Mixing
Green's Theorem for 3 dimensions
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I'm reading Introduction to Fourier Optics - J. Goodman and got to this statements which is referred to as Green's Theorem:
Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, and let $S$ be a closed surface surrounding a volume $V$. If $U$, $G$, and their first and second partial derivatives are single-valued and continuous within and on $S$, then we have $$iiint_V Bigr(U nabla^2 G - G nabla^2 UBigr) dv = iint_S Bigr(Ufrac{partial G}{partial n} -Gfrac{partial U}{partial n}Bigr)ds$$ where $frac{partial}{partial n}$ signifies a partial derivative in the outward normal direction at each point on $S$.
I remembered to have learned about Green's Theorem in the mathematics courses but did not recall this form. I looked it up and got to what I remembered, that is a 2-dimensional case given by $$iint_S Bigr(frac{partial P}{partial x} - frac{partial Q}{partial y}Bigr)dxdy=int_C Qdx+ Pdy$$ which could be considered a particular case of Stokes' Theorem.
Are the two formulas related (which means I missed the link, in which case, can someone suggest some directions), or are they different formulas and the author in the book misfortunately used "Green's Theorem"?
greens-theorem
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
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add a comment |
$begingroup$
I'm reading Introduction to Fourier Optics - J. Goodman and got to this statements which is referred to as Green's Theorem:
Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, and let $S$ be a closed surface surrounding a volume $V$. If $U$, $G$, and their first and second partial derivatives are single-valued and continuous within and on $S$, then we have $$iiint_V Bigr(U nabla^2 G - G nabla^2 UBigr) dv = iint_S Bigr(Ufrac{partial G}{partial n} -Gfrac{partial U}{partial n}Bigr)ds$$ where $frac{partial}{partial n}$ signifies a partial derivative in the outward normal direction at each point on $S$.
I remembered to have learned about Green's Theorem in the mathematics courses but did not recall this form. I looked it up and got to what I remembered, that is a 2-dimensional case given by $$iint_S Bigr(frac{partial P}{partial x} - frac{partial Q}{partial y}Bigr)dxdy=int_C Qdx+ Pdy$$ which could be considered a particular case of Stokes' Theorem.
Are the two formulas related (which means I missed the link, in which case, can someone suggest some directions), or are they different formulas and the author in the book misfortunately used "Green's Theorem"?
greens-theorem
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
$endgroup$
add a comment |
$begingroup$
I'm reading Introduction to Fourier Optics - J. Goodman and got to this statements which is referred to as Green's Theorem:
Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, and let $S$ be a closed surface surrounding a volume $V$. If $U$, $G$, and their first and second partial derivatives are single-valued and continuous within and on $S$, then we have $$iiint_V Bigr(U nabla^2 G - G nabla^2 UBigr) dv = iint_S Bigr(Ufrac{partial G}{partial n} -Gfrac{partial U}{partial n}Bigr)ds$$ where $frac{partial}{partial n}$ signifies a partial derivative in the outward normal direction at each point on $S$.
I remembered to have learned about Green's Theorem in the mathematics courses but did not recall this form. I looked it up and got to what I remembered, that is a 2-dimensional case given by $$iint_S Bigr(frac{partial P}{partial x} - frac{partial Q}{partial y}Bigr)dxdy=int_C Qdx+ Pdy$$ which could be considered a particular case of Stokes' Theorem.
Are the two formulas related (which means I missed the link, in which case, can someone suggest some directions), or are they different formulas and the author in the book misfortunately used "Green's Theorem"?
greens-theorem
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
$endgroup$
I'm reading Introduction to Fourier Optics - J. Goodman and got to this statements which is referred to as Green's Theorem:
Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, and let $S$ be a closed surface surrounding a volume $V$. If $U$, $G$, and their first and second partial derivatives are single-valued and continuous within and on $S$, then we have $$iiint_V Bigr(U nabla^2 G - G nabla^2 UBigr) dv = iint_S Bigr(Ufrac{partial G}{partial n} -Gfrac{partial U}{partial n}Bigr)ds$$ where $frac{partial}{partial n}$ signifies a partial derivative in the outward normal direction at each point on $S$.
I remembered to have learned about Green's Theorem in the mathematics courses but did not recall this form. I looked it up and got to what I remembered, that is a 2-dimensional case given by $$iint_S Bigr(frac{partial P}{partial x} - frac{partial Q}{partial y}Bigr)dxdy=int_C Qdx+ Pdy$$ which could be considered a particular case of Stokes' Theorem.
Are the two formulas related (which means I missed the link, in which case, can someone suggest some directions), or are they different formulas and the author in the book misfortunately used "Green's Theorem"?
greens-theorem
greens-theorem
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
asked Nov 27 '17 at 10:26
Victor PaleaVictor Palea
301312
301312
add a comment |
add a comment |
1 Answer
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I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
$endgroup$
add a comment |
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$begingroup$
I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
$endgroup$
add a comment |
$begingroup$
I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
$endgroup$
add a comment |
$begingroup$
I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
$endgroup$
I searched further and after some luck with the terms used in the search engine I got to what are called Green's Identities. The formula I had trouble with is the second identity.
Now that I found the "correct" name of it, I could look for links to the Stokes theorem and got to this material, so if someone has a similar problem, you can start from these.
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
answered Nov 28 '17 at 10:33
Victor PaleaVictor Palea
301312
301312
add a comment |
add a comment |
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