Example 5

Given w = xyz, express the increment Δw in the form

Δw = dw + ε₁ Δx + ε₂ Δy + ε₃ Δz.

We first find Δw and dw,

Δw = (x + Δx)(y + Δy)(z + Δz) - xyz

= yz Δx + xz Δy + xy Δz + x Δy Δz + y Δx Δz + z Δx Δy + Δx Δy Δz

dw = yz Δx + xz Δy + xy Δz.

Thus

Δw = dw + (y Δz + z Δy) Δx + (x Δz) Δy + (Δx Δy) Δz.

Figure 11.4.8 pictures dw and Δw.

Figure 11.4.8