17.3 Tangent plane approximations

To extend the notion of a linear approximation, we apply the tangent plane approximation at the point \(x=a\), \(y=b\):

\[\begin{equation} L(x,y) = f(a,b) + f_{x}(a,b) \cdot (x-a) + f_{y}(a,b) \cdot (y-b), \end{equation}\]

where \(f_{x}\) is the partial derivative of \(f(x,y)\) with respect to \(x\) and \(f_{y}\) is the partial derivative of \(f(x,y)\) with respect to \(y\). For simplicity let’s work with the equilibrium solution at \((0,0)\). If we say that \(f(H,L)=.3 H - .1 HL\), we know \(f(0,0)=0\). Furthermore the locally linear approximation is:

\[\begin{equation} \begin{split} f_{H} = .3 - .1L & \rightarrow f_{H}(0,0)=.3 \\ f_{L} = -.1H & \rightarrow f_{L}(0,0)=0 \end{split} \end{equation}\]

With this information we are able to take this and then write down the locally linear approximation for \(f(H,L)\):

\[\begin{equation} f(H,L) \approx 0 + .3 \cdot (H-0) - 0 \cdot (L-0) = .3H \end{equation}\]

Likewise if we consider \(g(H,L)= .05HL -.2L\), then we have:

\[\begin{equation} \begin{split} g_{H} &= .05L-.2L \rightarrow g_{H}(0,0)=0 \\ g_{L} &= .05H-.2 \rightarrow f_{L}(0,0)=-.2 \end{split} \end{equation}\]

With these results, our locally linear approximation for \(g(H,L)\) is:

\[\begin{equation} g(H,L) \approx 0 + 0 \cdot (H-0) -.2 \cdot (L-0) = 0 - .2L. \end{equation}\]

So, at the equilibrium solution \((0,0)\), Equation (17.2) behaves like the following system of equations:

\[\begin{align} \frac{dH}{dt} &= .3H\\ \frac{dL}{dt} &= - .2L \tag{17.3} \end{align}\]

Notice how Equation (17.3) is an entirely decoupled linear system of equations that can be solved easily by separation of variables. For an initial condition \((H_{0},L_{0})\) Equation (17.3) has a solution \(H(t)=H_{0}e^{.3t}\) and \(L(t)=L_{0}e^{-.2t}\).