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a) Compute the curl of \( {\bf F} = e^{x}yz \ {\bf i} + 2 e…

a) Compute the curl of \( {\bf F} = e^{x}yz \ {\bf i} + 2 e ^{x}yz^{2} \ {\bf j} + e^{x}y^{2}z^{2} \ {\bf k}\), and thereby deduce that \({\bf F} \) is not conservative. b) Show \({\bf G} = y^{2}z^{3} \ {\bf i} + 2xyz^{3} \ {\bf j} + 3xy^{2}z^{2} \ {\bf k} \) is conservative by finding an explicit potential function \( \phi \), so that \( {\bf G} = \nabla \phi\).

Calculate \( \int_{C} {\bf F} \cdot {\bf dr} \) where \( {\b…

Calculate \( \int_{C} {\bf F} \cdot {\bf dr} \) where \( {\bf F}= \langle 2xy, x^{2}+3y^{2} \rangle\) and \(C\) given by                                                 \( {\bf r}(t)= (\ln (t^{3}+1) \sin(t)) {\bf i} + (\cos(t) t ){\bf j},\) where \(0\leq t \leq \pi\).