Schwere, Elektricität und Magnetismus:380

VECTOR ANALYSIS.

<section begin=t1 />two components, ${\beta }^{\prime }$ and ${\beta }^{″}$, respectively parallel and perpendicular to $\alpha$. Then

Vorlage:MathForm1

Vorlage:MathForm1

Vorlage:Idt227. General relation of the vector products of three factors.—In the triple product $\alpha \times [\beta \times \gamma ]$ we may set

Vorlage:MathForm1

unless $\beta$ and $\gamma$ have the same direction. Then

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Vorlage:MathForm1

Vorlage:MathForm1

which is evidently true, when $\beta$ and $\gamma$ have the same directions. It may also be written

Vorlage:MathForm1

Vorlage:Idt228. This principle may be used in the transformation of more complex products. It will be observed that its application will always simultaneously eliminate, or introduce, two signs of skew multiplication.

Vorlage:Idt2The student will easily prove the following identical equations, which, although of considerable importance, are here given principally as exercises in the application of the preceding formulae.

Vorlage:Idt229. $\alpha \times [\beta \times \gamma ]+\beta \times [\gamma \times \alpha ]+\gamma \times [\alpha \times \beta ]=0.$

Vorlage:Idt230. $[\alpha \times \beta ]\cdot [\gamma \times \delta ]=(\alpha \cdot \gamma )(\beta \cdot \delta )-(\alpha \cdot \delta )(\beta \cdot \gamma ).$

Vorlage:Idt231. $[\alpha \times \beta ]\times [\gamma \times \delta ]=(\alpha \cdot \gamma \times \delta )\beta -(\beta \cdot \gamma \times \delta )\alpha =(\alpha \cdot \beta \times \delta )\gamma -(\alpha \cdot \beta \times \gamma )\delta .$

Vorlage:Idt232. $\alpha \times [\beta \times [\gamma \times \delta ]]=(\alpha \cdot \gamma \times \delta )\beta -(\alpha \cdot \beta )\gamma \times \delta =(\beta \cdot \delta )\alpha \times \gamma -(\beta \cdot \gamma )\alpha \times \delta .$

Vorlage:Idt233. $[\alpha \times \beta ]\times [\gamma \times \delta ]\times [\epsilon \times \zeta ]=(\alpha \cdot \beta \times \delta )(\gamma \cdot \epsilon \times \zeta )-(\alpha \cdot \beta \times \gamma )(\delta \cdot \epsilon \times \zeta )$

$=-(\alpha \cdot \beta \times \zeta )(\epsilon \cdot \gamma \times \delta )+(\alpha \cdot \beta \times \epsilon )(\zeta \cdot \gamma \times \delta )$

$=(\gamma \cdot \delta \times \alpha )(\beta \cdot \epsilon \times \zeta )+(\gamma \cdot \delta \times \beta )(\alpha \cdot \epsilon \times \zeta ).$

Vorlage:Idt234. $[\alpha \times \beta ]\cdot [\beta \times \gamma ]\times [\gamma \times \alpha ]=(\alpha \cdot \beta \times \gamma {)}^2.$

<section end=t1 /> Vorlage:References