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Differential of a C2 partial Legendre transform on a local inverse chart

Statement

Let $O \subseteq R^{k} \times R^{m}$ be open with $k, m \geq 1$ , and let $f : O \to R$ be $C^{2}$ . Suppose the map $(u, z) \mapsto (x, z) : = (D_{u} f (u, z), z)$ restricts to a $C^{1}$ diffeomorphism between open neighborhoods, with inverse $(x, z) \mapsto (g (x, z), z)$ . Define

$\hat{f} (x, z) : = f (g (x, z), z) - x \cdot g (x, z) .$

Then $\hat{f}$ is $C^{2}$ and

$d \hat{f} = - g \cdot d x + D_{z} f (g (x, z), z) \cdot d z .$

This is a local statement. Existence of the inverse chart is a hypothesis; the lemma asserts no global inverse or boundary transform.

Facts & Assumptions

Given: An open domain, a $C^{2}$ function $f$ , and a $C^{1}$ inverse chart $g$ for $(u, z) \mapsto (D_{u} f (u, z), z)$ .

[F1]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : Differentiating a composition of totally differentiable maps gives the composed derivative.

[F2]

$C^{k}$ Euclidean maps and diffeomorphisms: A $C^{1}$ diffeomorphism is a bijection between open neighborhoods whose map and inverse are both $C^{1}$ .

[F3]

Chain sum product and composition rules for Banach derivatives: Differentiation of a bounded-bilinear product obeys the sum of the two product terms.

Proof

technique · direct

1.1givenF1F2

The inverse identity gives $D_{u} f (g (x, z), z) = x$ . By [F1], differentiating the composition in $\hat{f}$ gives $d [f (g (x, z), z)] = x \cdot d g + D_{z} f (g (x, z), z) \cdot d z$ .

2.1F3step 1.1algebra

Apply [F3] componentwise to the finite scalar products in $x \cdot g$ to obtain $d [x \cdot g] = g \cdot d x + x \cdot d g$ . Subtracting this from step 1.1 cancels the $x \cdot d g$ terms and yields the stated differential.

3.1givenF1F2step 2.1∎

The coefficient functions $g$ and $D_{z} f (g, z)$ are $C^{1}$ , since $g$ is $C^{1}$ , $f$ is $C^{2}$ , and [F1] applies to their compositions. The first partial derivatives of $\hat{f}$ are therefore $C^{1}$ , so $\hat{f}$ is $C^{2}$ .

Depends on

Dependency tree · 2 levels

Sources

Daniel Arovas, Thermodynamics and Statistical Mechanics, Appendix II, §2.15