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Tju [1.3M]
3 years ago
15

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) = (yex + sin(y))i + (ex + x cos(y))j
Mathematics
1 answer:
gavmur [86]3 years ago
3 0

If

\nabla f=(ye^x+\sin y)\,\vec\imath+(e^x+x\cos y)\,\vec\jmath

then

\dfrac{\partial f}{\partial x}=ye^x+\sin y\implies f(x,y)=ye^x+x\sin y+g(y)

Differentiating with respec to y gives

\dfrac{\partial f}{\partial y}=e^x+x\cos y=e^x+x\cos y+g'(y)

\implies g'(y)=0\implies g(y)=C

So F is indeed conservative, and

f(x,y)=ye^x+x\sin y+C

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