Conservative force/Proofs

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Main article: Conservative force

[edit] Properties of conservative forces

The main page states that the following three properties are equivalent for conservative forces:

The curl of F is zero:

$\nabla \times \vec{F} = 0. \,$
The work, W, is zero for any simple closed path:

$W = \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$
The force can be written as the gradient of a potential, $Φ$ :

$\vec{F} = -\nabla \Phi. \,$

Proof: Let C be any simple closed path and consider a surface S of which C is the boundary. Then Stokes' theorem gives that

$\int_S (\nabla \times \vec{F}) \cdot \mathrm{d}\vec{a} = \oint_C \vec{F} \cdot \mathrm{d}\vec{r}$

If the curl of F is zero the left hand side is zero - therefore statement 2 is true.

Now assume that statement 2 holds. Let c be a simple curve from the origin to a point $x$ and define a function

$\Phi(x) = \int_c \vec{F} \cdot \mathrm{d}\vec{r}.$

From the fundamental theorem of calculus now follows that

$\vec{F} = \nabla \Phi.$

The minus sign is included for convenience in physical calculations. So statement 2 implies statement three.

Finally, assume that the third statement is true. For this we use the following rule:

Lemma For any twice continuously differentiable function f, curl(grad f) = 0.

Proof Writing out the definitions yields

$\nabla \times \nabla f = \left( \frac{\partial^2 f}{\partial y \partial z} - \frac{\partial^2 f}{\partial z \partial y} \right) \vec{e}_x + \left( \frac{\partial^2 f}{\partial z \partial x} - \frac{\partial^2 f}{\partial x \partial z} \right) \vec{e}_y + \left( \frac{\partial^2 f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x} \right) \vec{e}_z$