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=== dr Reformulation === |
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| Lastly, if a vector field 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: | Again parameterize ''F'' using ''r(t)'' for ''a ≤ t ≤ b'' to get: |
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| Or parameterize ''f'' using ''r(t)'' for ''a ≤ t ≤ b'' to get: | === 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: |
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| If ''C'' is closed and circular, i.e. there are no endpoints, the integral is notated like | Or parameterize ''f'' using ''r(t)'' for ''a ≤ t ≤ b'' to get: |
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| and is generally evaluated using '''Green's theorem'''. ---- |
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| === Circular Integral === | |
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| == Green's Theorem == | If ''C'' is closed, i.e. there are no endpoints, the integral is notated like: |
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| '''Green's theorem''' enables conversion between [[Calculus/Integral|double integrals]] and line integrals along a closed and circular curve ''C''. | {{attachment:circ.svg}} |
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| Consider a [[Calculus/VectorField|vector field]] expressed as ''F = <P(x,y), Q(x,y)>'' and a closed circular curve ''C'' [[Calculus/ParametricEquation|parameterized]] as ''r(t) = <x(t), y(t)>''. The '''circulation form''' of the theorem states that: \oint_C F \cdot T ds = \oint_C Pdx + Qdy = \iint_D(Q_x - P_y)\,dA 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 can also be expressed in terms of [[Calculus/Curl|curl]], leading to it sometimes being called the '''curl form'''. \iint_D \mathrm{curl} \, F \cdot k \, dA The '''[[Calculus/Flux|flux]] form''' or '''normal form''' of the theorem states that: \oint_C F \cdot N ds = \iint_D(P_x + Q_y) \, dA |
See the [[Calculus/CirculationIntegral|circulation]] and [[Calculus/FluxIntegral#Closed_Line_Integrals|normal forms]] of '''Green's theorem'''. |
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, i.e. there are no endpoints, the integral is notated like:
See the circulation and normal forms of Green's theorem.