Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the function hx,y in Table 4.11.1 has a differential at the origin, and hence is differentiable at the origin.
For h to be differentiable at the origin, Δ h≡h0+h,0+k−h0,0=hh,k must assume the form
hx0,0 h+hy0,0 k+ηh,k⋅h2+k2
where η→0 as h,k→0,0. Since hx0,0=hy0,0=0 from Example 4.11.7, it follows that
= h⁢k h2−k2h2+k23/2⋅h2+k2
where λx,y=h⁢k h2−k2h2+k23/2. To show that λ→0 as h,k→0,0, make the following estimate.
=h k h2−k2h2+k23/2
≤h k h2+k2h2+k23/2
where Inequalities 4 and 5 from Table 3.2.1 have been invoked. Hence, setting η=λ implies that h is differentiable at the origin.
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