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Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. Green’s theorem has two forms: a circulation form and a flux form, both of which require region 𝐷 in the double integral to be simply connected. However, we will extend Green’s theorems to regions that are not simply connected.

Rectangular domain

Semi-infinite strip

Polar coordinates

Curvilinear coordinates

General case

Green's Theorems

Recall that the Fundamental Theorem of Calculus says that
\[ \int_a^b f' (x)\,{\text d}x = f(b-0) - f(a+0) , \]
where \( \displaystyle \quad f(b-0) = \lim_{\varepsilon \downarrow 0} f(b - \varepsilon ) \ \) is the left limit value of function f at point b. Similarly, \( \displaystyle \quad f(a+0) = \lim_{\varepsilon \downarrow 0} f(a + \varepsilon ) \ \) is the right limit value at point 𝑎. As a geometric statement, this equation says that the integral over the region below the graph of positive function f( x) and above the line segment [𝑎,𝑏] depends only on the value of f at the endpoints 𝑎 and 𝑏 of that segment. Since the numbers 𝑎 and 𝑏 are the boundary of the line segment [𝑎,𝑏], the theorem says we can calculate integral ∫𝑏𝑎𝐹′(𝑥)𝑑𝑥 based on information about the boundary of line segment [𝑎,𝑏]. The same idea is true of the Fundamental Theorem for Line Integrals:
\[ \int_C \nabla f \bullet \,{\text d}{\bf r} = f(\mathbf{r}(b-0)) - f(\mathbf{r}(a+0) . \]

Figure 16.4.2 at https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.04%3A_Greens_Theorem

Green’s theorem takes this idea and extends it to calculating double integrals via line integrals over boundary of the domain oriented in positive way. The first form of Green’s theorem that we examine is the circulation form. This form of the theorem relates the vector line integral over a simple, closed plane curve 𝐶 to a double integral over the region enclosed by 𝐶. Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa.

Green's Theorem 1 (Circulation form): Let R be an open, simply connected region with a boundary curve C = ∂R that is a piecewise smooth, simple closed curve oriented counterclockwise. Let 𝐅 =⟨𝑃, 𝑄⟩ be a vector field with component functions that have continuous partial derivatives on R. Then,
\begin{equation} \label{EqGreen.1} \oint_{\partial R} \mathbf{F} \bullet {\text d}\mathbf{r} = \oint_{\partial R} P\,{\text d}x + Q\,{\text d} y = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) {\text d}A , \end{equation}
where dA is is the area element of domain R.

(Figure 16.4.2 https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.04%3A_Greens_Theorem

Notice that the left-hand side of Eq.\eqref{EqGreen.1} can be rewritten as the tangential form of Green’s theorem

\begin{equation} \label{EqGreen.2} \oint_{\partial R} \mathbf{F} \bullet \mathbf{T}\, {\text d}s = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) {\text d}A , \end{equation}
where ds is the element of arclength.

Using rotor operator curl

\[ \mbox{curl}(\mathbf{v}) = \nabla \times \mathbf{v} = \left( \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) {\bf i} + \left( \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) {\bf j} + \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) {\bf k} \]
for a vector v = (u, v, w), we can rewrite the right-hand side of Eq.\eqref{EqGreen.1} as
\[ \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) {\text d}A = \iint_R \mbox{curl} \left( \mathbf{F} \right) \bullet {\bf k} \,{\text d}A \]
because the dot product k • k = 1 and vector field is F = (P,Q, 0) has zero component in the applicant direction.

Proof of Theorem 1 will be given in the last section.

   
Example 1:    ■
End of Example 1

Green's Theorem 2 (Flux or normal form): Let R be an open, simply connected region with a boundary curve C = ∂R that is a piecewise smooth, simple closed curve oriented counterclockwise. Let 𝐅 =⟨𝑃, 𝑄⟩ be a vector field with component functions that have continuous partial derivatives on R. Then,
\begin{equation} \label{EqGreen.3} \oint_{\partial R} \left( \mathbf{F} \bullet \mathbf{n} \right) {\text d}s = \oint_{\partial R} P\,{\text d}y - Q\,{\text d}x = \iint_R \left( \frac{\partial Q}{\partial y} + \frac{\partial P}{\partial x} \right) {\text d}A , \end{equation}
where n is the unit normal to ∂R vector directed outside domain R.

(Figure 16.4.7)

 

 

  1. Apostol, T.M., Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass.–London–Don Mills, Ont., 1974
  2. Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: ‎ 978-0471000075.
  3. do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976
  4. C.H. Edwards; Advanced Calculus of Several Variables, Academic Press (a subsidiary of Harcourt Brace Jovanovich), New York–London, 1973.
  5. Fikhtengol'ts, G.M., The Fundamentals of Mathematical Analysis: International Series of Monographs in Pure and Applied Mathematics, Volume 2, Pergamon, 2013.
  6. Fleming, W., Functions of Several Variables, Undergraduate Texts in Mathematics, Second edition, Springer-Verlag, New York–Heidelberg, 1977.
  7. Hubbard, J.H. and Hubbard, B.B., Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Matrix Editions; 5th edition, 2015; ISBN-13: ‎ 978-0971576681
  8. Kaplan, Advanced Calculus, Pearson; 5th edition, 2002.
  9. Marsden, J.E., Tromba, A.J., Vector Calculus, third edition, Freeman and Company, New York, 2003.