I was reading an article, it really looks like The Chain Rule and F.T.C is used. My calculus is terrible. Can anyone explicitly point out the steps in the following equality.
Also it would be helpful if someone could point out how much 'smoothness/differentiability' the following function needs to have.
Let $V: \mathbb{R}^d \to \mathbb{R}^d $ (be at-least twice differentiable), for all $x,y\in \mathbb{R}^d$ we have
$$ \frac{V(x)-V(y)}{\|x-y\|}=\int_0^1 DV\big(y+r(x-y)\big) \frac{x-y}{\|x-y\|} dr. $$
Where $DV$ is the Jacobian of $V$.
Can someone please show me the working in each step above.
First of all the $||x-y||$ at the denominator can be simplified.
Fix than a component $V_i$. You have:
$V_i(x)-V_i(y)=\int_{\gamma} \nabla V_i \cdot dL $
where $\gamma$ is any path going from y to x. Just choose the straight one with parametrization $\tau(r)=y+r(x-y)$
Than:
$\int_{\gamma} \nabla V_i \cdot dL = \int_0^1 \nabla V_i( \tau(r))\cdot \tau'(r)dr = \sum_j \int_0^1 \partial_j V_i( \tau(r)) \tau_j'(r)dr $
can you conclude now inserting the expression for $\tau(r),\tau'(r)$ and the definition of the jacobian ?
PS: note that $\tau'(r)$ does not depend on $r$ and therefore can be taken out of the integration.