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How to deduce Green's formulas from Gauss-Green's theorem?
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Gauss-Green Theorem
Gauss-Green
Green's Formulas
Green
What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware of that the process would be similar to integration by parts in 1-dimension, but the format of these formulas still seem different. So I'm still stucked at where to start the integration, what should be the u and what should be the dv of the process of integration by parts, or I'm totally at the wrong track.
Could anyone please give me a hint? Thanks in advance!
greens-theorem
edited Jul 26 at 13:19
Bernard
110k635102
asked Jul 26 at 11:43
Evelyn Venne
143
add a comment |Â
up vote
0
down vote
favorite
Gauss-Green Theorem
Gauss-Green
Green's Formulas
Green
What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware of that the process would be similar to integration by parts in 1-dimension, but the format of these formulas still seem different. So I'm still stucked at where to start the integration, what should be the u and what should be the dv of the process of integration by parts, or I'm totally at the wrong track.
Could anyone please give me a hint? Thanks in advance!
greens-theorem
edited Jul 26 at 13:19
Bernard
110k635102
asked Jul 26 at 11:43
Evelyn Venne
143
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Gauss-Green Theorem
Gauss-Green
Green's Formulas
Green
What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware of that the process would be similar to integration by parts in 1-dimension, but the format of these formulas still seem different. So I'm still stucked at where to start the integration, what should be the u and what should be the dv of the process of integration by parts, or I'm totally at the wrong track.
Could anyone please give me a hint? Thanks in advance!
greens-theorem
edited Jul 26 at 13:19
Bernard
110k635102
asked Jul 26 at 11:43
Evelyn Venne
143
Gauss-Green Theorem
Gauss-Green
Green's Formulas
Green
What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware of that the process would be similar to integration by parts in 1-dimension, but the format of these formulas still seem different. So I'm still stucked at where to start the integration, what should be the u and what should be the dv of the process of integration by parts, or I'm totally at the wrong track.
Could anyone please give me a hint? Thanks in advance!
greens-theorem
edited Jul 26 at 13:19
Bernard
110k635102
asked Jul 26 at 11:43
Evelyn Venne
143
edited Jul 26 at 13:19
Bernard
110k635102
edited Jul 26 at 13:19
Bernard
110k635102
edited Jul 26 at 13:19
Bernard
110k635102
110k635102
asked Jul 26 at 11:43
Evelyn Venne
143
asked Jul 26 at 11:43
Evelyn Venne
143
asked Jul 26 at 11:43
Evelyn Venne
143
143
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since
$$ partial_x_i(f(x)g(x)) = partial_x_i(f(x))g(x) + partial_x_i(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $Delta f$ and $fracpartial fpartial nu$.
answered Jul 26 at 13:28
ertl
445110
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since
$$ partial_x_i(f(x)g(x)) = partial_x_i(f(x))g(x) + partial_x_i(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $Delta f$ and $fracpartial fpartial nu$.
answered Jul 26 at 13:28
ertl
445110
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
add a comment |Â
up vote
0
down vote
accepted
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since
$$ partial_x_i(f(x)g(x)) = partial_x_i(f(x))g(x) + partial_x_i(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $Delta f$ and $fracpartial fpartial nu$.
answered Jul 26 at 13:28
ertl
445110
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since
$$ partial_x_i(f(x)g(x)) = partial_x_i(f(x))g(x) + partial_x_i(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $Delta f$ and $fracpartial fpartial nu$.
answered Jul 26 at 13:28
ertl
445110
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since
$$ partial_x_i(f(x)g(x)) = partial_x_i(f(x))g(x) + partial_x_i(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $Delta f$ and $fracpartial fpartial nu$.
answered Jul 26 at 13:28
ertl
445110
answered Jul 26 at 13:28
ertl
445110
answered Jul 26 at 13:28
ertl
445110
answered Jul 26 at 13:28
ertl
445110
445110
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
add a comment |Â
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
Thanks for your answer. But on the right hand side of the first formula, where does the minus sign come from? It turns out to be a plus from the product rule. Does it matter?
– Evelyn Venne
Jul 26 at 14:06
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
It's on the other side of the inequality now, that's why there is a minus sign.
– ertl
Jul 26 at 14:11
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
Ooooh I was just about to delete the comment as I got it myself.. Yea I'm a bit slow on it, thank you :)
– Evelyn Venne
Jul 26 at 14:15
add a comment |Â
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