Let C be a curve in the xy plane which connects points and , (see Fig. 6-2). Let and be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by . Call and and suppose that points are chosen so that they are situated on C between points and . Form the sum
The limit of this sum as in such a way that all quantities approaches zero, if such limit exists, is called a line integral along C and is denoted by
or
The limit does exist if P and Q are continuous (or piecewise continuous) at all points of C. The value of the integral depends in general on P, Q, the particular curve C, and on the limits and .
In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as
where , and are functions of , and .
Other types of line integrals, depending on particular curves, can be defined. For example, if denotes the arc length along curve C in the above figure between points and , then
is called the line integral of along curve C. Extensions to three (or higher) dimensions are possible.
and
, (see Fig. 6-2). Let
and
be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by
. Call
and
and suppose that points
are chosen so that they are situated on C between points
and
. Form the sum
in such a way that all quantities
approaches zero, if such limit exists, is called a line integral along C and is denoted by
or
and
.
![\displaystyle \lim\limits_{n \rightarrow\infty}\sum_{k=1}^{n}\left\{ A_1(\xi_k, \eta_k, \zeta_k)\Delta x_k + A_2(\xi_k, \eta_k, \zeta_k)\Delta y_k + A_3(\xi_k, \eta_k, \zeta_k)\Delta z_k \right\} \\ = \int_C \left[ A_1dx + A_2dy + A_3dz \right] \cdots(15)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clim%5Climits_%7Bn+%5Crightarrow%5Cinfty%7D%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%5Cleft%5C%7B+A_1%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+x_k+%2B+A_2%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+y_k+%2B+A_3%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+z_k++%5Cright%5C%7D+%5C%5C+%3D+%5Cint_C+%5Cleft%5B+A_1dx+%2B+A_2dy+%2B+A_3dz+%5Cright%5D+%5Ccdots%2815%29&bg=T&fg=000000&s=0 "\displaystyle \lim\limits_{n \rightarrow\infty}\sum_{k=1}^{n}\left\{ A_1(\xi_k, \eta_k, \zeta_k)\Delta x_k + A_2(\xi_k, \eta_k, \zeta_k)\Delta y_k + A_3(\xi_k, \eta_k, \zeta_k)\Delta z_k \right\} \\ = \int_C \left[ A_1dx + A_2dy + A_3dz \right] \cdots(15)")
,
and
are functions of
,
and
.
denotes the arc length along curve C in the above figure between points
, then
along curve C. Extensions to three (or higher) dimensions are possible.
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)](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%5Cleft%5B+Pdx+%2B+Qdy+%5Cright%5D%5Ccdots%2814%29&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)")
