University Mathematics Handbook (2015)
IX. Vector Analysis
Chapter 9. Line Integrals
9.1 Definition
Given: Curve 
And vector field 
a. Line 
is a smooth curve if 
are continuous vector functions.
b. The positive direction on is the same as the direction of parameter 
increase. If is a closed line, then we define its equation such that when the parameter increases, its direction on is such that during the movement along in this direction, the finite domain is bounded by it remains on the left.
c. Line integral, or integral along is
d. If 
is a planar vector field, and curve is given in its explicit form, 
, then:
e. The physical meaning of line integral is work done by field 
of moving a particle from point 
to 
along .
9.2 Properties of Line Integrals
a. 
b. 
, when 
is the length of .
c. If curve 
consists of a finite number of curves 
, then 
.
d. The line integral depends on the direction defined on . If the same route is passed in the opposite direction, the integral will change its sign: 
.
9.3 Green's Theorem: Conservative Field in the Plane
a. The positive direction on the boundary curve 
(consisting of a finite number of closed and piecewise smooth lines) of region 
is such that if we move along all parts of in this direction, region will be on the left.
b. Green's Theorem: Let be a connected and open domain with boundary in the positive direction, and let 
vector field of 
in . Then:
Or, in a vector form: 
c. If
is a vector field defined above planar domain , then all the following propositions are equivalent:
1. 
for every closed line in .
2. The integral 
is independent of the path connecting points 
and 
, which is entirely in .
3. There exists continuous scalar function 
holding 
, in other words, the expression 
is a full differential, 
, and there holds
4. If, in addition, field is of class and domain is simply connected, that is, its boundary consists of a single closed line not intersecting itself, then the last three propositions are equivalent to
d. The vector field 
is a conservative field in if there exists scalar field such that 
. Function 
is called a potential of field .
The function of a potential of a vector field is a unique up to a constant.
e. Formulas of potential function .
If is a conservative field in and 
, then:
or 
f. If is a bound domain with boundary 
and 
is a unit normal to directed outwards and 
, then:
