Question

Consider the vector field f(x,y,z)=(2z+3y)i+(3z+3x)j+(3y+2x)kf(x,y,z)=(2z+3y)i+(3z+3x)j+(3y+2x)k.

a.find a function ff such that f=∇ff=∇f and f(0,0,0)=0f(0,0,0)=0.

Answer 1

`(\partial f)/(\partial x)=2z+3y\implies f(x,y,z)=2xz+3xy+g(y,z)`


`(\partial f)/(\partial y)=3x+(\partial g)/(\partial y)=3z+3x`

`(\partial g)/(\partial y)=3z\implies g(y,z)=3yz+h(z)`

`\implies f(x,y,z)=3xz+3xy+3yz+h(z)`



`(\partial f)/(\partial z)=3x+3y+(\mathrm dh)/(\mathrm dz)=3y+2x`

`(\mathrm dh)/(\mathrm dz)=-x`

But we assume
`h` is a function of
`z` alone, so there is no solution for
`h` and hence no scalar function
`f` such that
`\\abla f=\mathbf f`.