Green's Theorem | Mathquery

Green's Theorem:

  Statement :-

          If a domain E, regular with respect to both the axes , is bounded by a contour C , and f and g are two single - valued functions which along with their partial derivatives ∂f/∂y and ∂g/∂x  are continuous on E , then 

     ∫∫  (∂g/∂x - ∂f/∂y) dx dy = ∫ (f dx + g dy )

       E                                         C

 where the line integral is taken in the positive direction .

Proof :-

        Let us first consider a function f which , alongwith its partial derivative ∂f/∂y ,is continuous on a region E , regular with respect to y-axis . Let E be bounded by contour C , consisting of the curves y= φ(x) , y= ψ(x) , x = a , x = b , such that 

                      φ(x) ≤ ψ(x) , ∀ x ∈ [a,b]

we have 

                  ∫∫ ∂f(x,y)/∂y dx dy 

                     E

                                      b     ψ(x)

                                  = ∫ dx ∫ ∂f(x,y)/∂y dy

                                     a     φ(x)

                            b                         b

                         = ∫ f(x,ψ(x)) dx - ∫ f(x,φ(x)) dx

                            a                        a

     The two integrals on the right are line integrals along the contour from A to B and from C to D , respectively .(The portions BC and DA of the contour coincide with the lines x = a and x=b respectively .)

 Also contour 

                       C = C₁ + C₂ + C₃ + C₄

  Now 

             b

             ∫ f(x,ψ(x)) dx = ∫ f(x,y) dx = -∫f(x,y)dx

             a                          -c₁                  c₁

              ∫ f(x,y) dx = 0= ∫ f(x,y) dx .

               c₂                       c₄

               b

              ∫ f(x,φ(x)) dx = ∫ f(x,y) dx

              a                         c₃

∴       ∫∫ ∂f/∂y dx dy = 

            E

                -∫ f dx - ∫ f dx - ∫f dx -∫ f dx

                 c₁           c₂         c₃        c₄

                  = -∫ f dx

                      c

    In order to extend the result to any region , let us first suppose the region E to be piecewise regular with regard to the       y-axis , i.e., E can be split up into a finite number of sub-regions, E₁,E₂,....,Eₙ, the contour of each of which is cut in at most two points by a line parallel to y-axis .

    Since result(1) holds for each sub-region , let us apply it to all the sub-regions and add. The double integrals add up to the double integral on the whole region E.

         Since the line integrals along the partition lines cancel each other because along each line the integral is taken twice in opposite directions, therefore the line integrals (in positive sense ) along the contours of sub-regions also add up to the line integral along the contour C of the whole region E.

        Hence for any region E, piecewise regular with respect to y-axis, we have 

                    ∫∫ ∂f/∂y dx dy = -∫ f dx .....(2)

                      E                         c

        It can similarly be proved that for any region E , piecewise regular with respect to x-axis ,

                    ∫∫ ∂g/∂x dx dy = ∫ g dy .......(3)

                     E                          c

       The for any region E, which is regular with respect to both the axes , we have

      ∫∫ (∂g/∂x - ∂f/∂y) dx dy = ∫ f dx + g dy

        E                                        c

                                                    ...............(4)

 Hence the theorem .

Note :-

     The theorem holds even when the domain is enclosed by only two curves ,    y= φ(x) and y = ψ(x), between x=a and x=b , the contours C₂ and C₄ reducing to zero.

Corollary:-

        If f(x,y) = y , we see from (2) that the area of a domain which is piecewise regular with respect to y-axis 

                              = -∫ y dx                  ..........(5)

                                   c

     Similarly putting g(x,y) = x, we see from(3) that the area of a domain , piecewise regular with respect to x-axis 

                              = -∫ x dy                 .............(6)                                    c

    By adding the above results, or putting
g(x,y) = x and f(x,y) = -y , we see from(4) that 

    = 1/2 ∫(x dy - y dx) = ∫∫ dx dy = area of E
             c                            E

 Thus the area of a domain E    (with contour C), regular with respect to both the axes

                = 1/2 ∫(x dy - y dx)   .................(7)
                          c

 Example :-

          With the help of Green's formula , compute the difference between the line integrals 

       I₁ = ∫ {(x+y)² dx - (x-y)² dy}

             ACB

       I₂ = ∫ {(x+y)² dx - (x-y)² dy}

            ADB

      where ACB and ADB are respectively the straight line y = x , and the parabolic are     y = x² , joining the points A(0,0) and B(1,1) .

Solution :-

          Now , 

              I₁-I₂ = ∫ {(x+y)² dx - (x-y)² dy}

                    ADBCA

      The line integral is along the closed contour ADBCA which encloses a domain E, bounded by y = x and y= x² from (0,0) to (1,1) and quadratic with respect to y-axis (in fact both axes) . Therefore by               Green's Theorem ,

   I₁-I₂ = ∫∫{-∂(x-y)²/∂x - ∂(x+y)²/∂y } dx dy

                E

           = -4 ∫∫ x dx dy 

                    E

                   1         x

           = -4 ∫ x dx ∫ dy = 1/3 

                  0          x²

∴ |I₂-I₁| = 1/3 

 Hence the required solution .

                    

About The Scientists :

 

         George Green (mathematician) ... Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work on potential theory ran parallel to that of Carl Friedrich Gauss. For more .