Fluxintegralet (Ortogonalt fladeintegral) PDF

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Videre forklaring af Flux, og denne gang mere dybdegående, ved forelæser Michael Pedersen....

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&ůƵdžŝŶƚĞŐƌĂůĞƚ;KƌƚŽŐŽŶĂůƚĨůĂĚĞŝŶƚĞŐƌĂůͿ ϮϳĂƉƌŝůϮϬϮϭ

ϭϭϭϯ

 sŽůƵŵĞŶĞŬƐƉĂŶƐŝŽŶ  'ĂƵƐƐΖĚŝǀĞƌŐĞŶƐƐčƚŶŝŶŐ

sŽůƵŵĞŶĞŬƐƉĂŶƐŝŽŶ

'ĂƵƐƐΖĚŝǀĞƌŐĞŶƐƐčƚŶŝŶŐ

sŝŚĂƌĞŶĨůĂĚĞ‫ܨ‬௥  sŝŚĂƌŽŐƐĊĞŶƌčŬŬĞĨůŽǁŬƵƌǀĞƌ

ȳ ൌ ŵĂƐƐŝǀƚƌƵŵůŝŐƚŽŵƌĊĚ

DŽƚŝǀĂƚŝŽŶ

߲ȳ ĞƌĞŶůƵŬŬĞƚĨůĂĚĞ

sŝŚĂƌĞƚǀŝŶĚƵĞǀŝƐĞƌĚĞƚĨƌĂƐŝĚĞŶĞƌĞƌůLJƐŝŶĚĨĂůĚƌĞƉƌčƐĞŶƚĞƌĞƚǀĞĚ ĞƚǀĞŬƚŽƌĨĞůƚ sŝǀŝůŐĞƌŶĞŵĊůĞǀĂƌŵĞƐƚƌĊůŝŶŐĞŶƐŽŵŬŽŵŵĞƌŝŶĚŝůŽŬĂůĞƚůčŶŐĚĞŶĂĨ ƉƌŽũĞŬƚŝŽŶĞŶĂĨǀĞŬƚŽƌĨĞůƚĞƚŝŶĚƉĊĞŶŶŽƌŵĂůǀĞŬƚŽƌǀŝŶŬĞůƌĞƚƉĊƌƵĚĞŶ หܸ ȉ ݊ห ൌ หܸห ‫ כ‬ሺ‫ݒ‬ሻ නܸ ȉ ݊ ݀ߤ ி

&ůƵdžŝŶƚĞŐƌĂů sŝŚĂƌĞŶĨůĂĚĞŵĞĚĞŶƉĂƌĂŵĞƚƌŝƐĞƌŝŶŐ‫ܨ‬௥ ‫ݎ‬ሺ‫ݑ‬ǡ ‫ ݒ‬ሻ ൌ‫ݔ‬ሺ൫ ‫ݑ‬ǡ ‫ݒ‬ሻǡ ‫ ݕ‬ሺ‫ݑ‬ǡ ‫ ݒ‬ሻǡ ‫ݖ‬ሺ‫ݑ‬ǡ ‫ ݒ‬ሻ൯ ‫ א ݑ‬ሾ ܽǡ ܾሿ ‫ א ݒ‬ሾܿǡ ݀ሿ

ĞƚŚĞƌĞƌĚĞƚƐǀčƌĞŽŐǀŝƐƉƌŝŶŐĞƌŐĞůŝŶĚĞŽǀĞƌĚĞƚ sŝŚĂƌĞƚǀĞŬƚŽƌĨĞůƚŽŐĞŶĨůĂĚĞŶŽƌŵĂůǀĞŬƚŽƌ݊ &ůĂĚĞŶŽƌŵĂů ݊

ᇲ ௥ೣᇲ ൈ௥೤ ൌቐ ȁ௥ೣᇲ ൈ௥೤ᇲȁቑ

Ͳ

ǡ

ŚǀŝƐ‫ݎ‬௫ᇱ

ൈ

‫ݎ‬௬ᇱ

ൌͲ

ܸሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ ݖ‬ሻ ൌܸ൫ଵሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻǡ ܸଶሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻǡ ܸଷሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ൯ ‫ܬ‬௥ሺ ‫ݑ‬ǡ ‫ݒ‬ሻ ൌ ȁ‫ݎ‬௨ᇱ ൈ ‫ݎ‬௩ᇱ ȁ

ௗ ௕ ‫ݎ‬௨ᇱ ൈ ‫ݎ‬௩ᇱ ȉ ᇱ ‫ܸݔݑܨ‬ǡ൫ ‫ܨ‬௥൯ൌ න ܸ ȉ ݊ ݀ߤ ൌ න න ܸ ቀ‫ ݎ‬ሺ‫ݑ‬ǡ ‫ ݒ‬ሻቁ ‫ ܬ כ‬ሺ‫ݑ‬ǡ ‫ ݒ‬ሻ݀‫ݒ݀ݑ‬ ȁ‫ݎ‬௫ ൈ ‫ݎ‬௩ᇱ ȁ ௥ ݈ ௖ ௔ ிೝ ௗ

௕

ȉ ‫ݎ‬௨ᇱ ൈ ‫ݎ‬௩ᇱ ݀‫ݒ݀ݑ‬ ൌ න න ܸ ቀ‫ݎ‬ሺ ‫ݑ‬ǡ ‫ݒ‬ሻቁ ௖

௔

ǣ ǡǡൌǦǡÞൌ sŝŬŝŐŐĞƌŬƵŶ ƉĊĚĞŶŬƌƵŵŵĞŽǀĞƌĨůĂĚĞ ‫ܨ‬௥ ǣ ‫ ݎ‬ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ൌ ሺܴ ‫ כ‬ሺ ‫ݒ‬ሻ ǡ ܴ ‫ כ‬ሺ‫ݒ‬ሻ ǡ ‫ݒ‬ሻ ‫ א ݑ‬ሾͲǡʹߨሿ ‫ א ݒ‬ሾͲǡ ݄ሿ െܴ ‫ כ‬ሺ‫ ݑ‬ሻ ൌ ൥ ܴ ‫ כ‬ሺ‫ݑ‬ሻ ൩ Ͳ Ͳ ‫ݎ‬௩ᇱ ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ൌ ൥ Ͳ൩ ͳ ‫ݎ‬௨ᇱ ሺ‫ݑ‬ǡ ‫ݒ‬ሻ

ܴ ‫ כ‬ ሺ‫ݑ‬ሻ ‫ݎ‬௨ᇱ ൈ ‫ݎ‬௩ᇱ ൌ ൥ ܴ ‫ כ‬ሺ‫ݑ‬ሻ൩ Ͳ

ͳሻ ܸሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ ൌ ሺ ͳǡͲǡͲሻ ௛

ଶగ

‫ܸݔݑܨ‬ǡ൫ ‫ܨ‬௥ൌ ൯ න න ሺͳǡͲǡͲሻ ȉ ሺܴ ‫ כ‬ ሺ‫ݑ‬ሻ ǡ ܴ ‫ כ‬ሺ‫ݑ‬ሻ ǡ Ͳ ሻ ݀‫ݒ݀ ݑ‬ ݈ ଴ ଴ ௛

ൌන න ଴

ଶగ

଴

ܴ ‫ כ‬ሺ‫ݑ‬ሻ ݀‫ ݒ݀ ݑ‬ൌ Ͳ

,ǀŽƌĨŽƌďůŝǀĞƌĚĞƚϬ ĞƌĞƌůŝŐĞŵĞŐĞƚĚĞƌƐƚƌƆŵŵĞƌŝŶĚƐŽŵĚĞƌƐƚƌƆŵŵĞƌƵĚŵĞĚĚĞƚ ǀĞŬƚŽƌĨĞůƚ WƌŽũĞŬƚŝŽŶĞŶĂĨǀĞŬƚŽƌsŝŶĚƉĊĚĞŶǀŝŶŬĞůƌĞƚƚĞŶŽƌŵĂůǀĞŬƚŽƌǀŝůŐĊƵĚ ŵĞĚĚĞŶŵŽĚƐĂƚƌĞƚƚĞĚĞ

߲ȳ ൌ ƌĂŶĚĞŶĂĨȳ

>ƆďĞƌŽŐƐĊŝŐĞŶŶĞŵĨůĂĚĞŶŽŐĨůLJĚĞƌŽŐƐĊůĂŶŐƐĨůŽǁŬƵƌǀĞƌŶĞ&ůĂĚĞŶ ŐĞŶŶĞŵƐƚƌƆŵŵĞƌĞƚƌƵŵŵĞůŝŐƚŽŵƌĊĚĞƐŽŵŐĞŶŶĞŵƐƚƌƆŵŵĞƐĂĨĨůĂĚĞƌŶĞƚŝů ƚŝĚĞƌŶĞƚсϬŽŐƚ ݉݅ ݉݀ ȳሺ‫ ݐ‬ሻ ൌ ܴ‫ݐ݈݉݅݃ݑ‬ǡ ‫ݎ݋ ݐݒݏݏ‬¤

ĞƚĞƌĚĞƚŽŵƌĊĚĞƐŽŵ‫ܨ‬௥ ŐĞŶŶĞŵƐƚƌƆŵŵĞƌƚŝůƚŝĚĞŶƚ &ůŽǁŬƵƌǀĞƌĨŽƌǀĞŬƚŽƌĨĞůƚĞƚ ܸ ݀ ሻ൯ ቁ ‫ܸ ݔݑܨ‬ǡ ቀܸ‫݈݋‬ ȳ൫ሺ ‫ݐ‬ൌ ൫ ‫ܨ‬௥ ൯ ݀‫ݐ‬ ݈ ‫ݐ‬ൌͲ ŬƐĞŵƉĞů &ŝŶĚĨůŽǁŬƵƌǀĞƌĨŽƌǀĞŬƚŽƌĨĞůƚ ܸሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ ݖ‬ሻ ൌ ሺ‫ݔ‬ǡ ͲǡͲሻ ‫ ݔ‬ᇱሺ ‫ݐ‬ሻ ‫ ݔ‬ሺ‫ݐ‬ሻ ቎‫ ݕ‬ᇱሺ ‫ݐ‬ሻ ቏ ൌ ൥ Ͳ ൩ Ͳ ‫ ݖ‬ᇱ ሺ ‫ݐ‬ሻ ‫ݔ‬ሺ ‫ݐ‬ሻ ݇ଵ ݁ ௧ ቎‫ݕ‬ሺ ‫ݐ‬ሻ ቏ ൌ ቎ ݇ଶ ቏ ‫ݖ‬ሺ ‫ݐ‬ሻ ݇ଷ

&ůŽǁŬƵƌǀĞŐĞŶŶĞŵሺ ݄ǡ ͲǡͲሻ ƚŝů‫ ݐ‬ൌ Ͳ ൥

݄ ‫݁ כ‬௧ Ͳ ൩ Ͳ

൥

݄ ‫݁ כ‬௧ ܾ ൩ ݈

&ůŽǁŬƵƌǀĞŐĞŶŶĞŵሺ ݄ǡ ܾǡ ݈ሻ ƚŝů‫ ݐ‬ൌ Ͳ

ܸ‫݈݋‬൫ ȳ ሺ‫ ݐ‬ሻ൯ ൌ ሺ݄ ‫ ݁ כ‬௧ െ ݄ ሻ ‫ ݈ כ ܾ כ‬ൌ ݄ ‫ ݁ כ‬௧ ‫ ݈ כ ܾ כ‬െ ݄ ‫݈ כ ܾ כ‬ ݀ ȳ൫ሺ‫ ݐ‬ሻ൯ ൌ ݁ ௧ ‫݄ܾ݈ כ‬ ݀‫ݐ‬

‫ݐ‬ൌͲ ݀ ȳ൫ሺ‫ݐ‬ሻ൯ ൌ ݈ܾ݄ ݀‫ݐ‬

/ŬŬĞĨůĞƌĞŶŽƚĞƌŵĞĚŵŝŶĚƌĞĚĞƚĞƌŶƆĚǀĞŶĚŝŐƚŬŝŐƉĊƐŝĚƐƚĞŐĂŶŐ/d, >ŽǀĞƵΎфϯ...