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Multiple integral, curl, vector fields

Source: 2019 South Korea USMC P3

August 13, 2020
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Problem Statement

Two vector fields F,G\mathbf{F},\mathbf{G} are defined on a three dimensional region W={(x,y,z)∈R3:x2+y2≤1,∣z∣≤1}W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}. \mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),  \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0) Evaluate the following integral. ∭W(G⋅curl(F)−F⋅curl(G))dV\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV
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