Multivariate analogue of triple product rule

Question

I was doing some thermodynamic calculations and I found the need for a multivariable analogue of the triple product rule. Basically I had a set of $m$ functions $z_i(y, x_1, x_2, \ldots, x_m)$, and I needed to calculate partial derivatives of changes in $x$ resulting from a change in $y$ where all $z_i$ values are simultaneously held constant, i.e., I want to calculate:

$$ \left(\frac{\partial x_1}{\partial y}\right)_{z_1, z_2, \ldots, z_m}, \ldots, \left(\frac{\partial x_m}{\partial y}\right)_{z_1, z_2, \ldots, z_m} . $$

As a reminder, the regular triple product rule would tell us the following, which unfortunately is not helpful in the multivariable case:

$$ \left(\frac{\partial x}{\partial y}\right)_{z} = - \frac{\left(\frac{\partial z}{\partial y}\right)_{x}}{\left(\frac{\partial z}{\partial x}\right)_{y}} \qquad \qquad \text{when $m=1$}. $$

To compactify, let's use linear algebra notation and write our problem as us knowing $\mathbf z (y, \mathbf x)$, and wanting to find:

$$ \left(\frac{\partial \mathbf x}{\partial y}\right)_{\mathbf z} = ~ ??? $$

I believe I have the answer to this (which I will post as an Answer), but I am unsure as I cannot find it in standard places. This raises a follow-up question for me, which is whether this has a name?

---

Example

I came across the need for the above rule when trying to work out heat capacity in the grand canonical ensemble, in the case of a multi-component system (multiple chemical potentials $\mu_1, \ldots, \mu_m$ and multiple particle counts $N_1, \ldots, N_m$). In this ensemble our natural variables are temperature $T$ and $\boldsymbol \mu$, but heat capacity is a constant-$\mathbf N$ derivative. We have:

\begin{align} C_V & = T \left(\frac{\partial S}{\partial T}\right)_{V, \mathbf N} \\ & = T \left(\frac{\partial S}{\partial T}\right)_{V, \boldsymbol \mu} + T { \left(\frac{\partial S}{\partial \boldsymbol \mu}\right)_{V, T} }^T \left(\frac{\partial \boldsymbol \mu}{\partial T}\right)_{V, \mathbf N} \qquad \text {(using regular multidimensional chain rule)} \end{align}

So the question is then how to get $ \left(\frac{\partial \boldsymbol \mu}{\partial T}\right)_{V, \mathbf N} $, i.e., here $\mathbf x = \boldsymbol \mu$ and $\mathbf z = \mathbf N $. Besides grand canonical ensemble, similar examples would appear if we are considering ensembles involving vectorial quantities, such as magnetization or angular momentum.

Answer 1

Essentially we are asking, when $y$ changes, what movement in $\mathbf x$ is needed to ensure that $\mathbf z$ stays constant. We have this exact differential for changes in $\mathbf z$:

$$\mathrm d\mathbf z = \mathbf J \, \mathrm d\mathbf x+ \left(\frac{\partial \mathbf z}{\partial y}\right)_{\mathbf x} \, \mathrm dy $$

where $\mathbf J$ is a Jacobian-like matrix, with elements given by:

$$ J_{ij} = \left(\frac{\partial z_i}{\partial x_j}\right)_{y,x_1,\ldots,x_n \text{ excluding } x_i}. $$

Now just set $\mathrm d \mathbf z = 0$, and isolate $\mathrm d\mathbf x/\mathrm d y$ to get the answer:

$$ \left(\frac{\partial \mathbf x}{\partial y}\right)_{\mathbf z} = - \mathbf J ^{-1} \left(\frac{\partial \mathbf z}{\partial y}\right)_{\mathbf x}. $$

Alternatively, we might write this as:

$$ \left(\frac{\partial \mathbf x}{\partial y}\right)_{\mathbf z} = - [(\boldsymbol \nabla_{\mathbf x} \mathbf z)_y] ^{-1} \left(\frac{\partial \mathbf z}{\partial y}\right)_{\mathbf x}. $$

---

So for my heat capacity example, we get:

$$ C_V = T \left(\frac{\partial S}{\partial T}\right)_{V, \boldsymbol \mu} + T { \left(\frac{\partial S}{\partial \boldsymbol \mu}\right)_{V, T} }^T \left[ (\nabla_{\boldsymbol \mu}\mathbf N)_{V, T} \right]^{-1} \left(\frac{\partial \mathbf N}{\partial T}\right)_{V, \boldsymbol \mu} . $$

What's interesting is that the matrix $(\nabla_{\boldsymbol \mu}\mathbf N)_{V, T} $ is actually the covariance matrix of particle numbers, multiplied by $kT$. The same matrix also shows up when we try to calculate isothermal compressibility $\beta_T$, where $1/\beta_T = V (\partial p / \partial V)_{T, \boldsymbol N}$ -- here it is slightly less of a surprise since it's know that particle number fluctuations are exactly related to compressibility, but it's interesting that the inverse of the covariance matrix (i.e., the precision matrix) is what appears in these calculations.