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The ordinary mean value theorem gives ${\Delta }_{h}f(a)=h{f}^{\prime }(b_1)$ for some $b_1\in (a,a+h)$. More generally, applying it to the difference of two translated functions shows inductively thatfor some $b_k\in (a,a+kh)$. Taking $k=n$ proves the result.

The mean value theorem says that for continuous $f$ on $[a,b]$, differentiable inside, $f(b)-f(a)=f^{\prime }(c)(b-a)$ for some $c$. Applied to $\log$ on $[b,a]$,which gives $(a-b)/a<\log (a/b)<(a-b)/b$.

On times $\tau \gg ϵ$, the quasi-steady-state approximation setsThe second equation givesIn the first equation the terms $-\beta u{v}_1+\gamma v_2$ then cancel, leavingSolving these two equations givesSince $\dot{p}=k_3{c}_2=k_3{e}_0{v}_2$,

At small substrate concentration this allosteric rate is quadratic, $dp/dt\sim (A/\alpha )u^2$, so its graph leaves the origin with zero slope. At large $u$ it saturates at $A\gamma /\beta =k_3{e}_0$. A Michaelis-Menten equation also saturates, but its rate $V_{\max }u/(K_M+u)$ is linear near the origin. The allosteric curve is therefore more sigmoidal:This is the quasi-steady rate law for a two-substrate allosteric enzyme.

Use enzyme conservation to writeSince $\tau =k_1{e}_{0}t$, $u=s/s_0$, $v_i=c_i/e_0$, and $ϵ=e_0/s_0$, substitution into the mass-action equations givesandwhereThe factors of $ϵ$ multiplying $d{v}_i/d\tau$ identify the complex concentrations as fast variables in a singular perturbation.

Adding the equations for the free enzyme and its two complexes givesThusThis conservation law says that the enzyme is neither created nor destroyed: each enzyme molecule is either free, bound in $C_1$, or bound in $C_2$.

The first reaction is reversible, with reverse rate $k_{-1}{c}_1$. Applying the law of mass action to each reaction givesEach term has the sign dictated by whether the corresponding species is consumed or produced.

The Gershgorin circle theorem states that every eigenvalue $\lambda$ of a complex matrix $A=(a_{ij})$ lies in at least one diskTo prove it, choose a nonzero eigenvector $x$ for $\lambda$ and an index $i$ such that $|x_i|={\max }_j|x_j|>0$. The $i$th component of $Ax=\lambda x$ givesTaking absolute values and using $|x_j|\le |x_i|$ yieldsDivision by $|x_i|$ proves the theorem.

For one-dimensional homentropic flow,and $dp=c^{2}d\rho$. For a perfect gas, $dc/c=(\gamma -1)d\rho /(2\rho )$. Combining the two equations therefore shows that the Riemann invariantssatisfyTheir characteristic curves are parametrized byand $R_{\pm }$ is constant along the corresponding curve until characteristic intersection creates a shock.

Put $z=x-ct$ and $h(x,t)=f(z)$. The conservation law becomesOne integration gives the required first-order autonomous equationwhere $C$ is a constant fixed by the far-field film heights or other boundary data.

The volume flux per unit transverse width isIntegrating incompressibility across the layer and using the wall and free-surface kinematic conditions gives local mass conservation $h_t+q_x=0$. Hence the gravity-driven thin film on an incline obeys

Integrating the normal momentum equation from the free surface gives the hydrostatic pressureThus $p_x=\rho g\cos \alpha h_x$. Twice integrating the downslope equation and applying $u(0)=0$ and $u_y(h)=0$ gives

The leading lubrication momentum equations and incompressibility condition areAt the solid plane $y=0$, no slip and no penetration give $u=v=0$. At the free surface $y=h(x,t)$, neglecting surface tension and ambient viscous stress givesto leading order. The kinematic boundary condition for a free-surface graph is

Balancing the downslope gravitational force density $\rho g\sin \alpha$ against the viscous term $\mu u_{yy}$ gives the velocity scaleIf the streamwise variation scale is $L\sim h/|h_x|$, the lubrication theory requirement is firstThe ratio of streamwise inertia $\rho U^2/L$ to transverse viscous stress $\mu U/h^2$ isThus self-consistency also requires

LetThe Euler-Lagrange equation givesSince ${\dot{x}}^{\mu }=Ld{x}^{\mu }/ds$ and $d/d\lambda =Ld/ds$, this becomesExpanding the derivative, multiplying by the inverse metric, and using the Christoffel symbolgives the geodesic equation