向量分析 (III)

1.3 向量積分

1.3.1 線、面以及體積分

In electrodynamics we encounter several different kinds of integrals, among which the most important are line (or path) integrals, surface integrals (or flux), and volume integrals.
(a) Line Integrals. A line integral is an expression of the form
\[\int^\mathbf{b}_{\mathbf{a}\mathcal{P}}\mathbf{v}\cdot \mathrm{d}\mathbf{l}, \]
where $\mathbf{v}$ is a vector function, $d\mathbf{l}$ is the infinitesimal displacement vector, and the integral is to be carried out along a prescribed path $\mathcal{P}$ from point $\mathbf{a}$ to point $\mathbf{b}$. If the path in question forms a closed loop (that is, if $\mathbf{b}=\mathbf{a}$), we shall put a circle on the integral sign:
\[\oint\mathbf{v}\cdot d\mathbf{l}. \]
To a physicist, the most familiar example of a line integral is the work done by a force $\mathbf{F}$:
\[\int\mathbf{F}\cdot d\mathbf{l}. \]
Ordinarily, the value of a line integral depends critically on the particular path taken from $\mathbf{a}$ to $\mathbf{b}$, but there is an important special class of vector functions for which the line integral is independent of the path, and is
determined entirely by the end points. A force that has this property is called conservative.

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