Seite:Prinzipien der Dynamik des Elektrons (1903).djvu/9

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	Max Abraham: Prinzipien der Dynamik des Elektrons (1903)

$\Delta {\mathfrak {A}}$ ist der Vektor mit den Komponenten:

{\frac {\partial ^{2}{\mathfrak {A}}_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{x}}{\partial z^{2}}},\ \ \ \ {\frac {\partial ^{2}{\mathfrak {A}}_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{y}}{\partial z^{2}}},

{\frac {\partial ^{2}{\mathfrak {A}}_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}{\mathfrak {A}}_{z}}{\partial z^{2}}}.

$({\mathfrak {A}}\nabla ){\mathfrak {B}}$ ist der Vektor, dessen Komponenten sind:

{\mathfrak {A}}_{x}{\frac {\partial {\mathfrak {B}}_{x}}{\partial x}}+{\mathfrak {A}}_{y}{\frac {\partial {\mathfrak {B}}_{x}}{\partial y}}+{\mathfrak {A}}_{z}{\frac {\partial {\mathfrak {B}}_{x}}{\partial z}},\ \ \ \ {\mathfrak {A}}_{x}{\frac {\partial {\mathfrak {B}}_{y}}{\partial x}}+{\mathfrak {A}}_{y}{\frac {\partial {\mathfrak {B}}_{y}}{\partial y}}+{\mathfrak {A}}_{z}{\frac {\partial {\mathfrak {B}}_{y}}{\partial z}},

{\mathfrak {A}}_{x}{\frac {\partial {\mathfrak {B}}_{z}}{\partial x}}+{\mathfrak {A}}_{y}{\frac {\partial {\mathfrak {B}}_{z}}{\partial y}}+{\mathfrak {A}}_{z}{\frac {\partial {\mathfrak {B}}_{z}}{\partial z}}{\textbf {.}}

Auch der mit der Vektoranalysis nicht Vertraute überzeugt sich durch Nachrechnen leicht von der Gültigkeit folgender

Rechnungsregeln:

$\alpha )\ \ ({\mathfrak {A\ B}})=({\mathfrak {B\ A}}).$

$\beta )\ \ [{\mathfrak {A\ B}}]=-[{\mathfrak {B\ A}}].$

$\gamma )\ \ ([{\mathfrak {A\ B}}],{\mathfrak {C}})=({\mathfrak {A}},[{\mathfrak {B\ C}}]).$

$\delta )\ \ [{\mathfrak {A}},[{\mathfrak {B\ C}}]]={\mathfrak {B}}({\mathfrak {A\ C}})-{\mathfrak {C}}({\mathfrak {A\ B}}).$

$\epsilon )\ \ \mathrm {div} \ \varphi \ {\mathfrak {A}}=\varphi \ \mathrm {div} \ {\mathfrak {A}}-({\mathfrak {A}}\ \mathrm {grad} \ \varphi );$ vermöge des Gaußschen Satzes kann man diese Rechnungsregel auch schreiben:

\int \int \ do\ \varphi \ {\mathfrak {A}}_{\nu }=\int \int \int \ dv\ \varphi \ \mathrm {div} \ {\mathfrak {A}}-\int \int \int \ dv\ ({\mathfrak {A}}\ \mathrm {grad} \ \varphi ).

$\zeta )\ \ \mathrm {div} \ [{\mathfrak {A\ B}}]=({\mathfrak {B}}\ \mathrm {curl} \ {\mathfrak {A}})-({\mathfrak {A}}\ \mathrm {curl} \ {\mathfrak {B}});$ der Gauß’sche Satz ergibt:

\int \int \ do\ [{\mathfrak {A\ B}}]_{\nu }=\int \int \int \ dv({\mathfrak {B}}\ \mathrm {curl} \ {\mathfrak {A}})-\int \int \int \ dv({\mathfrak {A}}\ \mathrm {curl} \ {\mathfrak {B}}).

$\eta )\ \ \mathrm {curl} \ [{\mathfrak {A\ B}}]=({\mathfrak {B}}\nabla ){\mathfrak {A}}-({\mathfrak {A}}\nabla ){\mathfrak {B}}+{\mathfrak {A}}\ \mathrm {div} \ {\mathfrak {B}}-{\mathfrak {B}}\ \mathrm {div} \ {\mathfrak {A}}.$

$\vartheta )\ \ -\mathrm {grad} \ ({\mathfrak {A\ B}})=[{\mathfrak {A}}\ \mathrm {curl} \ {\mathfrak {B}}]+[{\mathfrak {B}}\ \mathrm {curl} \ {\mathfrak {A}}]+({\mathfrak {A}}\nabla ){\mathfrak {B}}+({\mathfrak {B}}\nabla ){\mathfrak {A}}.$

$\ \iota )\ \ \mathbf {\mathrm {div} } \ \mathrm {grad} \ \varphi =-\Delta \varphi .$

$\varkappa )\ \ \mathbf {\mathrm {curl} } \ \mathrm {curl} \ {\mathfrak {A}}=-\mathrm {grad} \ \mathrm {div} \ {\mathfrak {A}}-\Delta {\mathfrak {A}}.$