(a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors $$ \vec{A}=2 \vec{i}- $$ $$ 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} $$ and $$ \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} $$. (b) i. If $$ \vec{F} $$ is a conservative field, prove that curl $$ \vec{F}=
abla imes \vec{F}=\overrightarrow{0} $$ (i.e., $$ \vec{F} $$ is irrotational). (2 marks) 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 $$ t=0 $$

Question

(a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors $$ \vec{A}=2 \vec{i}- $$ $$ 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} $$ and $$ \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} $$. (b) i. If $$ \vec{F} $$ is a conservative field, prove that curl $$ \vec{F}=
abla 	imes \vec{F}=\overrightarrow{0} $$ (i.e., $$ \vec{F} $$ is irrotational). (2 marks) 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 $$ t=0 $$

Schofield Hodgson · Accepted Answer

(a) The volume of the parallelepiped is 7 cubic units. (b) i. If $$ \vec{F} $$ is conservative, then $$ 
abla 	imes \vec{F} = \overrightarrow{0} $$. (b) ii. If $$ 
abla 	imes \vec{F} = \overrightarrow{0} $$, then $$ \vec{F} $$ is conservative.

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