Across a thin timelike matter layer, the induced metric is continuous while the jump in its extrinsic curvature is determined by the surface stress-energy tensor. Choose one common spacelike normal and $K_{ab}=\frac{1}{2}{\partial }_y{h}_{ab}$ in Gaussian normal coordinates. The singular tangential Einstein tensor is $-([K_{ab}]-h_{ab}[K])\delta (y)$; equating it to ${\kappa }_D^2{S}_{ab}\delta (y)$ gives the displayed conditions. Reversing the normal or the sign convention for extrinsic curvature changes the corresponding jump signs consistently. These conditions determine the cosmological expansion law of a thin gravitating brane.

An exchange economy specifies consumers, their preference relations, their consumption sets and their initial endowments. A consumption allocation is feasible when aggregate consumption equals aggregate endowment; free-disposal models instead allow a coordinatewise inequality. These conventions must be distinguished when specifying supporting prices.

At any price vector, the total value of aggregate excess demand is zero when consumers exhaust their budgets. After normalization to a price simplex, choosing a price that maximizes the value of a candidate excess-demand vector is a useful Kakutani fixed-point theorem construction.

A Daniell cell uses the zinc/${\mathrm{Zn}}^{2+}$ and ${\mathrm{Cu}}^{2+}$/copper pairs. In the separated-compartment teaching version, a zinc electrode contacts an aqueous chemical solution containing ${\mathrm{Zn}}^{2+}$, and a copper electrode contacts one containing ${\mathrm{Cu}}^{2+}$. A salt bridge supplies an ionic connection; a wire and load supply an electronic connection. During discharge,The zinc electrode loses material and copper is deposited at the cathode. Each half-reaction conserves chemical atom counts and electric charge: the first has net charge zero on each side, the second $+2e-2e=0$ on the left and zero on the right. The net reaction has $+2e$ on both sides, with no electrons left in its chemical equation. The Zn/Cu teaching cell and these half-reactions are described in Florida State University, Redox Reactions in Voltaic Cells: Construction of a Potential Series, lab background.

Creating one mole of ${\mathrm{Zn}}^{2+}$ adds $2F$ of positive electric charge to the anode compartment's ionic inventory; consuming one mole of ${\mathrm{Cu}}^{2+}$ removes $2F$ of positive electric charge from the cathode compartment. Here $F$ is the Faraday constant. In an ideal bookkeeping example using a potassium nitrate salt bridge, two moles of nitrate anions entering the anode compartment compensate the first change, and two moles of potassium cations entering the cathode compartment compensate the second. Other ions in the cell can share the actual transport. The two compensating inventory changes do not mean that $4F$ flowed through the load: the same series electric current transports $2F$ through each cross-section of the circuit per mole of net reaction. See salt bridge.

Under standard-state conditions near $298\mathrm{K}$, tabulated reduction potentials give a cell electromotive force of approximately $1.10\mathrm{V}$. This is a reversible reference value, not a promised loaded reading; actual composition matters through thermodynamic activity. The tabulated copper and zinc values and their subtraction are given in Purdue University, Electrochemical Reactions: standard-state reduction potentials. For the net chemical equation as written, the molar Gibbs free energy change obeysThus $\mathcal{E}=1.10\mathrm{V}$ corresponds to about $-212\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of net reaction. At fixed temperature and pressure, $-{\Delta }_{r}G$ is the maximum reversible electrical work per mole for this cell reaction. It is not automatically all the heat released by a finite-current cell. See 17.4 Potential, Free Energy, and Equilibrium (§17.4: cell potential and free energy).

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Additional cited passages: §2.8, Definition 2.8.1 and equation (2.8.5); §2.8.1, Theorem 2.8.1 and equation (2.8.6), right-hand-side perturbation bound. The online edition is undated; Tobin A. Driscoll and Richard J. Braun are the whole-book coauthors, with no separate section byline.

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