Determine whether or not F is a conservative vector field. If it is, find a func

Mylo O'Moore 2021-09-30 Answered
Determine whether or not F is a conservative vector field. If it is, find a function f such that F=delf. \(\displaystyle{F}{\left({x},{y}\right)}={\left({3}{x}^{{2}}-{2}{y}^{{2}}\right)}{i}+{\left({4}{x}{y}+{3}\right)}{j}\)

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davonliefI
Answered 2021-10-01 Author has 23163 answers
The vector field \(\displaystyle{F}=\Pi+{Q}{j}\) is convervative if and only if
\(\displaystyle{\frac{{{d}{P}}}{{{\left.{d}{y}\right.}}}}={\frac{{{d}{Q}}}{{{\left.{d}{x}\right.}}}}\)
The given vector field is \(\displaystyle{F}{\left({x},{y}\right)}={\left({3}{x}^{{2}}-{2}{y}^{{2}}\right)}{i}+{\left({4}{x}{y}+{3}\right)}{j}\)
Here \(\displaystyle{P}={3}{x}^{{2}}-{2}{y}^{{2}}\), Therefore \(\displaystyle{\frac{{{d}{P}}}{{{\left.{d}{y}\right.}}}}=-{4}{y}\)
And \(\displaystyle{Q}={4}{x}{y}+{3}\), Therefore \(\displaystyle{\frac{{{d}{Q}}}{{{\left.{d}{x}\right.}}}}={4}{y}\)
\(\displaystyle{\frac{{{d}{P}}}{{{\left.{d}{y}\right.}}}}\ne{\frac{{{d}{Q}}}{{{\left.{d}{x}\right.}}}}\)
Therefore F is NOT conservative.
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