(b) i. If $$ F $$ is a conservative field, prove that curl $$ \vec{F}=
abla imes \vec{F}=\overrightarrow{0} $$ (i.e., $$ \vec{F} $$ is irrotational). ii. Conversely, if $$
abla imes \vec{F}=\overrightarrow{0} $$ (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. $$ \int_{C} \vec{F} imes d \vec{r} $$. "Mathematics Can Smile!" ( 2 mar

Question

(b) i. If $$ F $$ is a conservative field, prove that curl $$ \vec{F}=
abla 	imes \vec{F}=\overrightarrow{0} $$ (i.e., $$ \vec{F} $$ is irrotational). ii. Conversely, if $$ 
abla 	imes \vec{F}=\overrightarrow{0} $$ (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. $$ \int_{C} \vec{F} 	imes d \vec{r} $$. "Mathematics Can Smile!" ( 2 mar

Frank Carroll · Accepted Answer

(b) i. If $$ F $$ is a conservative field, its curl is zero, making it irrotational. Conversely, if the curl is zero, $$ F $$ is conservative. (c) i. The line integral of $$ \phi $$ along the curve $$ C $$ is $$ \frac{4}{5} $$.

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