Calculus 3 Quiz
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VECTOR FIELDS & LINE/SURFACE INTEGRALS
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Consider the vector field F(x,y,z)=⟨xy,z2,x+y⟩\mathbf{F}(x,y,z) = \langle xy, z^2, x+y \rangle and the surface SS defined by z=4−x2−y2z = 4 - x^2 - y^2 for z≥0z \geq 0, oriented with upward-pointing normal vectors. If the flux integral ∬SF⋅dS\iint_S \mathbf{F} \cdot d\mathbf{S} is evaluated using the parameterization r(u,v)=⟨u,v,4−u2−v2⟩\mathbf{r}(u,v) = \langle u, v, 4-u^2-v^2 \rangle, what is the integrand after computing F(r(u,v))⋅(ru×rv)\mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v)?
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