Schwere, Elektricität und Magnetismus:395

VECTOR ANALYSIS.

<section begin=t1 /> from the nature of the formula itself. The most important discontinuities of scalars are those which occur at surfaces: in the case of vectors, discontinuities at surfaces, at lines, and at points, should be considered.

Vorlage:Idt74. From equation (3) we obtain

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where the accents distinguish the quantities relating to the limits of the line-integrals. We are thus able to reduce a line-integral of the form $\int \mathrm{\nabla }t\cdot d\rho$ to the form $-\int t\mathrm{\nabla }u\cdot d\rho$ with quantities free from the sign of integration.

Vorlage:Idt75. From equation (5) we obtain

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where, as elsewhere in these 'equations, the line-integral relates to the boundary of the surface integral.

Vorlage:IdtFrom this, by substitution of $\mathrm{\nabla }t$ for $\omega$, we may derive as a particular case

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Vorlage:Idt76. From equation (4) we obtain

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where, as elsewhere in these equations, the surface-integral relates to the boundary of the volume-integrals.

Vorlage:IdtFrom this, by substitution of $\mathrm{\nabla }t$ for $\omega$, we derive as a particular case

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which is Green’s Theorem. The substitution of $s\mathrm{\nabla }t$ for $\omega$ gives the more general form of this theorem which is due to Thomson, viz:—

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Vorlage:Idt77. From equation (6) we obtain

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A particular ease is

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Integration of Differential Equations.

<section begin=t1 />Vorlage:Idt78. If throughout any continuous space (or in all space)

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