<Hilbert Book Model Project/nl
De ruimtelijke nabla en quaternionische nabla zijn bijzondere operatoren die een belangrijke rol spelen bij de partiële differentiaalvergelijkingen die het gedrag bepalen van velden in het Hilbert Book Model.
Hier behandelen we drie soorten nabla operatoren.
- ruimtelijke nabla
![{\textstyle {\vec {\nabla }}={\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce3c6b1a126a2f621bad03170d10b9931714006)
- quaternionische nabla
![{\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}=\nabla _{r}+{\vec {\nabla }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f54e7980f5c6c1aeedf6b5c2fbf29ec2b10605ed)
- Dirac nabla
![{\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+\mathbb {I} \left\{{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}\right\}=\nabla _{r}+\mathbb {I} \,{\vec {\nabla }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93963f68138e688f66671f86856f46974d742c6)
De Dirac nabla speelt een rol bij de interpretatie van de Dirac vergelijking.
Het nabla product is niet zonder meer associatief. Dus
-
![{\displaystyle \nabla (\nabla ^{*}\psi )=\nabla ^{*}(\nabla \psi )=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi \neq (\nabla \nabla ^{*})\psi =(\nabla ^{*}\nabla )\psi =(\nabla _{r}\nabla _{r}-{\vec {\nabla }}{\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \mp {\vec {\nabla }}\times {\vec {\nabla }})\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8413b9335f71837a061cc39eaa7f241401990056)
|
|
(1)
|
De ruimtelijke nabla bestaat in verschillende coördinatenstelsels. In deze sectie wordt de weergave van de quaternionische nabla voor cartesische coördinatensystemen en polaire coördinatensystemen.
-
![{\displaystyle {\vec {\nabla }}=\{{\frac {\partial {}}{\partial {x}}},\,{\frac {\partial {}}{\partial {y}}},\,{\frac {\partial {}}{\partial {z}}}\}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f082e4b053658e65d48cadba7a49583ab3e35948)
|
|
(2)
|
-
![{\displaystyle {\vec {\nabla }}a_{r}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}={\frac {\partial {a_{r}}}{\partial {\rho }}}{\vec {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial {a_{r}}}{\partial {\theta }}}{\vec {\hat {\theta }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{r}}}{\partial {\phi }}}{\vec {\hat {\phi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b024e87fac47254614db1667ff6fad9cc252cfa)
|
|
(3)
|
Here
zijn de coördinaten met
als coördinaat assen.
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {a}}\rangle ={\frac {\partial {a_{x}}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {a_{y}}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {a_{z}}}{\partial {z}}}{\vec {\hat {z}}}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e85351037438952a65060d519757065282cd4f)
|
|
(4)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle f_{r}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}{\frac {\partial {f_{r}}}{\partial \rho }})}}{\partial {\rho }}}+{\frac {1}{\rho ^{2}\sin(\theta )}}{\frac {\partial {(\sin(\theta ){\frac {\partial {f_{r}}}{\partial {\theta }}}}}{\partial {\theta )}}}+{\frac {1}{\rho ^{2}\sin ^{2}(\theta )}}{\frac {\partial ^{2}{f_{r}}}{\partial {\phi ^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9064ff6e977aa4bd0affab6284cdf51ea2699544)
|
|
(5)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {a}}=({\frac {\partial {a_{z}}}{\partial {y}}}-{\frac {\partial {a_{y}}}{\partial {z}}}){\vec {\hat {x}}}+({\frac {\partial {a_{x}}}{\partial {z}}}-{\frac {\partial {a_{z}}}{\partial {x}}}){\vec {\hat {y}}}+({\frac {\partial {a_{y}}}{\partial {x}}}-{\frac {\partial {a_{x}}}{\partial {y}}}){\vec {\hat {z}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc85e03a1398198302faf6cea682f62ce4d0d3e)
|
|
(6)
|
-
![{\displaystyle \qquad ={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d055c2071896b81f101988d7e0ff8378b352c5f)
|
|
(7)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {a}}={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38207649bf90628d31b12defa68f0b9373a2ee69)
|
|
(8)
|
De ruimtelijke nabla operator blijkt gedrag geldt voor alle quaternionische functies waarvoor de eerste orde partiële differentiaalvergelijking bestaat.
Hier gehoorzaamt het quaternionische veld
aan de eis dat de eerste orde partiële differentiaal
bestaat.
-
![{\displaystyle \Delta \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \langle {\vec {\nabla }},{\vec {\nabla }}\rangle \ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\partial {}^{2}}{\partial {x}^{2}}}+{\frac {\partial {}^{2}}{\partial {y}^{2}}}+{\frac {\partial {}^{2}}{\partial {z}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4214dcb6bd98c00cd0d4d27f75615d36d932b9b)
|
|
(9)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}a_{r}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=\nabla ^{2}a_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aad9dc1129b62ae31a8b21e386a7ca830ccbbf2)
|
|
(10)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}{\vec {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77de08d6cfa1b204556e048366ec4723ced8ff32)
|
|
(11)
|
-
![{\displaystyle \langle {\vec {\nabla }}\times {\vec {\nabla }},{\vec {a}}\rangle =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e74fcc576fe56053671fbd7ae444c415e5facc7)
|
|
(12)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}a_{r}={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4eb1dbf26ac35f1c50e7670c9cec9bdcfb2c67)
|
|
(13)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {a}}\rangle =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8bc3c2e7a0ac9b68c72a0fa80425ebbefe2abb)
|
|
(14)
|
-
![{\displaystyle {\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9967070d8917b183b968753fa6486cc27b89211)
|
|
(15)
|
-
![{\displaystyle {\vec {\nabla }}({\vec {\nabla }}a)={\vec {\nabla }}({\vec {\nabla }}\times a-\langle {\vec {\nabla }},{\vec {a}}\rangle +{\vec {\nabla }}a_{r})={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72dce07d9bf72922dc60d6db68f55d8e2a83105c)
|
|
(16)
|
Voor constante
en parameter
geldt
-
![{\displaystyle {\vec {\nabla }}\langle {\vec {k}},{\vec {q}}\rangle ={\vec {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd700a31d1d3c1c5a384cc6cf8de1a520a76ee46)
|
|
(17)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {q}}\rangle =3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0290e6185776b2f945d30d1693c5d6c249ee87b7)
|
|
(18)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {q}}={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b93c360faf8c91426d46e80dc0205981c63a2459)
|
|
(19)
|
-
![{\displaystyle {\vec {\nabla }}\,|{\vec {q}}|={\frac {\vec {q}}{|{\vec {q}}|}};\quad \nabla \,|q|={\frac {q}{|q|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c51724eb2e882972c633d2440547c30873254b9)
|
|
(20)
|
-
![{\displaystyle {\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}};\ \nabla {\frac {1}{|q-q^{'}|}}=-{\frac {q-q^{'}}{|q-q^{'}|^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc91841f90205944a643aa87211da10c02ac3211)
|
|
(21)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-\langle {\vec {\nabla }},{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}}\rangle =4\,\phi \,\delta ({\vec {q}}-{\vec {q}}^{'})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/401ec946bd3007f034ba71919601ec4a277f0975)
|
|
(22)
|
Dit geeft de overeenkomsten tussen de Poissonvergelijking en Green's functie..
-
![{\displaystyle \nabla \ q=1-3;\quad \nabla ^{*}q=1+3;\quad \nabla q^{*}=1+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3669ea862c95ae25b50de5d54033d4c7557e0b)
|
|
(23)
|
The term
geeft the kromming van veld
.
The term
geeft de stress van het veld
-
![{\displaystyle \varphi =\varphi _{r}+{\vec {\varphi }}=\nabla \psi =(\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d7e3f921183fe0d339c9dd5d23837ad055afe0)
|
|
(24)
|
De vergelijking is een quaternionische eerste orde partiële differentiaalvergelijking.
De vijf termen aan de rechterzijde geven de componenten die de volledige eerste orde verandering vormen.
Zij vertegenwoordigen deelgebieden van het veld
, en vaak krijgen ze speciale namen en symbolen.
is de gradient van
is de divergentie van
.
is de rotatie van
-
![{\displaystyle \varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c578b3ebf77fe9d22ab5f55b207f409d1a45bbd)
|
|
(25)
|
(Vergelijking (25) heeft geen equivalent in de vergelijkingen van Maxwell!)
-
![{\displaystyle {\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}=\mp {\vec {B}}-{\vec {E}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1927dc7470f71612d499272822371ea07824838)
|
|
(26)
|
-
![{\displaystyle {\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f87913177c18e0ba6ad870719c4e14d2bee6604)
|
|
(27)
|
-
![{\displaystyle {\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57d6226f55769887ac5fbd7f9755f4f21fcc8c37)
|
|
(28)
|
-
![{\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e969286b9ea9aa0e3a5edb803e29aa4268d6070)
|
|
(29)
|
Met behulp van de eigenschappen van de ruimtelijke nabla operator voert een interessante tweede orde partiële differentiaalvergelijking.
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {B}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c20148bfe3d321f437091e20bcc78411065dff43)
|
|
(30)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {E}}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\psi _{r}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b7fbaf4152526fca05cd326218416bf13b8cba)
|
|
(31)
|
-
![{\displaystyle \langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle =\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa6f0ae620e510ba1d35c1e5ac717007bce977d)
|
|
(32)
|
-
![{\displaystyle \nabla _{r}{\vec {E}}=-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32eb2d9558aadbddc3e056d7dbacd6ca6d726897)
|
|
(33)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})-{\vec {\nabla }}\times ({\vec {\nabla }}\psi _{r})=-\nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}{\vec {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78a2aca42687758ea4bf15f734443937eb52d0ae)
|
|
(34)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {B}}={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {\psi }})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790e839154e6d6b9e4a379d68fd12a3fbb4c45b3)
|
|
(35)
|
-
![{\displaystyle {\vec {\nabla }}\times {\vec {B}}-{\vec {\nabla }}\times {\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8bf0630abc69b3041e15194737d8dacc958cfe6)
|
|
(36)
|
-
![{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a1635b030daca2f9ea670f68e17534feb94ba1)
|
|
(37)
|
-
![{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a888b92ba605f2264e059af846e03a28f95da7)
|
|
(38)
|
-
![{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle -\nabla _{r}({\vec {B}}-{\vec {E}})=\nabla ^{*}{\vec {\varphi }}=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/102f3e3145430890de1898ec207c68d13f3a31ce)
|
|
(39)
|
Verder
-
![{\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle ){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ced2a6346b18fa5913f536f3b7c68c0456e981a)
|
|
(40)
|
Uit bovenstaande formules volgt dat de Maxwell vergelijkingen geen volledige set vormen. Natuurkundigen gebruiken gauge vergelijkingen om Maxwell vergelijkingen completer te maken.
Afleiden van tweede orde partiële differentiaalvergelijking 1
[edit | edit source]
-
![{\displaystyle \varphi =\nabla \psi =(\nabla _{r}+{\vec {\nabla }})\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df856be75bebe0c5fd26c24c81e0ae6461c57c00)
|
|
(41)
|
-
![{\displaystyle \varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c578b3ebf77fe9d22ab5f55b207f409d1a45bbd)
|
|
(42)
|
-
![{\displaystyle {\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3447e8447edfad036fb58b0a79cccdbc6399dfb2)
|
|
(43)
|
-
![{\displaystyle {\vec {\nabla }}\varphi _{r}={\vec {\nabla }}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5b9d78614d8bc1eb7f522116cedc8f6d28891f)
|
|
(44)
|
-
![{\displaystyle \nabla _{r}\varphi _{r}=\nabla _{r}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a32a37857ae61e542d51292e989625d2fdc1a68e)
|
|
(45)
|
-
![{\displaystyle \nabla ^{*}\varphi _{r}=-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/39b0c1aebe16b0f8aa705aa569f29cbbb44e8756)
|
|
(46)
|
-
![{\displaystyle {\vec {\nabla }}{\vec {\varphi }}={\vec {\nabla }}(\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}\rangle )=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}\mp \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle +{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/705c8852f4816bcac0b1342fb553dbcd722a045b)
|
|
(47)
|
-
![{\displaystyle {\vec {\nabla }}{\vec {\varphi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8204b112f2b79a174554bef7e076815c386c0b1c)
|
|
(48)
|
-
![{\displaystyle \nabla _{r}{\vec {\varphi }}=\nabla _{r}(\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }})=\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}\pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd75be386711585a28953aa1ea4ab28f35d3098f)
|
|
(49)
|
-
![{\displaystyle \nabla ^{*}{\vec {\varphi }}=\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}\pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \mp \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7893107e46783239ec728b143e2a20e8e42c6149)
|
|
(50)
|
-
![{\displaystyle \nabla ^{*}{\vec {\varphi }}=\nabla _{r}\nabla _{r}{\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8811099f16217d2c5295b093a96d2feafa63e59)
|
|
(51)
|
-
![{\displaystyle \nabla ^{*}\varphi =-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}{\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b746935fc8a94713b307a90a476e46fc15fdc0)
|
|
(52)
|
-
![{\displaystyle \chi =\nabla ^{*}\varphi =(\nabla _{r}-{\vec {\nabla }})\varphi =(\nabla _{r}-{\vec {\nabla }})(\nabla _{r}+{\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )(\psi _{r}+{\vec {\psi }})+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}{\vec {\nabla }}\psi _{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e452ad31f3016a510ba2f0791b2063a3617098c2)
|
|
(53)
|
Het merendeel van de termen verdwijnen.
-
![{\displaystyle \chi =\nabla ^{*}(\nabla \psi )=(\nabla _{r}-{\vec {\nabla }})((\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }}))=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aff9b4e956061b3e292853620af86abde5684cd4)
|
|
(54)
|
Afleiden van tweede orde partiële differentiaalvergelijking 2
[edit | edit source]
We voegen de complexe imaginaire basegetal
bij de ruimtelijke nabla operator
.
-
![{\displaystyle g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10dc9cb7dcd3246a63fbaa2a44b71ede4c24de39)
|
|
(55)
|
-
![{\displaystyle g_{r}=\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c29ddb56f0819ef2e734e75aeae52077b9a8ca2c)
|
|
(56)
|
-
![{\displaystyle {\vec {g}}=\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ef41013ba1277f5043bd7f9f9e442759cf7140)
|
|
(57)
|
-
![{\displaystyle \mathbb {I} {\vec {\nabla }}g_{r}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd5bf02975bf22d3fd75499af77a2915b5bc696)
|
|
(58)
|
-
![{\displaystyle \nabla _{r}g_{r}=\nabla _{r}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9d456f900b5ee6c3cd318f334c568af12028c7)
|
|
(59)
|
-
![{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }})g_{r}=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/79764f9c1da9d157da970de6e2af45c14230ad32)
|
|
(60)
|
-
![{\displaystyle \mathbb {I} {\vec {\nabla }}{\vec {g}}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle )=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}\mp \langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle +\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bde27291dd5f591c2a09a30ca6fa9828625f51e1)
|
|
(61)
|
-
![{\displaystyle \mathbb {I} {\vec {\nabla }}{\vec {g}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2080b112bf97690df8b66e5d91ff937e6198962c)
|
|
(62)
|
-
![{\displaystyle \nabla _{r}{\vec {g}}=\nabla _{r}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}})=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82aed295407fbf20405a046f3364dc25bc3b9587)
|
|
(63)
|
-
![{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \mp \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee6f66e7d4aeae9f05124c7537b5825611d6543)
|
|
(64)
|
-
![{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7245a181c68b9683e913c66077d9d679d83aac8)
|
|
(65)
|
-
![{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1bcd295dcd731ef8a4fa48a133524f9a4354d57)
|
|
(66)
|
-
![{\displaystyle \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c591a8a531eba91882392ed42d9b6d266241378)
|
|
(67)
|
-
![{\displaystyle \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +\mathbb {I} (\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}{\vec {\nabla }}f_{r}+\nabla _{r}{\vec {\nabla }}f_{r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58b6c70f5a3911f0e31f3c09056500b2c1fa15be)
|
|
(68)
|
-
![{\displaystyle \zeta =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c36c8171cd822e749d31a578b048aa77ffcc99)
|
|
(69)
|
Dus ook deze quaternionische tweede orde partiële differentiaalvergelijking splitst in twee eerste orde partiële differentiaalvergelijkingen. Maar dit zijn geen quaternionische partiële differentiaalvergelijkingen
-
![{\displaystyle g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f;\qquad \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcb80ea1d753cb6c292d93de15652b32c7b55fb)
|
|
(70)
|