Conservative force/Proofs

Properties of conservative forces

The main page states that the following three properties are equivalent for conservative forces:
#The curl of "F" is zero:
#:$abla\; imes\; vec\{F\}\; =\; 0.\; ,$
#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, $Phi$:
#:$vec\{F\}\; =\; -\; abla\; Phi.\; ,$

Proof:

1 implies 2: Let "C" be any simple closed path and consider a surface "S" of which "C" is the boundary. Then Stokes' theorem gives that:$int\_S\; (\; abla\; imes\; vec\{F\})\; cdot\; mathrm\{d\}vec\{a\}\; =\; oint\_C\; vec\{F\}\; cdot\; mathrm\{d\}vec\{r\}$If the curl of F is zero the left hand side is zero - therefore statement 2 is true.

2 implies 3: Assume that statement 2 holds. Let "c" be a simple curve from the origin to a point $x$ and define a function:$Phi(x)\; =\; int\_c\; vec\{F\}\; cdot\; mathrm\{d\}vec\{r\}.$The fact that this function is well-defined (independent of the choice of "c") follows from statement 2. Anyway, from the fundamental theorem of calculus now follows that :$vec\{F\}\; =\; abla\; Phi.$The minus sign is included for convenience in physical calculations. So statement 2 implies statement three.

3 implies 1: 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::$abla\; imes\; abla\; f\; =\; left(\; frac\{partial^2\; f\}\{partial\; y\; partial\; z\}\; -\; frac\{partial^2\; f\}\{partial\; z\; partial\; y\}\; ight)\; vec\{e\}\_x\; +\; left(\; frac\{partial^2\; f\}\{partial\; z\; partial\; x\}\; -\; frac\{partial^2\; f\}\{partial\; x\; partial\; z\}\; ight)\; vec\{e\}\_y\; +\; left(\; frac\{partial^2\; f\}\{partial\; x\; partial\; y\}\; -\; frac\{partial^2\; f\}\{partial\; y\; partial\; x\}\; ight)\; vec\{e\}\_z$:where the "e_x" etc. denote unit vectors. Now use equality of mixed partial derivatives to see that each component vanishes.The proof now consists only of applying the above theorem to F - which is a gradient by assumption - to see that its curl is zero.

This shows that statement 1 implies 2, 2 implies 3 and 3 implies 1, therefore the proof is complete.

The equivalence of 1 and 3 is also known as (one aspect of) Helmholtz's theorem.