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Area Physics for electrotechnology / Relativistic physics for electrotechnology

IEV ref 113-07-43

Symbol
Grad

en
four-gradient
four-dimensional generalization of three-dimensional gradient

Gradρ:=( ρ x 0 , ρ x 1 , ρ x 2 , ρ x 3 )=( ρ j c 0 t ;gradρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaai4raiaackhacaGGHb Gaaiizaiaaeg8acaGG6aGaeyypa0ZaaeWaaeaadaWcaaqaaiabgkGi 2kaaeg8aaeaacqGHciITcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaaki aacYcadaWcaaqaaiabgkGi2kaaeg8aaeaacqGHciITcaWG4bWaaSba aSqaaiaaigdaaeqaaaaakiaacYcadaWcaaqaaiabgkGi2kaaeg8aae aacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiaacYcadaWc aaqaaiabgkGi2kaaeg8aaeaacqGHciITcaWG4bWaaSbaaSqaaiaaio daaeqaaaaaaOGaayjkaiaawMcaaiabg2da9maabmaabaWaaSaaaeaa cqGHciITcaaHbpaabaGaeyOaIyRaiyjGbQgacaWGJbWaaSbaaSqaai aaicdaaeqaaOGaamiDaaaacaGG7aGaai4zaiaackhacaGGHbGaaiiz aiaaeg8aaiaawIcacaGLPaaaaaa@6775@ where ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaaqyWdaaa@34F0@ is an arbitrary scalar field quantity

Note 1 to entry: Four-gradient is useful in STR. In general theory of relativity (GTR) for non-flat space-time, a more sophisticated method is used.

Note 2 to entry: Four-gradient of a tensor quantity Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamyuaaaa@347C@ of order n0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamOBaiabgwMiZkaaic daaaa@3719@ is a direct product of four-nabla and quantity Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamyuaaaa@347C@ : GradQ=Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaai4raiaackhacaGGHb GaaiizaiaadgfacqGH9aqpcqGHelc4caWGrbaaaa@3C38@ yielding a tensor quantity of order n+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamOBaiabgUcaRiaaig daaaa@3636@ .

Note 3 to entry: In the International System of Quantities, the dimension of four-gradient is L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaciitamaaCaaaleqaba GaeyOeI0IaaGymaaaaaaa@35AD@ .


fr
quadrigradient, m
généralisation quadridimensionelle du gradient tridimensionnel

Gradρ:=( ρ x 0 , ρ x 1 , ρ x 2 , ρ x 3 )=( ρ j c 0 t ;gradρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaai4raiaackhacaGGHb Gaaiizaiaaeg8acaGG6aGaeyypa0ZaaeWaaeaadaWcaaqaaiabgkGi 2kaaeg8aaeaacqGHciITcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaaki aacYcadaWcaaqaaiabgkGi2kaaeg8aaeaacqGHciITcaWG4bWaaSba aSqaaiaaigdaaeqaaaaakiaacYcadaWcaaqaaiabgkGi2kaaeg8aae aacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiaacYcadaWc aaqaaiabgkGi2kaaeg8aaeaacqGHciITcaWG4bWaaSbaaSqaaiaaio daaeqaaaaaaOGaayjkaiaawMcaaiabg2da9maabmaabaWaaSaaaeaa cqGHciITcaaHbpaabaGaeyOaIyRaiyjGbQgacaWGJbWaaSbaaSqaai aaicdaaeqaaOGaamiDaaaacaGG7aGaai4zaiaackhacaGGHbGaaiiz aiaaeg8aaiaawIcacaGLPaaaaaa@6775@ ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaaqyWdaaa@34F0@ est une grandeur de champ scalaire arbitraire

Note 1 à l’article: Le quadrigradient est utile en relativité restreinte. En relativité générale (GTR) pour un espace-temps non plat, une méthode plus élaborée est utilisée.

Note 2 à l’article: Le quadrigradient d’une grandeur tensorielle Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamyuaaaa@347C@ d’ordre n0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamOBaiabgwMiZkaaic daaaa@3719@ est le produit direct du quadrinabla et de la grandeur Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamyuaaaa@347C@ : GradQ=Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaai4raiaackhacaGGHb GaaiizaiaadgfacqGH9aqpcqGHelc4caWGrbaaaa@3C38@ ce qui donne une grandeur tensorielle d’ordre n+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGaciGacmaaceGaaiGacaabaaGcbaGaamOBaiabgUcaRiaaig daaaa@3636@ .

Note 3 à l’article: Dans le Système international de grandeurs, la dimension du quadrigradient est L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaciitamaaCaaaleqaba GaeyOeI0IaaGymaaaaaaa@35AD@ .


Publication date: 2022-06
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