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A note on Green's Theorem
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| Zeitschriftentitel: | Journal of the Australian Mathematical Society |
|---|---|
| Personen und Körperschaften: | |
| In: | Journal of the Australian Mathematical Society, 4, 1964, 3, S. 289-292 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Cambridge University Press (CUP)
|
| Schlagwörter: |
| author_facet |
Craven, B. D. Craven, B. D. |
|---|---|
| author |
Craven, B. D. |
| spellingShingle |
Craven, B. D. Journal of the Australian Mathematical Society A note on Green's Theorem General Earth and Planetary Sciences General Environmental Science |
| author_sort |
craven, b. d. |
| spelling |
Craven, B. D. 0004-9735 Cambridge University Press (CUP) General Earth and Planetary Sciences General Environmental Science http://dx.doi.org/10.1017/s1446788700024058 <jats:p>Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of <jats:italic>Q</jats:italic><jats:sub>x</jats:sub>—<jats:italic>P</jats:italic><jats:sup>y</jats:sup> is assumed, where <jats:italic>P</jats:italic>(<jats:italic>x, y</jats:italic>) and <jats:italic>Q</jats:italic>(<jats:italic>x, y</jats:italic>) are the functions involved, but not the integrability of the individual partial derivatives Q<jats:sub>x</jats:sub>, and p<jats:sub>y</jats:sub> this latter assumption being made by other authors. However, <jats:italic>P</jats:italic> and <jats:italic>Q</jats:italic> are assumed differentiable, at points interior to the curve.</jats:p> A note on Green's Theorem Journal of the Australian Mathematical Society |
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Journal of the Australian Mathematical Society |
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A note on Green's Theorem |
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A note on Green's Theorem |
| title_full |
A note on Green's Theorem |
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A note on Green's Theorem |
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A note on Green's Theorem |
| title_short |
A note on Green's Theorem |
| title_sort |
a note on green's theorem |
| topic |
General Earth and Planetary Sciences General Environmental Science |
| url |
http://dx.doi.org/10.1017/s1446788700024058 |
| publishDate |
1964 |
| physical |
289-292 |
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<jats:p>Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of <jats:italic>Q</jats:italic><jats:sub>x</jats:sub>—<jats:italic>P</jats:italic><jats:sup>y</jats:sup> is assumed, where <jats:italic>P</jats:italic>(<jats:italic>x, y</jats:italic>) and <jats:italic>Q</jats:italic>(<jats:italic>x, y</jats:italic>) are the functions involved, but not the integrability of the individual partial derivatives Q<jats:sub>x</jats:sub>, and p<jats:sub>y</jats:sub> this latter assumption being made by other authors. However, <jats:italic>P</jats:italic> and <jats:italic>Q</jats:italic> are assumed differentiable, at points interior to the curve.</jats:p> |
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| description | <jats:p>Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of <jats:italic>Q</jats:italic><jats:sub>x</jats:sub>—<jats:italic>P</jats:italic><jats:sup>y</jats:sup> is assumed, where <jats:italic>P</jats:italic>(<jats:italic>x, y</jats:italic>) and <jats:italic>Q</jats:italic>(<jats:italic>x, y</jats:italic>) are the functions involved, but not the integrability of the individual partial derivatives Q<jats:sub>x</jats:sub>, and p<jats:sub>y</jats:sub> this latter assumption being made by other authors. However, <jats:italic>P</jats:italic> and <jats:italic>Q</jats:italic> are assumed differentiable, at points interior to the curve.</jats:p> |
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| imprint | Cambridge University Press (CUP), 1964 |
| imprint_str_mv | Cambridge University Press (CUP), 1964 |
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| physical | 289-292 |
| publishDate | 1964 |
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| publisher | Cambridge University Press (CUP) |
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| series | Journal of the Australian Mathematical Society |
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| spelling | Craven, B. D. 0004-9735 Cambridge University Press (CUP) General Earth and Planetary Sciences General Environmental Science http://dx.doi.org/10.1017/s1446788700024058 <jats:p>Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of <jats:italic>Q</jats:italic><jats:sub>x</jats:sub>—<jats:italic>P</jats:italic><jats:sup>y</jats:sup> is assumed, where <jats:italic>P</jats:italic>(<jats:italic>x, y</jats:italic>) and <jats:italic>Q</jats:italic>(<jats:italic>x, y</jats:italic>) are the functions involved, but not the integrability of the individual partial derivatives Q<jats:sub>x</jats:sub>, and p<jats:sub>y</jats:sub> this latter assumption being made by other authors. However, <jats:italic>P</jats:italic> and <jats:italic>Q</jats:italic> are assumed differentiable, at points interior to the curve.</jats:p> A note on Green's Theorem Journal of the Australian Mathematical Society |
| spellingShingle | Craven, B. D., Journal of the Australian Mathematical Society, A note on Green's Theorem, General Earth and Planetary Sciences, General Environmental Science |
| title | A note on Green's Theorem |
| title_full | A note on Green's Theorem |
| title_fullStr | A note on Green's Theorem |
| title_full_unstemmed | A note on Green's Theorem |
| title_short | A note on Green's Theorem |
| title_sort | a note on green's theorem |
| title_unstemmed | A note on Green's Theorem |
| topic | General Earth and Planetary Sciences, General Environmental Science |
| url | http://dx.doi.org/10.1017/s1446788700024058 |