Simpson Nichols

02/16/2023 · escuela secundaria

(b) i. If \( F \) is a conservative field, prove that curl \( \vec{F}=\nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational). ii. Conversely, if \( \nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational), prove that \( \vec{F} \) is conservative. (c) If \( \phi=2 x y z^{2}, \quad \vec{F}=x y \vec{i}-z \vec{j}+x^{2} \vec{k} \) and C is the curve \( x=t^{2}, \quad y=2 t, \quad z=t^{3} \) from \( t=0 \) to \( t=1 \), evaluate the line integrals: i. \( \int_{C} \phi d \vec{r} \). ii. \( \int_{C} \vec{F} \times d \vec{r} \). "Mathematics Can Smile!" ( 2 mar