Green's theorem for flux, Multivariable Calculus

We look at Green's theorem relating the flux across a boundary curve enclosing a region in the plane to the total divergence across the enclosed region. Green's Theorem for flux is applicable to a closed, bounded region 𝐷 in the xy-plane. Here are the components we need to identify:

- The boundary of 𝐷, call it 𝐶, is a collection of finitely many, positively oriented, continuously differentiable curves.
- The unit normal vector 𝑛̂ along 𝐶 which points outward from 𝐷.
- The vector field 𝐹⃗ is a continuously differential vector field on ℝ^2.

Here are the two "sides" of Green's Theorem for flux:
- The flux integral, ∮ 𝐹⃗ ⋅𝑛̂ 𝑑𝑠, measuring the outward flux across 𝐷's boundary (C)
- The divergence of 𝐹⃗ over 𝐷, ∬∇⋅𝐹⃗ 𝑑𝐴, representing the total outward or inward fluid flow, indicating expansion or contraction in 𝐷.

What Green's Theorem for flux tells us is that with the right hypotheses, these two measurements are equal: ∮ 𝐹⃗ ⋅𝑛̂ 𝑑s =∬ ∇⋅𝐹⃗ 𝑑𝐴, where the domain is C on the left and D on the right. This is Multivariable Calculus Unit 7 Lecture 3.

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[kofhk]

12:57 why the answer is 8pi not 4pi. pi(4)^2/4=4pi. The finial answer by green theorem is not match with the simple geometry calculation.