I am reading Landau and Lifshitz and I am confused about two steps in the Fluctuation theory chapter. They occur just before Eqn. 3 in "Fluctuations of the fundamental thermodynamic quantities". Here they are:
First we expand $\Delta E$ in a power series, I'm good with this: $$\Delta E -T\Delta S+ P \Delta V = \frac{1}{2}\left(\frac{\partial^2 E}{\partial S^2}\Delta S^2 + 2\frac{\partial^2 E}{\partial S \partial V}\Delta S\Delta V + \frac{\partial^2 E}{\partial V^2}\Delta V^2\right)$$
but then they rewrite this as
$$\frac{1}{2}\left(\Delta S\Delta\left(\frac{\partial E}{\partial S}\right)_V+\Delta V\Delta\left(\frac{\partial E}{\partial V}\right)_S\right)$$
This part doesn't make sense to me. And lastly they rewrite this as
$$\frac{1}{2}(\Delta S \Delta T - \Delta P\Delta V)$$
This seems plausible given the previous equation, I just don't know why the deltas are there. In other words, why isn't it
$$\frac{1}{2}(T \Delta S - P\Delta V)$$
Any help would be greatly appreciated.