I am given the following form of the first law of thermodynamics (with magnetisation) $$ \mathrm d E=T\mathrm dS-M\mathrm dB$$ with $E$ the internal energy, $M$ the magnetisation, and $B$ the applied magnetic field, and I am asked at some point to use the identities that \begin{align} \left.\frac{\partial S}{\partial T}\right|_M = \left.\frac{\partial S}{\partial T}\right|_B + \left.\frac{\partial S}{\partial B}\right|_T \left.\frac{\partial B}{\partial T}\right|_M \tag{1} \end{align} and \begin{align} \left.\frac{\partial x}{\partial y}\right|_z \left.\frac{\partial y}{\partial z}\right|_x \left.\frac{\partial z}{\partial x}\right|_y = -1, \tag{2} \end{align} which I would like to be able to rigorously prove.
I have managed to make sense of the second identity in the following way. Implicitly, the values $(x,y,z)\in\mathbb R^3$ are constrained to a hypersurface $\mathcal M\subset\mathbb R^3$ according to physical laws, which allow us to express $x=x(y,z)$ by the implicit function theorem and likewise for $y$ and $Z$ (assuming nondegeneracy), after which this identity becomes a quick computation, which is easily seen to hold on $\mathcal M$. However, I'm struggling to adapt this technique to prove the first identity.
Whether the variables $(E,T,S,M,B)$ are constrained to live on a codimension $1$ or $2$ manifold, we still have the problem that $S$ requires at least three variables in order to be specified, so that $\left.\frac{\partial S}{\partial T}\right|_M$ is poorly defined. So I am led to suspect that we're working on a codimension $3$ manifold, although this seems physically poorly justified, and to interpret the law $ \mathrm d E=T\mathrm dS-M\mathrm dB $ as a relation between $1$-forms on $\mathcal M$. So how in fact do we prove (and make sense of) the first identity?