Fundamental theorem of calculus

Green’s theorem

Green’s theorem is a generalisation of the Fundamental theorem of calculus for evaluating a Double integral. It is a special case of Stokes’s theorem and Острогра́дский’s divergence theorem in two dimensions, relating “two-dimensional curl” and Circulation. It is similar to the Fundamental theorem for line integrals, in that one integral is lost by moving from a field to its potential, but in this case it is a vector potential.

Let 𝐷 be a region bounded by a positively oriented, piecewise smooth, simple closed curve in the plane. If 𝑃 and 𝑄 have continuous partial derivatives on an open region that contains 𝐷, then calculus

∬𝐷(𝜕𝑄𝜕𝑥−𝜕𝑃𝜕𝑦)𝑑𝐴=∮𝜕𝐵𝑃𝑑𝑥+𝑄𝑑𝑦

Note that the right hand side is equivalent to the circulation integral Γ.¹

Extended version

Green’s theorem may be extended to a set difference of simply connected regions 𝐴 and 𝐵 where 𝐴 ⊂𝐵

∬𝐵∖𝐴(𝜕𝑄𝜕𝑥−𝜕𝑃𝜕𝑦)𝑑𝐴=∮𝜕𝐵𝑃𝑑𝑥+𝑄𝑑𝑦−∮𝜕𝐴𝑃𝑑𝑥+𝑄𝑑𝑦

the principle here being that a “cut” may be made which cancels itself out (since it is traversed between 𝜕𝐵 and 𝜕𝐴 in both directions).

Practice problems

• 2016. Calculus, pp. 1141–1143 (§16.4 exercises)
• 2023. Advanced Mathematical Methods, p. 44 (§2 problems 1–4)

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Footnotes

1. 2016. Calculus, p. 1136 ↩