Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem. Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and having continuous partial derivatives there, then where the path of integration along is anticlockwise. In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals , stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. Proof when "D" is a simple region. The following is a proof of half of the theorem for the simplified area "D", a type I region where "C" and "C" are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the  theorem when "D" is a type II region where "C" and "C" are curves connected by horizontal lines (again, possibly of zero length). Putting these two parts together, the theorem is thus proven for regions of type III (defined as regions which are both type I and type II). The general case can then be deduced from this special case by decomposing "D" into a set of type III regions. If it can be shown that and are true, then Green's theorem follows immediately for the region D. We can prove () easily for regions of type I, and () for  regions of type II. Green's theorem then follows for regions of type III. Assume region "D" is a type I region and can thus be characterized, as pictured on the right, by formula_1 where "g" and "g" are continuous functions on . Compute the double integral in (): Now compute the line integral in (). "C" can be rewritten as the union of four curves: "C", "C", "C", "C". With "C", use the parametric equations: "x" = "x", "y" = "g"("x"), "a" ≤ "x" ≤ "b". Then formula_2 With "C", use the parametric equations: "x" = "x", "y" = "g"("x"), "a" ≤  x ≤ "b". Then formula_3 The integral over "C" is negated because it goes in the negative direction from "b" to "a", as "C" is oriented positively (anticlockwise). On "C" and "C", "x" remains constant, meaning formula_4 Therefore, Combining () with (), we get () for regions of type I. A similar treatment yields () for regions of type II. Putting the two together, we get the result for regions of type III. Proof for rectifiable Jordan curves. We are going to prove the following We need the following lemmas whose proofs can be found in: Now we are in position to prove the  theorem: Proof of Theorem. Let formula_5 be an arbitrary positive real number. By continuity of formula_6, formula_7 and compactness of formula_8, given formula_9, there exists formula_10 such that whenever two points of formula_8 are less than formula_12 apart, their images under formula_13 are less than formula_5 apart. For this formula_15, consider the decomposition given by the previous Lemma. We have formula_16 Put formula_17. For each formula_18, the curve formula_19 is a positively oriented square, for which Green's formula holds. Hence formula_20 Every point of a border region is at a distance no greater than formula_12 from formula_22. Thus, if  formula_23 is the union of all border regions, then formula_24; hence formula_25, by Lemma 2. Notice that formula_26 This yields formula_27 We may as well choose formula_15 so that the RHS of the last inequality is formula_29 The remark in the beginning of this proof implies that the oscillations of formula_6 and formula_7 on every border region is at most formula_5. We have formula_33 By Lemma 1(iii), formula_34 Combining these, we finally get formula_35 for some formula_36. Since this is true for every formula_37, we are done. Validity under different hypotheses. The hypothesis of the last theorem are not the  only ones under which Green's formula is true. Another common set of conditions is the following: The functions formula_38 are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of formula_39. This implies the existence of all directional derivatives, in particular formula_40, where, as usual, formula_41 is the canonical ordered basis of formula_42. In addition, we require the function formula_43 to be Riemann-integrable over formula_39. As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves: Multiply-connected regions. Theorem. Let formula_45 be positively oriented rectifiable  Jordan curves in formula_46 satisfying formula_47 where formula_48 is the inner region of formula_19. Let formula_50 Suppose formula_51 and formula_52 are continuous functions whose restriction to formula_53 is Fréchet-differentiable. If the function formula_54 is Riemann-integrable over formula_53, then formula_56 Relationship to Stokes' theorem. Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the formula_57-plane. We can augment the two-dimensional field into a three-dimensional field with a "z" component that is always 0. Write F for the vector-valued function formula_58. Start with the left side of  Green's theorem: formula_59 The Kelvin–Stokes theorem: formula_60 The surface formula_61 is just the region in the plane formula_53, with the unit normal formula_63 defined (by convention) to have a positive z component in order to match the "positive orientation" definitions for both theorems. The expression inside the integral becomes formula_64 Thus we get the right side of Green's theorem formula_65 Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: formula_66 Relationship to the divergence theorem. Considering only two-dimensional vector fields, Green's theorem is equivalent to  the two-dimensional version of the divergence theorem: where formula_67 is the divergence on the two-dimensional vector field formula_68, and formula_63 is the outward-pointing unit normal vector on the boundary. To see this, consider the unit normal formula_63 in the right side of the equation. Since in Green's theorem formula_71 is a vector pointing tangential along the curve, and the curve "C" is the positively oriented (i.e. anticlockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right of this; one choice would be formula_72. The length of this  vector is formula_73 So formula_74 Start with the left side of Green's theorem: formula_75 Applying the two-dimensional divergence theorem with formula_76, we get the right side of Green's theorem: formula_77 Area calculation. Green's theorem can be used to compute area by line integral. The area of a planar region formula_53 is given by formula_79 Choose formula_80 and formula_81 such that formula_82, the area is given by formula_83 Possible formulas for the area of formula_53 include formula_85 History. It is named after George Green, who stated a similar result in an 1828 paper titled "An Essay on  the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". In 1846, Augustin-Louis Cauchy published a paper stating Green's theorem as the penultimate sentence. This is in fact the first printed version of Green's theorem in the form appearing in modern textbooks. Bernhard Riemann gave the first proof of Green's theorem in his doctoral dissertation on the theory of functions of a complex variable.