As an undergrad I learned a lot about partial derivatives in physics classes. But they told us rules as needed, without proving them. This rule completely freaked me out. If derivatives are kinda like fractions, shouldn’t this equal 1?
Let me show you why it’s -1.
First, consider an example:
This example shows that the identity is not crazy. But in fact it
holds the key to the general proof! Since is a coordinate system we can assume without loss of generality that . At any point we can approximate to first order as for some constants . But for derivatives the constant doesn’t matter, so we can assume it’s zero.
Then just compute!
There’s also a proof using differential forms that you might like
better. You can see it here, along with an application to
But this still leaves us yearning for more intuition — and for me, at least, a more symmetrical, conceptual proof. Over on Twitter, someone named
Postdoc/cake provided some intuition using the same example from thermodynamics:
Using physics intuition to get the minus sign:
- increasing temperature at const volume = more pressure (gas pushes out more)
- increasing temperature at const pressure = increasing volume (ditto)
- increasing pressure at const temperature = decreasing volume (you push in more)
Jules Jacobs gave the symmetrical, conceptual proof that I was dreaming of:
As I’d hoped, the minus signs come from the anticommutativity of the wedge product of 1-forms, e.g.
Since the space of 2-forms at a point in the plane is 1-dimensional, we can divide them. In fact a ratio like
is just the Jacobian of the tranformation from coordinates to coordinates. We also need that these ratios obey the rule
where are nonzero 2-forms at a point in the plane. This seems obvious, but you need to check it. It’s not hard. But to put it in fancy language, it follows from the fact that nonzero 2-forms at a point in the plane are a ‘torsor’! I explain that idea here:
Torsors are widespread in physics, and the nonzero vectors in any 1-dimensional real vector space form a torsor for the multiplicative group of nonzero real numbers.
Jules Jacob’s proof is more sophisticated than my simple argument, but it’s very pretty, and it generalizes to higher dimensions in ways that’d be hard to guess otherwise.