Learning Objectives
- Apply the flux form of Green’s theorem.
theorem: Green’s theorem, flux form
Let D be an open, simply connected region with a boundary curve C that is a piecewise smooth, simple closed curve that is oriented counterclockwise (Figure 1). Let F=⟨P,Q⟩ be a vector field with component functions that have continuous partial derivatives on an open region containing D. Then,
∮F⋅Nds=∬DPx+QydA.

Figure 1. The flux form of Green’s theorem relates a double integral over region D to the flux across curve C.
Because this form of Green’s theorem contains unit normal vector N, it is sometimes referred to as the normal form of Green’s theorem.
Proof
Recall that ∮F⋅Nds=∮C−Qdx+Pdy. Let M=−Q and N=P. By the circulation form of Green’s theorem,
∮C−Qdx+Pdy=∮CMdx+nDy=∬DNx−MydA=∬DPx−(−Q)ydA=∬DPx+QydA.
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Example: Applying Green’s theorem for flux across a circle
Let C be a circle of radius r centered at the origin (Figure 2) and let F(x,y)=⟨x,y⟩. Calculate the flux across C.

Figure 2. Curve C is a circle of radius r centered at the origin.
Example: applying green’s theorem for flux across a triangle
Let S be the triangle with vertices (0,0), (1,0), and (0,3) oriented clockwise (Figure 3). Calculate the flux of F(x,y)=⟨P(x,y),Q(x,y)⟩=⟨x2+ey,x+y⟩ across S.

Figure 3. Curve S is a triangle with vertices (0,0), (1,0), and (0,3) oriented clockwise.
try it
Calculate the flux of F(x,y)=⟨x3,y3⟩ across a unit circle oriented counterclockwise.
Watch the following video to see the worked solution to the above Try It
Example: applying green’s theorem for water flow across a rectangle
Water flows from a spring located at the origin. The velocity of the water is modeled by vector field v(x,y)=⟨5x+y,x+3y⟩ m/sec. Find the amount of water per second that flows across the rectangle with vertices (−1,−2), (1,−2), (1,3), and (−1,3), oriented counterclockwise (Figure 4).

Figure 4. Water flows across the rectangle with vertices (−1,−2), (1,−2), (1,3), and (−1,3), oriented counterclockwise.
Recall that if vector field F is conservative, then F does no work around closed curves—that is, the circulation of F around a closed curve is zero. In fact, if the domain of F is simply connected, then F is conservative if and only if the circulation of F around any closed curve is zero. If we replace “circulation of F” with “flux of F,” then we get a definition of a source-free vector field. The following statements are all equivalent ways of defining a source-free field F=⟨P,Q⟩ on a simply connected domain (note the similarities with properties of conservative vector fields):
- The flux ∮CF⋅Nds across any closed curve C is zero.
- If C1 and C2 are curves in the domain of F with the same starting points and endpoints, then ∫C1F⋅Nds=∫C2F⋅Nds. In other words, flux is independent of path.
- There is a stream function g(x,y) for F. A stream function for F=⟨P,Q⟩ is a function g such that P=gy and Q=−gx. Geometrically, F(a,b) is tangential to the level curve of g at (a,b). Since the gradient of g is perpendicular to the level curve of g at (a,b), stream function g has the property F(a,b)∙∇g(a,b)=0 for any point (a,b) in the domain of g. (Stream functions play the same role for source-free fields that potential functions play for conservative fields.)
- Px+Qy=0
Example: finding a stream function
Verify that rotation vector field F(x,y)=⟨y,−x⟩ is source free, and find a stream function for F.
try it
Find a stream function for vector field F(x,y)=⟨xsiny,cosy⟩.
Vector fields that are both conservative and source free are important vector fields. One important feature of conservative and source-free vector fields on a simply connected domain is that any potential function f of such a field satisfies Laplace’s equation fxx+fyy=0. Laplace’s equation is foundational in the field of partial differential equations because it models such phenomena as gravitational and magnetic potentials in space, and the velocity potential of an ideal fluid. A function that satisfies Laplace’s equation is called a harmonic function. Therefore any potential function of a conservative and source-free vector field is harmonic.
To see that any potential function of a conservative and source-free vector field on a simply connected domain is harmonic, let f be such a potential function of vector field F=⟨P,Q⟩. Then, fx=P and fx=Q because ∇f=F. Therefore, fxx=Px and fyy=. Since F is source free, fxx+fyy=Px+Qy=0, and we have that f is harmonic.
Example: satisfying laplace’s equation
For vector field F(x,y)=⟨exsiny,excosy⟩, verify that the field is both conservative and source free, find a potential function for F, and verify that the potential function is harmonic.
try it
Is the function f(x,y)=ex+5y harmonic?
Candela Citations
- CP 6.36. Authored by: Ryan Melton. License: CC BY: Attribution
- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction