One now tries to find a connection between a scalar field and a vector field.

The Definition of the gradient is:

df⋅h⃗=(∇⃗f)∙h⃗

df ist the total differential of f.

In cartesian coordinates:

(∇⃗(ϕ(x⃗)))_i=÷{∂}{∂x_i}(ϕ(x⃗))

The result is a vector. The parameter is a scalar field.

In the following, the scalar field is ϕ and the vector field is F.

F⃗=-∇⃗ϕ

Because calculations using the scalar field are simpler, one wants to use the scalar field.

So, given a F⃗, we want to get ϕ (once).

F⃗(x⃗)=-∇⃗ϕ(x⃗)
ϕ = -∫F_1⋅dx_1
ϕ = -∫F_2⋅dx_2

usw...

ϕ is supposed to be unique. The integration constant after integration is a function of the other parameters (the ones that were not integrated over).

Hence solve a system of equation like this for ϕ:

ϕ = -∫\limits_{a}^{b} F_1⋅dx_1 + f(x_2)
ϕ = -∫\limits_{c}^{d} F_2⋅dx_2 + g(x_1)