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Corollary 2
Our second corollary to the Increment Theorem shows that the tangent plane closely follows the surface. COROLLARY 2 Suppose z = f(x, y) is smooth at (a, b). Then for every point (x,y) at an infinitesimal distance
from (a, b), the change in z on the tangent plane is infinitely close to the change in z along the surface compared to Δs, i.e.,
PROOF We have Δz = dz + ε1 Δx + ε2 Δy. Both Δx/Δs and Δy/Δs are finite, so
In Figure 11.4.6, we see that the piece of the surface seen through an infinitesimal microscope aimed at (a, b, f(a, b)) is infinitely close to a piece of the tangent plane, compared to the field of view of the microscope.
Figure 11.4.6
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Last Update: 2010-11-25