z = f[x, y] & dz = ∂f/∂x dx + ∂f/∂y dy

The differential approximation in 2 variables

f[x + dx, y + dy] - f[x, y] = ∂f/∂x[x, y] · dx + ∂f/∂y[x, y] · dy + ε · | dX |

means an explicit graph looks like the local linear graph under infinite magnification:

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[Graphics:../HTMLFiles/Lect3_5.gif]

Figure 1.1.1: A Tangent to a Paraboloid

Figure 1

Created by Mathematica  (July 2, 2004)