Find a potential function for F or determine that F is not conservative. (If F is not conservative, enter NOT CONSERVATIVE.) F = cos(z), 10y, - x sin(z)

Question

asked Oct 15, 2019 24.9k views

1 Answer

Lucahuy · Answer 1 · 2019-10-22T10:10:34+0000

For
to be conservative, we need to have

$(\partial f)/(\partial x)=\cos z$

$(\partial f)/(\partial y)=10y$

$(\partial f)/(\partial z)=-x\sin z$

Integrate the first PDE with respect to
:

$\displaystyle\int(\partial f)/(\partial x)\,\mathrm dx+\int\cos z\,\mathrm dx\implies f(x,y,z)=x\cos z+g(y,z)$

Differentiate with respect to
:

$(\partial f)/(\partial y)=10y=(\partial g)/(\partial y)\implies g(y,z)=5y^2+h(z)$

Now differentiate
with respect to
:

$(\partial f)/(\partial z)=-x\sin z=-x\sin z+(\mathrm dh)/(\mathrm dz)\implies(\mathrm dh)/(\mathrm dz)=0\implies h(z)=C$

So we have

$f(x,y,z)=x\cos z+5y^2+C$

so
is indeed conservative.