Question 8 (20 points) Parts (a) and (b) are distinct from e…
Question 8 (20 points) Parts (a) and (b) are distinct from each other. State any theorems or formulas used. (a) Use Stokes’ Theorem to evaluate \(\displaystyle \int_C \vec{F} \cdot d\vec{r}\), where \(\vec{F}(x,y,z)= \langle z^2, y^2, xy \rangle\) and \(C\) is the triangle with vertices (1,0,0), (0,1,0) and (0,0,2) oriented counterclockwise as viewed from above. (b) Let \(\vec{F}(x,y,z)=\langle x^3+e^{y^2+z^2}, \,2xz^2, \, 3y^2z \rangle\) be a vector field on \(\mathbb{R^3}\), \(E\) be the part of the solid bounded above by the paraboloid \(z=4-x^2-y^2\) and below by the xy-plane , and \(S\) be the boundary of \(E\) oriented outward. Using the Divergence Theorem find the total flux, \(\displaystyle \iint_S \vec{F} \cdot d\vec{S}\), of \(\vec{F}\) through \(S\).
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