Question

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = â‡f. (If the vector field is not conservative, enter DNE.) F(x, y, z) = eyzi + xzeyzj + xyeyzk f(x, y, z) = Incorrect: Your answer is incorrect.

Answer 1

We have to have

$\dfrac{\partial f}{\partial x}=e^{yz}$

$\dfrac{\partial f}{\partial y}=xze^{yz}$

$\dfrac{\partial f}{\partial z}=xye^{yz}$

Integrate the first PDE with respect to $x$:

$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int e^{yz}\,\mathrm dx\implies f(x,y,z)=xe^{yz}+g(y,z)$

Differentiate with respect to $y$:

$\dfrac{\partial f}{\partial y}=xze^{yz}=xze^{yz}+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

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

$\dfrac{\partial f}{\partial z}=xye^{yz}=xye^{yz}+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

So we have

$f(x,y,z)=xe^{yz}+C$

which means $F$ is conservative.