[tex]\nabla f=\mathbf f=x\,\mathbf i+y\,\mathbf j+z\,\mathbf k[/tex]
[tex]\dfrac{\partial f}{\partial x}=x\implies f(x,y,z)=\dfrac{x^2}2+g(y,z)[/tex]
[tex]\dfrac{\partial f}{\partial y}=y=\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies g(y,z)=\dfrac{y^2}2+h(z)[/tex]
[tex]\dfrac{\partial f}{\partial z}=z=\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]\implies h(z)=\dfrac{z^2}2+C[/tex]
[tex]\implies f(x,y,z)=\dfrac{x^2}2+\dfrac{y^2}2+\dfrac{z^2}2+C[/tex]
[tex]f(0,0,0)=0\implies C=0[/tex]
[tex]\implies f(x,y,z)=\dfrac{x^2}2+\dfrac{y^2}2+\dfrac{z^2}2[/tex]
