= Line Integral =

A '''line integral''' is an [[Calculus/Integral|integral]] along a smooth curve ''C''.

<<TableOfContents>>

----



== 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.

{{attachment:scalar1.svg}}

[[Calculus/ParametricEquation|Parameterize]] ''f'' using ''r(t)'' for ''a ≤ t ≤ b'' to get:

{{attachment:scalar2.svg}}

This gives a straightforward calculation for arc length:

{{attachment:arc.svg}}

----



== Vector Line Integral ==

A [[Calculus/VectorField|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:

{{attachment:vector1.svg}}

where ''T'' is the unit tangent vector.

[[Calculus/ParametricEquation|Parameterize]] ''F'' using ''r(t)'' for ''a ≤ t ≤ b'' to get:

{{attachment:vector2.svg}}



=== dr Reformulation ===

Another common notation follows from reformulating ''r'(t)'' as:

{{attachment:dr1.svg}}

{{attachment:dr2.svg}}

Therefore ''dr'' can be substituted into the above equation.

{{attachment:vector3.svg}}

Again parameterize ''F'' using ''r(t)'' for ''a ≤ t ≤ b'' to get:

{{attachment:vector4.svg}}



=== Conservative Reformulation ===

If vector field ''F'' is [[Calculus/VectorField#Conservative_Fields|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:

{{attachment:vector5.svg}}

Or parameterize ''f'' using ''r(t)'' for ''a ≤ t ≤ b'' to get:

{{attachment:vector6.svg}}



=== Circular Integral ===

If ''C'' is closed, i.e. there are no endpoints, the integral is notated like:

{{attachment:circ.svg}}

See the [[Calculus/CirculationIntegral|circulation]] and [[Calculus/FluxIntegral#Closed_Line_Integrals|normal forms]] of '''Green's theorem'''.



----
CategoryRicottone