meanings of Green's Theorem

Various Meanings of "Green's Theorem" PhL 6.30.12 The phrase Green's Theorem seems to mean different things to different people. I will try here to make a short list. 1. First, we at least state what are usually called the two Green's Identities: ∫V dV ψ 2φ = ∫S dS ψ( ∂φ/∂n) – ∫V dV {ψ φ} (5a) Green #1 ∫V dV [ ψ 2φ – φ 2ψ ] = ∫S dS [ ψ( ∂φ/∂n) – φ( ∂ψ/∂n) ] (6) Green #2 = ∫S dS [ ψφ – φψ ] Do people refer to either of these as "Green's Theorem". Yes! On page 14 green Jackson refers to the 2nd identity both by that name, and by the term Green's Theorem. And on p 161 M&M also refer to both of the above as "Green's Theorems". And Chap 7 p 806 M&F also refer to Green #2 as Green's Theorem. 2. Stakgold seems to have a special definition of "Green's Theorem" in his 5.72 and 5.73 p 40 vLu - uL*v = div J (5.72) ∫V dV (vLu - uL*v) = ∫S dS n J (5.73) So in Stak's version, L can be any linear differential operator! And of course the space holding dV can have any number of dimensions N. So his I think is the most general. If you set L = 2, then we know from Stak page 40 that J = u v - v u and in that case Stak's Green's Theorem becomes the second identity. So really Stak's thing is a generalization of the second identity. 3. Lai on page 411 uses the term "Green's Theorem" to describe the "integral of a gradient theorem" in 2D space. That is to say, from my sheet [ I think this requires that C be "convex" ] ∫V dV φ = ∫S dS φ = ∫S dS n φ // "integral of a gradient theorem" and in 2D this becomes ∫A dA∂xφ = ∫C ds nx φ = ∫C dy φ ∫A dA∂yφ = ∫C ds ny φ = – ∫C dx φ 4. Wiki: Now suppose we consider the 1st line above applied to φ = M and the 2nd line applied to φ = L, ∫A dA∂xM = ∫C ds nx M = ∫C dy M ∫A dA∂yL = ∫C ds ny L = – ∫C dx L Now subtract the second line from the first to get ∫A dA ( ∂xM - ∂yL) = ∫C dy M + ∫C dx L = ∫C (dy M + dx L) According to wiki, this is yet another "Green's Theorem" that differs from all of the above!! Suppose the two functions form a vector of functions M = fy L = fx then the above theorem becomes ∫A dA ( ∂x fy - ∂y fx) = ∫C dy fy + ∫C dx fx = ∫C (dy fy + dx fx) = ∫C dx f My PDF called "chap3 integral theorems" refers to this as "Green's Theorem in a Plane" and also "Stokes Theorem in a Plane".