An intuitive explanation for Green theorem and Divergence theorem
$begingroup$
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the divergence. If we imagine a vector field as the flow of an incompressible fluid, the divergence of a certain domain in space would be linked to 'how much' matter is created inside the domain. If matter is created within a certain region and the fluid is incompressible, it would make sense that the fluid would have to escape this region. In this case, what the divergence theorem says is that the amount of matter that appears (respectively, disappears) within a domain is equal to the amount of matter that exits (respectively, enters) said domain. So, here is my first question : is this a correct way to think about the divergence theorem ?
Now, on to the second question. No matter how much I rack my brain, I can't seem to find a similar reasoning for Green theorem (or Stokes theorem, for that matter). Do you have some ideas on how to think about those 2 theorems in the situation of incompressible fluid ? Or any other situation that make it seem more intuitive. Thank you in advance !
PS : I'm afraid that this question might be a duplicate, but I looked at other similar questions and couldn't find any answer that made sense to me... I'm sorry if the same question has already been asked somewhere else...
And I hope the question makes sense despite my questionable english...
vector-analysis divergence stokes-theorem greens-theorem curl
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
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add a comment |
$begingroup$
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the divergence. If we imagine a vector field as the flow of an incompressible fluid, the divergence of a certain domain in space would be linked to 'how much' matter is created inside the domain. If matter is created within a certain region and the fluid is incompressible, it would make sense that the fluid would have to escape this region. In this case, what the divergence theorem says is that the amount of matter that appears (respectively, disappears) within a domain is equal to the amount of matter that exits (respectively, enters) said domain. So, here is my first question : is this a correct way to think about the divergence theorem ?
Now, on to the second question. No matter how much I rack my brain, I can't seem to find a similar reasoning for Green theorem (or Stokes theorem, for that matter). Do you have some ideas on how to think about those 2 theorems in the situation of incompressible fluid ? Or any other situation that make it seem more intuitive. Thank you in advance !
PS : I'm afraid that this question might be a duplicate, but I looked at other similar questions and couldn't find any answer that made sense to me... I'm sorry if the same question has already been asked somewhere else...
And I hope the question makes sense despite my questionable english...
vector-analysis divergence stokes-theorem greens-theorem curl
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
$endgroup$
add a comment |
$begingroup$
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the divergence. If we imagine a vector field as the flow of an incompressible fluid, the divergence of a certain domain in space would be linked to 'how much' matter is created inside the domain. If matter is created within a certain region and the fluid is incompressible, it would make sense that the fluid would have to escape this region. In this case, what the divergence theorem says is that the amount of matter that appears (respectively, disappears) within a domain is equal to the amount of matter that exits (respectively, enters) said domain. So, here is my first question : is this a correct way to think about the divergence theorem ?
Now, on to the second question. No matter how much I rack my brain, I can't seem to find a similar reasoning for Green theorem (or Stokes theorem, for that matter). Do you have some ideas on how to think about those 2 theorems in the situation of incompressible fluid ? Or any other situation that make it seem more intuitive. Thank you in advance !
PS : I'm afraid that this question might be a duplicate, but I looked at other similar questions and couldn't find any answer that made sense to me... I'm sorry if the same question has already been asked somewhere else...
And I hope the question makes sense despite my questionable english...
vector-analysis divergence stokes-theorem greens-theorem curl
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
$endgroup$
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the divergence. If we imagine a vector field as the flow of an incompressible fluid, the divergence of a certain domain in space would be linked to 'how much' matter is created inside the domain. If matter is created within a certain region and the fluid is incompressible, it would make sense that the fluid would have to escape this region. In this case, what the divergence theorem says is that the amount of matter that appears (respectively, disappears) within a domain is equal to the amount of matter that exits (respectively, enters) said domain. So, here is my first question : is this a correct way to think about the divergence theorem ?
Now, on to the second question. No matter how much I rack my brain, I can't seem to find a similar reasoning for Green theorem (or Stokes theorem, for that matter). Do you have some ideas on how to think about those 2 theorems in the situation of incompressible fluid ? Or any other situation that make it seem more intuitive. Thank you in advance !
PS : I'm afraid that this question might be a duplicate, but I looked at other similar questions and couldn't find any answer that made sense to me... I'm sorry if the same question has already been asked somewhere else...
And I hope the question makes sense despite my questionable english...
vector-analysis divergence stokes-theorem greens-theorem curl
vector-analysis divergence stokes-theorem greens-theorem curl
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
edited Jan 10 at 9:16
Gabel Luc
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
asked Jan 9 at 14:54
Gabel LucGabel Luc
62
62
add a comment |
add a comment |
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