Line Integral
A line integral is an integral along a smooth curve C.
Contents
Scalar Line Integral
Given a smooth curve C, integrating a function f along C gives the scalar line integral. As the name implies, this returns a scalar value.
Parameterize f using r(t) for a ≤ t ≤ b to get:
This gives a straightforward calculation for arc length:
Vector Line Integral
A vector field F is defined for some domain D. Given a smooth curve C that exists entirely within D, the vector line integral is given by:
where T is the unit tangent vector.
Parameterize F using r(t) for a ≤ t ≤ b to get:
dr Reformulation
Another common notation follows from reformulating r'(t) as:
Therefore dr can be substituted into the above equation.
Again parameterize F using r(t) for a ≤ t ≤ b to get:
Conservative Reformulation
If vector field F is conservative, then there exists at least one function f satisfying F = ∇f. If given the start and end points A and B of the curve C, then:
Or parameterize f using r(t) for a ≤ t ≤ b to get:
Circular Integral
If C is closed and circular, i.e. there are no endpoints, the integral is notated like
and is generally evaluated using Green's theorem.
Green's Theorem
Circulation Form
Green's theorem enables conversion between double integrals and line integrals along a closed and circular curve C.
Consider a vector field expressed as F = <P(x,y), Q(x,y)> and a closed circular curve C parameterized as r(t) = <x(t), y(t)>. The circulation form of the theorem states that:
for a C that is oriented counter-clockwise. (For a clockwise oriented C, negate the formula.) In some cases, such an integral is expressed as being along the perimeter of a surface D rather than along curve C. In this case C is substituted for ∂D.
This formulation is particularly useful in cases where derivation eliminates all variables of P and Q, leaving an integral that is simply the area of region D (i.e., ∫∫D dA) multiplied by some scalar.
This can also be expressed in terms of curl, leading to it sometimes being called the curl form.
(Note that k is the unit vector.)
Note also that there is a normal form of the theorem.