Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the function gx,y in Table 4.11.1 has first partial derivatives everywhere.
For g to be differentiable at the origin, Δ g≡g0+h,0+k−g0,0=gh,k must assume the form
gx0,0 h+gy0,0 k+ηh,k⋅h2+k2
where η→0 as h,k→0,0. Since gx0,0=gy0,0=0 from Example 4.11.1, it follows that
where λx,y=h2+k2⁢sin1h2+k2. Since λ is the product of a bounded factor and a factor that goes to zero, λ→0 as h,k→0,0. Hence, setting η=λ implies that g 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
What kind of issue would you like to report? (Optional)