approach is correct?




surfaces surface-integrals




share|cite|improve this question













I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region
$$int_S fractextbfr.nr^2 dS, = int_H fracdHr^2 ,$$



My approach towards this question was that r = xi +yj + zk then if we try to find n it is equal to



$$fracgradphimid gradphimid = frac(1,1,1)sqrt3$$



Therefore if we open dS, we would end up with the following



$$int_S' fractextbfr.nr^2fracdS'textbfn.k ,$$



If we take k to be equal to (0,0,1), we would end up with the following equation,



$$int_S' fractextbfr.(1,1,1)r^2fracdS'1 ,$$



Then r could be changed to (x,y,z)



$$int_S' frac(x,y,z).(1,1,1)r^2dS' ,$$



Therefore



$$int_S' frac(x,y,z)r^2dS' ,$$



Then changing for region H



$$iiint_H frac(x,y,z)r^2dxdydz ,$$



is equal to



$$int_H fracdHr^2 ,$$



Can somebody please check if my approach is correct?





surfaces surface-integrals





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share|cite|improve this question




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edited 11 hours ago
























asked 12 hours ago









user7075815

11




11











  • Proof-verification tag is to be used when you provide a proof of something.
    – Arnaud Mortier
    12 hours ago
















  • Proof-verification tag is to be used when you provide a proof of something.
    – Arnaud Mortier
    12 hours ago















Proof-verification tag is to be used when you provide a proof of something.
– Arnaud Mortier
12 hours ago




Proof-verification tag is to be used when you provide a proof of something.
– Arnaud Mortier
12 hours ago










1 Answer
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Hint




$$nablacdotleft(hatrover rright) = 1over r^2$$







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  • Why would you take the divergence in this case?
    – user7075815
    11 hours ago










  • Using the divergence theorem gives a trivial proof for the problem
    – Davide Morgante
    11 hours ago










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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













Hint




$$nablacdotleft(hatrover rright) = 1over r^2$$







share|cite|improve this answer





















  • Why would you take the divergence in this case?
    – user7075815
    11 hours ago










  • Using the divergence theorem gives a trivial proof for the problem
    – Davide Morgante
    11 hours ago














up vote
0
down vote













Hint




$$nablacdotleft(hatrover rright) = 1over r^2$$







share|cite|improve this answer





















  • Why would you take the divergence in this case?
    – user7075815
    11 hours ago










  • Using the divergence theorem gives a trivial proof for the problem
    – Davide Morgante
    11 hours ago












up vote
0
down vote










up vote
0
down vote









Hint




$$nablacdotleft(hatrover rright) = 1over r^2$$







share|cite|improve this answer













Hint




$$nablacdotleft(hatrover rright) = 1over r^2$$








share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered 12 hours ago









Davide Morgante

1,574120




1,574120











  • Why would you take the divergence in this case?
    – user7075815
    11 hours ago










  • Using the divergence theorem gives a trivial proof for the problem
    – Davide Morgante
    11 hours ago
















  • Why would you take the divergence in this case?
    – user7075815
    11 hours ago










  • Using the divergence theorem gives a trivial proof for the problem
    – Davide Morgante
    11 hours ago















Why would you take the divergence in this case?
– user7075815
11 hours ago




Why would you take the divergence in this case?
– user7075815
11 hours ago












Using the divergence theorem gives a trivial proof for the problem
– Davide Morgante
11 hours ago




Using the divergence theorem gives a trivial proof for the problem
– Davide Morgante
11 hours ago












 

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