Line Integral
A line integral is an integral along a smooth curve C.
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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, i.e. there are no endpoints, the integral is notated like:
See the circulation and normal forms of Green's theorem.