Vector and scalar field Laplacian

Question

My vector field is $$F=(-6x^2yz^2+ye^x)i + (-2x^3z^2 + e^x)j - (4x^3yz)k$$How do i find the scalar field $$f,\quad where\quad F=del\,f$$Am i meant to integrate $F$? Also i need to find the laplacian of the scalar field $f$. Is this found by finding the del of F?

Answer 1

If $f$ does exist, we will require \begin{eqnarray*} \frac{ \partial f}{\partial x} &=&-6x^2 y z^2 +y e^x \\ \frac{ \partial f}{\partial y} &=&-2x^3 z^2 + e^x \\ \frac{ \partial f}{\partial z} &=&-4x^2 y z \\ \end{eqnarray*} $f=-2x^3yz^2+ye^x\color{yellow}{+C}$ is consistent with these equations. The Laplacian \begin{eqnarray*} \nabla^2 f = \frac{ \partial^2 f}{\partial x^2}+\frac{ \partial^2 f}{\partial y^2}+\frac{ \partial^2 f}{\partial z^2}= ... \end{eqnarray*}