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andreev551 [17]
3 years ago
9

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:
sineoko [7]3 years ago
7 0
\mathbf f(x,y)=(ye^x+\sin y)\,\mathbf i+(e^x+x\cos y)\,\mathbf j is continuous, so \mathbf f(x,y) is indeed conservative. So we know there is some scalar function f(x,y) such that \nabla f(x,y)=\mathbf f(x,y).

\dfrac{\partial f}{\partial x}=ye^x+\sin y
\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int(ye^x+\sin y)\,\mathrm dx
f(x,y)=ye^x+x\sin y+g(y)

\dfrac{\partial f}{\partial y}=\dfrac{\partial(ye^x+x\sin y+g(y))}{\partial y}
e^x+x\cos y=e^x+x\cos y+g'(y)
0=g'(y)
\implies g(y)=C

So we have

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