Question

Determine whether or not\(F\)is a conservative vectorfield. If it is, find a function\({\rm{f}}\)such that\({\rm{F = }}\nabla {\rm{f}}\).

\({\rm{F}}\left( {{\rm{x,y}}} \right){\rm{ = }}{{\rm{e}}^{\rm{x}}}{\rm{cos}}\left( {\rm{y}} \right){\rm{i + }}{{\rm{e}}^{\rm{x}}}{\rm{sin}}\left( {\rm{y}} \right){\rm{j}}\)

Short answer

The vector field is not conservative.

Step-by-step solution

Step 1:To determine whether the vector field is conservative

Known that,

\(\begin{array}{c}{\rm{P = }}{{\rm{e}}^{\rm{x}}}{\rm{cosy}}\\{\rm{Q = }}{{\rm{e}}^{\rm{x}}}{\rm{siny}}\\\frac{{\partial {\rm{Q}}}}{{\partial {\rm{x}}}}{\rm{ = - }}{{\rm{e}}^{\rm{x}}}{\rm{siny}}\\\frac{{\partial {\rm{P}}}}{{\partial {\rm{y}}}}{\rm{ = }}{{\rm{e}}^{\rm{x}}}{\rm{siny}}\end{array}\)

Result

So,

\(\frac{{\partial {\rm{Q}}}}{{\partial {\rm{x}}}} \ne \frac{{\partial {\rm{P}}}}{{\partial {\rm{y}}}}\)

Therefore, the vector field is not conservative.