Question

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)

Answer 1

For
`F` 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
`x`:

`\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
`y`:

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

Now differentiate
`f` with respect to
`z`:

`(\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
`F` is indeed conservative.