Not as painful as it looks, Part II

Calculus Level 4

If ∇ × F ⃗ = ( cos ⁡ ( z 2 ) − 2 x , y + z sin ⁡ ( x 2 ) , z − y sin ⁡ ( x 2 ) ) , \nabla \times \vec{F} =\left(\cos(z^2)-2x,y+z\sin(x^2),z-y\sin(x^2)\right), find the circulation of F ⃗ \vec{F} around the curve C C given by x 2 + y 2 = 1 x^2+y^2=1 and z = ∣ x ∣ z=|x| , oriented counterclockwise as viewed from above.


The answer is 4.

Otto Bretscher by Otto Bretscher , 65, USA

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1 solution

Otto Bretscher
Dec 6, 2018

The curve C C is the boundary of the cylinder S S given by y 2 + z 2 = 1 , z ≥ 0 , y^2+z^2=1,z\geq 0, and ∣ x ∣ ≤ z |x|\leq z , with upward unit normal n ⃗ = ( 0 , y , z ) \vec{n}=(0,y,z) . By Stokes' theorem, the circulation we seek is ∫ S ( ∇ × F ⃗ ) ⋅ n ⃗ d S = ∫ S d S \int_S (\nabla \times \vec{F}) \cdot \vec{n}\ dS=\int_S dS , the surface area of S S . This area is ∫ θ = 0 π 2 sin ⁡ ( θ ) d θ = 4 \int_{\theta=0}^{\pi}2\sin(\theta)\ d\theta=\boxed{4} .

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