Line integrals have properties which are analogous to those of ordinary integrals. For example:
Thus reversal of the path of integration changes the sign of the line integral.
where
is another point on C.
Similar properties hold for line integrals in space.
日: 2014年9月1日
![\displaystyle \int_C [P(x, y)dx + Q(x, y)dy] = \int_C P(x, y)dx + \int_C Q(x, y)dy](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5BP%28x%2C+y%29dx+%2B+Q%28x%2C+y%29dy%5D+%3D+%5Cint_C+P%28x%2C+y%29dx+%2B+%5Cint_C+Q%28x%2C+y%29dy&bg=T&fg=000000&s=0 "\displaystyle \int_C [P(x, y)dx + Q(x, y)dy] = \int_C P(x, y)dx + \int_C Q(x, y)dy")
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+-+%5Cint_%7B%28a_2%2C+b_2%29%7D%5E%7B%28a_1%2C+b_1%29%7D%5BPdx+%2B+Qdy%5D&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]")
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_3%2C+b_3%29%7D+%5BPdx+%2B+Qdy%5D+%2B+%5Cint_%7B%28a_3%2C+b_3%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy]")
線積分の属性
線積分には通常の積分と類似した属性があります．例えば，
ゆえに積分経路の逆転は線積分の符号の変化をもたらします．
ここで
空間内の線積分も同様の属性を有します．