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.
<< Previous Example Section 4.11
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document