“line integral”的概念、定义、翻译、参考文献-科学参考

line integral

Physics

If a vector function V(x,y,z) is defined between two points A and B on a curve, the curve is approximately given by a series of equal directed chords Δ‎₁l,Δ‎₂l,…Δ‎_nl. In each segment i of the curve it is possible to define the scalar product V_i·Δ‎_il. If the sum of the scalar products

$\sum_{i}^{n} = V_{i} \cdot Δ_{i} l$

is considered, the line integral is defined by

$\int_{A}^{B} V \cdot d l = lim_{n \to \infty} \sum_{i = 1}^{n} V \cdot Δ_{i} l .$

An example of a line integral is given by the case of a force F acting on a particle in the field of the force. The line integral $\int_{A}^{B} F \cdot d l$ is the work done on the particle as it moves from A to B because of the force.

If the line integral is taken round a closed path (loop), the line integral is denoted by ∫_C V·dl or ∮V·dl. A line integral for a scalar function ϕ‎(x,y,z) is defined in a similar way and is denoted $\int_{A}^{B} ϕ d l$ . It is also possible to define another type of line integral for a vector function V by $\int_{A}^{B} V \times d l$ .

Mathematics

An integral along a curve. Say a curve C is parametrized by r(t) where a ≤ t ≤ b. For a scalar function f, defined on C, we define the line integral

$\int_{C} f d s = \int_{a}^{b} f (r (t)) | \frac{d r}{d t} | d t,$

where s denotes arc length. It is independent of the parametrization. If f represents the density of a wire in the shape of C, then the line integral would equal the mass of the wire.

For a vector function F, defined on C, we define the line integral

$\int_{C} F \cdot d r = \int_{a}^{b} F (r (t)) \cdot \frac{d r}{d t} d t,$

which equals the work done by F along C. It is independent of the parametrization up to sign but changes sign if the orientation of the parametrization is reversed.

Line integrals are also called path integrals, particularly in complex analysis.

单词	line integral
释义	line integral Physics If a vector function V(x,y,z) is defined between two points A and B on a curve, the curve is approximately given by a series of equal directed chords Δ‎₁l,Δ‎₂l,…Δ‎_nl. In each segment i of the curve it is possible to define the scalar product V_i·Δ‎_il. If the sum of the scalar products $\sum_{i}^{n} = V_{i} \cdot Δ_{i} l$ is considered, the line integral is defined by $\int_{A}^{B} V \cdot d l = lim_{n \to \infty} \sum_{i = 1}^{n} V \cdot Δ_{i} l .$ An example of a line integral is given by the case of a force F acting on a particle in the field of the force. The line integral $\int_{A}^{B} F \cdot d l$ is the work done on the particle as it moves from A to B because of the force. If the line integral is taken round a closed path (loop), the line integral is denoted by ∫_C V·dl or ∮V·dl. A line integral for a scalar function ϕ‎(x,y,z) is defined in a similar way and is denoted $\int_{A}^{B} ϕ d l$ . It is also possible to define another type of line integral for a vector function V by $\int_{A}^{B} V \times d l$ . Mathematics An integral along a curve. Say a curve C is parametrized by r(t) where a ≤ t ≤ b. For a scalar function f, defined on C, we define the line integral $\int_{C} f d s = \int_{a}^{b} f (r (t)) \| \frac{d r}{d t} \| d t,$ where s denotes arc length. It is independent of the parametrization. If f represents the density of a wire in the shape of C, then the line integral would equal the mass of the wire. For a vector function F, defined on C, we define the line integral $\int_{C} F \cdot d r = \int_{a}^{b} F (r (t)) \cdot \frac{d r}{d t} d t,$ which equals the work done by F along C. It is independent of the parametrization up to sign but changes sign if the orientation of the parametrization is reversed. Line integrals are also called path integrals, particularly in complex analysis.
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