PROOF OF THE INCREMENT THEOREM

We break Δz into two parts by going first from (a, b) to (a + Δx, b) and then from (a + Δx, b) to (a + Δx, b + Δy), as shown in Figure 11.4.3,

Figure 11.4.3

Δz = [f(a + Δx,b + Δy) - f(a + Δx, b)] + [f(a + Δx, b) - f(a, b)].

Our plan is as follows. First, we regard f(a, b) as a one-variable function of a and show that

(1)

f(a + Δx, b) - f(a, b) = f_x(a, b) Δx + ε₁ Δx

for some infinitesimal ε₁. Second, we regard f(a + Δx, b) as a one-variable function of b and show that

(2)

f(a + Δx,b + Δy) - f(a + Δx, b) = f_y(a, b) Δy + ε₂ Δy

for some infinitesimal ε₂.

Once Equations 1 and 2 are established the proof will be complete because by adding Equations 1 and 2 we get the desired result

Δz = f_x(a, b) Δx + f_y(a, b) Δy + ε₁ Δx + ε₂ Δy = dz + ε₁ Δx + ε₂ Δy.

Equation 1 follows at once from the one-variable Increment Theorem since f_x(a, b) exists.

We now prove Equation 2. We regard f(a + Δx, y) as a one-variable function of y. For all y between b and b + Δy, the point (a + Δx, y) is infinitely close to (a, b), so f_y(a + Δx, y) is defined. By the one-variable Mean Value Theorem on the interval [b, b + Δy], there is a y₁ between b and b + Δy such that

Since f_y is continuous at (a, b),

f_y(a + Δx, y₁) =f_y (a, b) + ε₂, where ε₂ is infinitesimal.

Then

= f_y(a, b) + ε₂,

and Equation 2 follows.