Fundamental theorem of calculus

Stokes’s theorem

Let Σ be an oriented piecewise-smooth surface bounded by a positively oriented, piecewise smooth, simple closed curve 𝜕Σ. If ⃗𝐅 is a vector field differentiable in Σ then calculus

∮𝜕Σ⃗𝐅⋅𝑑⃗𝐫=∬Σ⃗∇×⃗𝐅⋅𝑑⃗𝐚

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

Practice problems

• 2016. Calculus, pp. 1179–1180 (§16.9 exercises)

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Footnotes

1. 2016. Calculus, p. 174 ↩