Consider the vector field f(x,y,z)=(4z+5y)i+(4z+5x)j+(4y+4x)k.a.find a function f such that f=∇f and f(0,0,0)=0.

[tex]\nabla f(x,y,z)=\mathbf f(x,y,z)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j+\dfrac{\partial f}{\partial z}\,\mathbf k[/tex]

[tex]\dfrac{\partial f}{\partial x}=4z+5y[/tex]
[tex]\implies f(x,y,z)=4xz+5xy+g(y,z)[/tex]

[tex]\dfrac{\partial f}{\partial y}=4z+5x=5x+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=4z\implies g(y,z)=4yz+h(z)[/tex]

[tex]\dfrac{\partial f}{\partial z}=4y+4x=4x+\dfrac{\partial g}{\partial z}[/tex]
[tex]\implies4y=4y+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]

[tex]f(x,y,z)=4xz+5xy+4yz+C[/tex]

[tex]f(0,0,0)=0\implies C=0\implies f(x,y,z)=4xz+5xy+4yz[/tex]