Fundamental theorem of calculus

Острогра́дский’s divergence theorem

Let Ω be a solid and 𝜕Ω be its oriented boundary. Let ⃗𝐅 be a vector field differentiable in Ω. Then calculus

⊂⊃∬𝜕Ω⃗𝐅⋅𝑑⃗𝐚=∭Ω⃗∇⋅⃗𝐅𝑑𝜏

Note the left hand side is equivalent to the flux through the surface of Ω, the right hand side refers to Divergence. Heuristically, if a region has no divergence, there is no nett in-flow or out-flow, and therefore the flux through the boundary is zero.

Corollaries

∭Ω(⃗∇×⃗𝐀)𝑑𝜏=−⊂⊃∬𝜕Ω⃗𝐀×𝑑⃗𝐚

Proof

For any vector ⃗𝐜 ∈ℝ3, we have

⃗𝐜⋅⊂⊃∬𝜕Ω⃗𝐀×⃗𝐝𝑎=−⊂⊃∬𝜕Ω(⃗𝐀×⃗𝐜)⋅𝑑⃗𝐚=−∭Ω⃗∇⋅(⃗𝐀×⃗𝐜)𝑑𝜏=⃗𝐜⋅∭Ω(⃗∇×⃗𝐀)𝑑𝜏

proving ^C1.

Practice problems

• 2016. Calculus, pp. 1185–1186 (§16.9 exercises)

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