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Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.) F(x, y) = (xy cos(xy) + sin(xy))i + (x2 cos(xy))j

Respuesta :

LammettHash

If

[tex]\nabla f=(xy\cos xy+\sin xy)\,\vec\imath+x^2\cos xy\,\vec\jmath[/tex]

then we should have

[tex]\dfrac{\partial f}{\partial y}=x^2\cos xy\implies f(x,y)=x\sin xy+g(x)[/tex]

Differentiating with respect to [tex]x[/tex] gives

[tex]\dfrac{\partial f}{\partial x}=xy\cos xy+\sin xy=\sin xy+xy\cos xy+g'(x)[/tex]

[tex]\implies g'(x)=0\implies g(x)=C[/tex]

So [tex]F[/tex] is indeed conservative, and

[tex]f(x,y)=x\sin xy+C[/tex]

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